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A detailed doping survey of the low-energy electrodynamics of YBa2Cu3O6+x Baglo, Jordan Carl 2014

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A Detailed Doping Survey of theLow-Energy Electrodynamics ofYBa2Cu3O6+xbyJordan Carl BagloB.Sc., The University of British Columbia, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2014c© Jordan Carl Baglo 2014AbstractThe rich phenomenology displayed by the high temperature (high-Tc) cuprate supercon-ductors has attracted intense experimental and theoretical attention for nearly thirty years.Despite steady and continued progress, a complete and consistent microscopic theory of thecuprates continues to elude researchers. However, recent work appears to be converging ona picture of separate spin and charge order phase transitions – well below and slightly aboveoptimal doping, respectively – along with associated Fermi surface reconstruction.As sensitive probes of the low-energy electrodynamics, microwave conductivity tech-niques are well-suited for characterizing the effects of such changes in electronic structure.For YBa2Cu3O6+x (YBCO), one of the cleanest and best-studied high-Tc cuprate super-conductors, previous measurements had focused on a relatively sparse set of dopings. Inthis thesis, detailed measurements of the doping dependence of both the microwave pen-etration depth and surface resistance of YBCO are combined with low-energy muon spinrotation measurements of the absolute magnetic penetration depth to produce a compre-hensive doping dependence survey of the microwave conductivity, spanning a wide range ofoxygen contents between 6+x = 6.49 and 6.998.The temperature derivatives of the magnetic penetration depth continue to decreaseat the highest dopings, challenging previous predictions of a peak in penetration depthnear the proposed quantum critical point in the overdoped regime. Additionally, a suddenincrease in the a–b anisotropy of low-temperature slopes was observed near optimal doping,suggesting the possibility of additional changes in electronic behaviour in this region.Microwave surface resistance measurements revealed a sharp increase in the scatteringrate – along the a axis only – for samples near 1/8th hole doping. Surprisingly, the a-axis scattering rate was discovered to decrease – not increase – after disordering the CuOchain layer oxygen configuration. The short a-axis correlation lengths in this doping range,combined with the strong scattering potential produced by chain order domain boundaries,are proposed as an explanation for this counterintuitive behaviour.iiPrefaceAll of the writing in this thesis is my own original, unpublished text. The need for a detailedmicrowave conductivity doping dependence survey of YBa2Cu3O6+x (YBCO) was identifiedin discussion between myself and Walter Hardy (my supervisor), Doug Bonn (group leaderand co-supervisor), and Ruixing Liang (master crystal grower); the doping survey projectwhich resulted became the crux of this thesis work. Although the low-energy µSR workhad been initiated by Rob Kiefl and collaborators and was already in its early stages by thetime I began my participation, I was involved in later experiment planning; the resultingabsolute penetration depth data played a crucial role in the microwave conductivity dataanalysis, and is an additional important research contribution presented in this thesis.The apparatus used for the microwave spectroscopy measurements described in thisthesis were previously constructed by Walter Hardy and Saeid Kamal (loop gap resonator)and Jake Bobowski (broadband bolometric spectrometer). The LabVIEW computer controlsystems were based on original versions by Jake Bobowski and Brad Ramshaw, along withmy own contributions and modifications to improve performance and usability. All of theYBCO samples described in this work have been grown and prepared by Ruixing Liang hereat UBC. The Ba(CoxFe1−x)2As2 sample (measurements of which are discussed in Chapter 6)was grown and prepared by Alex Thaler of the Canfield group at Ames Lab (Iowa StateUniversity).All of the microwave penetration depth data presented in this thesis (in Chapters 4and 6) were collected and analyzed myself. A portion of the YBCO 6.92 penetration depthdata have been previously been published in the proceedings of the µSR 2011 conference(M. D. Hossain et al., Physics Procedia 30, 235 (2012) [1]), on which I am a co-author. TheBa(CoxFe1−x)2As2 penetration depth data were also previously published (O. Ofer et al.,Phys. Rev. B 85, 060506(R) (2012) [2]).Nearly all of the surface resistance data presented in Chapter 5 were collected andanalyzed by myself independently. I collected the YBCO 6.49 dilution refrigerator surfaceiiiPrefaceresistance data jointly with James Day, but the analysis presented for this data is my own.The a-axis YBCO 6.993 surface resistance data presented in most of Chapter 5 (as a highdoping comparison point) were collected by Patrick Turner and Richard Harris, and theYBCO 6.993 width fit parameter data used in Section 5.4 were obtained by Jake Bobowski.The low-energy µSR (LEM) data presented in Chapter 6 were collected at the SwissMuon Source facility at the Paul Scherrer Institute (PSI) in Villigen, Switzerland. Thiswork was done in collaboration with the group of Rob Kiefl (Masrur Hossain and Oren Ofer)and the LEM group at PSI (including Zaher Salman, Thomas Prokscha, Andreas Suter,Bastian Wojek, Hassan Saadaoui, and Elvezio Morenzoni), as well as Sarah Dunsiger, whoeach contributed to the measurement or the analysis. I was an active participant in all of thesample preparation and LEM data collection at PSI (with the exception of the YBCO 6.60run, for which I prepared the sample mosaic, but did not travel to PSI), as well as onlineanalysis during beamtime. The LEM values of the absolute penetration depth λ(Tmeas)presented and used here are from the final analysis of Masrur Hossain. All microwavepenetration depth data for all samples were collected and analyzed by myself at UBC.The Ba(CoxFe1−x)2As2 LEM data have been published (O. Ofer et al., Phys. Rev. B 85,060506(R) (2012) [2]) as well as the YBCO 6.92 LEM data (R. F. Kiefl et al., Phys. Rev. B81, 180502(R) (2010) [3]; M. D. Hossain et al., Physics Procedia 30, 235 (2012) [1]); the restof the YBCO data remain unpublished. The γ-model fits presented here were carried outby Ruslan Prozorov of Ames Lab (Iowa State University) for the purposes of publication inthe Ofer et al. paper; all other fits and analysis presented are my own.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The high-Tc cuprate superconductors . . . . . . . . . . . . . . . . . . . . . 11.2 The cuprate superconductor YBa2Cu3O6+x . . . . . . . . . . . . . . . . . . 41.3 Changes from underdoped to overdoped . . . . . . . . . . . . . . . . . . . . 51.4 Quantum critical phenomenology . . . . . . . . . . . . . . . . . . . . . . . 61.5 A quantum critical point underneath the dome? . . . . . . . . . . . . . . . 91.6 Early experimental attempts . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Two quantum critical points underneath the dome? . . . . . . . . . . . . . 121.8 The conductivity doping scan . . . . . . . . . . . . . . . . . . . . . . . . . 132 Microwave electrodynamics of superconductors . . . . . . . . . . . . . . . 172.1 Electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 Linear response theory . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Response function causality, analyticity, and the Kramers-Kronig re-lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19vTable of Contents2.1.3 The Drude model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.4 The two-fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.5 Quasiclassical formalism for penetration depth . . . . . . . . . . . . 262.2 Surface impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.1 General expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.2 Surface impedance of a superconductor . . . . . . . . . . . . . . . . 292.3 Microwave cavity perturbation . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Introduction and phenomenology . . . . . . . . . . . . . . . . . . . 302.3.2 Derivation of the cavity perturbation equation . . . . . . . . . . . . 342.3.3 Thin platelet solution . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.4 Demagnetization effects . . . . . . . . . . . . . . . . . . . . . . . . . 433 Microwave spectroscopy tools and techniques . . . . . . . . . . . . . . . . 473.1 The cavity perturbation probe . . . . . . . . . . . . . . . . . . . . . . . . . 473.1.1 The 940 MHz loop-gap resonator . . . . . . . . . . . . . . . . . . . 473.1.2 The sample stage and environment . . . . . . . . . . . . . . . . . . 513.2 Swept-frequency cavity transmission measurement . . . . . . . . . . . . . . 543.3 The Robinson oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.1 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.2 Principles of oscillator operation . . . . . . . . . . . . . . . . . . . . 593.4 Time-domain measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5 Bolometric broadband spectroscopy . . . . . . . . . . . . . . . . . . . . . . 633.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5.2 Technique overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.5.3 Thermal stage considerations . . . . . . . . . . . . . . . . . . . . . . 673.5.4 Electronic detection and temperature control . . . . . . . . . . . . . 693.5.5 Computer control and data acquisition . . . . . . . . . . . . . . . . 714 YBa2Cu3O6+x conductivity doping dependence survey I:Penetration depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.1 The YBa2Cu3O6+x samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.1.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.1.2 Doping determination . . . . . . . . . . . . . . . . . . . . . . . . . . 76viTable of Contents4.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.1 Cryogenic procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.2 Measurement procedure: swept-frequency operation . . . . . . . . . 814.2.3 Measurement procedure: oscillator operation . . . . . . . . . . . . . 824.3 Data preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3.1 Data selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3.2 Oscillator data drift correction . . . . . . . . . . . . . . . . . . . . . 844.3.3 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.4 The penetration depth extraction code . . . . . . . . . . . . . . . . . . . . 864.5 Input parameters for ∆λ extraction . . . . . . . . . . . . . . . . . . . . . . 894.5.1 Absolute λa and λb . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.5.2 c-axis penetration depth . . . . . . . . . . . . . . . . . . . . . . . . 924.5.3 Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.4 Demagnetization effects . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.5 Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.6.1 Penetration depth shifts ∆λ(T ) . . . . . . . . . . . . . . . . . . . . 1024.6.2 The inverse squared penetration depth λ−2(T ) and related quantities 1024.6.3 The low-temperature slope . . . . . . . . . . . . . . . . . . . . . . . 1084.7 Theory of low-energy electrodynamics and transport in the cuprates . . . . 1104.7.1 Quasiparticle dispersion near the nodal points . . . . . . . . . . . . 1124.7.2 Low-energy electrodynamics and transport via Fermi liquid theory . 1144.7.3 The gap ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.8 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 YBa2Cu3O6+x conductivity doping dependence survey II:Surface resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.2.1 Basic processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.2.2 Self-consistent extraction of σ1 from Rs . . . . . . . . . . . . . . . . 1315.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133viiTable of Contents5.3.1 Rs(ω, T ) and σ1(ω, T ) results . . . . . . . . . . . . . . . . . . . . . . 1335.3.2 Conductivity width Γ(T ) . . . . . . . . . . . . . . . . . . . . . . . . 1365.3.3 Frequency exponent y(T ) . . . . . . . . . . . . . . . . . . . . . . . . 1365.3.4 Offset conductivity σ1D(T ) . . . . . . . . . . . . . . . . . . . . . . . 1415.3.5 Conductivity height σ0(T ) . . . . . . . . . . . . . . . . . . . . . . . 1445.3.6 Integrated spectral weight . . . . . . . . . . . . . . . . . . . . . . . 1445.4 The effects of oxygen chain disorder on conductivity . . . . . . . . . . . . . 1485.4.1 Production and characterization of the disordered sample . . . . . . 1485.4.2 The relationship to charge density wave order . . . . . . . . . . . . 1596 Low-energy µSR measurements of YBa2Cu3O6+x andBa(Co0.074Fe0.926)2As2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.1 Introduction to µSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.1.1 Muon decay asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 1646.1.2 Surface muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666.1.3 Spin precession and the muon polarization signal . . . . . . . . . . 1676.2 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.2.1 Low-energy muon production and transport . . . . . . . . . . . . . 1706.2.2 Muon implantation and the sample environment . . . . . . . . . . . 1746.2.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1756.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.1 TRIM.SP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.3.2 musrfit data a43nalysis . . . . . . . . . . . . . . . . . . . . . . . . 1776.4 YBCO measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.5 Measurements of Ba(Co0.074Fe0.926)2As2 . . . . . . . . . . . . . . . . . . . . 1846.5.1 Penetration depth measurement and a comparison of techniques . . 1846.5.2 Fits and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195viiiList of Tables6.1 Low-energy µSR a-axis penetration depth measurements for YBa2Cu3O6+x,along with microwave ∆λa(T ) extrapolations to T = 0, and the extractedvalues of the dead layer thickness (dead layer values in italics are from globalfits to both axes with a shared dead layer). . . . . . . . . . . . . . . . . . . 1836.2 Low-energy µSR b-axis penetration depth measurements for YBa2Cu3O6+x,along with microwave ∆λb(T ) extrapolations to T = 0, and the extractedvalues of the dead layer thickness (dead layer values in italics are from globalfits to both axes with a shared dead layer). . . . . . . . . . . . . . . . . . . 183ixList of Figures1.1 A schematic phase diagram of the high-Tc cuprate superconductors. . . . . 21.2 A unit cell of fully doped YBa2Cu3O7. . . . . . . . . . . . . . . . . . . . . . 31.3 The oxygen-ordered phases of the CuOx chain layer of YBa2Cu3O6+x. . . . 51.4 Cuprate phase diagram with quantum critical phenomenology. . . . . . . . 81.5 New cuprate phase diagram scenario with two quantum critical points. . . . 122.1 Illustration of the real and imaginary components of conductivity for theDrude model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Transmitted power and voltage phase shift through a microwave cavity. . . 312.3 A schematic of the microwave cavity geometry. . . . . . . . . . . . . . . . . 362.4 A schematic of the sample measurement geometry. . . . . . . . . . . . . . . 403.1 A photograph of the loop of the loop-gap resonator, with a side view schematicof the assembled resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Schematic view of loop-gap resonator assembly in operation. . . . . . . . . . 503.3 An example of swept-frequency data with a Lorentzian fit . . . . . . . . . . 553.4 The electronics setup for swept-frequency cavity perturbation measurement. 563.5 The Robinson oscillator circuitry. . . . . . . . . . . . . . . . . . . . . . . . . 583.6 A cutaway view of the assembled bolometric spectrometer. . . . . . . . . . . 643.7 Schematic of the amplification and bias circuitry for the bolometers. . . . . 703.8 Schematic of the electronics system configuration for the bolometry experiment. 724.1 The relationship between Tc and hole doping p for YBa2Cu3O6+x. . . . . . 774.2 Extrapolated T = 0 values of the in-plane magnetic penetration depths foreach axis, from LEM and Gd-ESR. . . . . . . . . . . . . . . . . . . . . . . . 904.3 Demonstration of the effects of λa,b(0 K) choices on λ−2a (T ). . . . . . . . . . 914.4 The c-axis penetration depth λc(0 K) for YBCO. . . . . . . . . . . . . . . . 92xList of Figures4.5 The normalized inverse-squared c-axis penetration depth λ2c(0)/λ2c(T ) of YBCOfrom several sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.6 Demonstration of the effects of c-axis penetration depth contributions on ∆λ. 954.7 Integrated thermal expansion ǫ(T ) as a function of axis, doping and temper-ature for YBCO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.8 Demonstration of the effects of thermal expansion on ∆λa. . . . . . . . . . 984.9 Demonstration of the effects of demagnetization on ∆λ. . . . . . . . . . . . 1004.10 The magnetic penetration depth shifts ∆λa(T ) and ∆λb(T ) at low temperature.1034.11 The magnetic penetration depth shifts ∆λa(T ) and ∆λb(T ) over the fulltemperature range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.12 The inverse squared penetration depths 1/λ2a(T ) and 1/λ2b(T ) over the fulltemperature range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.13 The normalized inverse squared penetration depths ρa ≡ λ2a(0)/λ2a(T ) andρb ≡ λ2b(0)/λ2b(T ) over the full range of T/Tc. . . . . . . . . . . . . . . . . . 1074.14 Estimated values of (ns/pnp)/(m∗/me), the effective number of supercon-ducting carriers of mass me per doped hole. . . . . . . . . . . . . . . . . . . 1094.15 The low-temperature slope of λ(T ) as a function of doping for both axes. . 1114.16 The modified nodal dispersion anisotropy ratio αsFL2 vFv2 as a function of doping.1174.17 A comparison of the modified anisotropy ratio αsFL2 vFv2 for different penetra-tion depth input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.18 A comparison of the modified anisotropy ratio αsFL2 vFv2 with thermal conduc-tivity measurements of vFv2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.19 The low-temperature slope of λ2(0)λ2(T ) for the a and b axes, as a function ofdoping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.20 Estimates of the gap ratio 2∆(0)/kBTc. . . . . . . . . . . . . . . . . . . . . 1225.1 The extracted a-axis Rs(ω, T ) for several dopings of YBCO. . . . . . . . . . 1345.2 The extracted b-axis Rs(ω, T ) for the measured dopings of YBCO. . . . . . 1355.3 The extracted a-axis σ1(ω, T ) for several dopings of YBCO. . . . . . . . . . 1375.4 The extracted b-axis σ1(ω, T ) for several dopings of YBCO. . . . . . . . . . 1385.5 Measurements of Γ(T ) in the a-axis, for free and constrained fits. . . . . . . 1395.6 Measurements of Γ(T ) in the b-axis, for free and constrained fits. . . . . . . 140xiList of Figures5.7 Extracted values of y(T ) in both axes and all dopings. . . . . . . . . . . . . 1425.8 Results for σ1D(T ) in both axes and all dopings. . . . . . . . . . . . . . . . 1435.9 Results for σ0(T ) in both axes and all dopings. . . . . . . . . . . . . . . . . 1455.10 Results for σ0(T ) in both axes, plotted as a function of doping. . . . . . . . 1465.11 Plot of the vertex correction factor βVC as a function of doping. . . . . . . . 1475.12 Results for the integrated spectral weight in both axes, plotted as a functionof temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.13 Results for the integrated spectral weight in both axes, plotted as a functionof doping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.14 Comparisons of integrated spectral weight from σ1 and ∆λ measurements forthe a axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.15 Comparisons of integrated spectral weight from σ1 and ∆λ measurements forthe b axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.16 Measurements of the time dependence of Tc for disordered ortho-VIII samplesreordering at room temperature. . . . . . . . . . . . . . . . . . . . . . . . . 1545.17 Comparison of the conductivity at 3 K for YBCO 6.67 before and after chainoxygen disordering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.18 A schematic picture contrasting extended and point defects. . . . . . . . . . 1575.19 Measurements of the x-ray correlation length in YBCO as a function of dop-ing for both axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.20 Plot of the conductivity width Γ as a function of doping for both axes. . . . 1606.1 Feynman diagram of the dominant decay process for positive muons. . . . . 1656.2 The asymmetric angular distribution of muon decay averaged over energy, asa function of polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656.3 An illustration of the spin alignment of the decay products of pion decay. . 1676.4 The asymmetric angular distribution of muon decay averaged over energy, asa function of polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.5 Muon implantation depth profiles as a function of implantation energy forYBCO 6.52. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.6 Example raw histogram data for low-energy µSR on ortho-II YBCO. . . . . 1796.7 Penetration depth temperature dependence data for YBCO 6.998. . . . . . 181xiiList of Figures6.8 The measured inverse squared penetration depths λ−2(0) for YBCO as afunction of doping, for both axes. . . . . . . . . . . . . . . . . . . . . . . . . 1856.9 Comparison of the a–b anisotropy for λ−2(0) as a function of doping. . . . . 1866.10 The average in-plane inverse squared penetration depth λ−2ab (0) vs. hole dop-ing p for YBa2Cu3O6+x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.11 Plot of Tc as a function of λ−2ab (0), a test of Uemura scaling. . . . . . . . . . 1886.12 A comparison of Ba(Co0.074Fe0.926)2As2 in-plane penetration depth measure-ments from different techniques. . . . . . . . . . . . . . . . . . . . . . . . . . 1906.13 Fits to the Ba(Co0.074Fe0.926)2As2 data with several models shown. . . . . . 192xiiiList of Symbolsa, b, c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Unit cell lattice constants (and directions)aˆ, bˆ, cˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lattice unit vectorsAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Area of surface SAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Area of sampleA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Spectral weightA(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Muon polarization signal asymmetry functionA(ωc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective spectral weight up to cutoff ωcB(r, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real-space magnetic fieldB˜(q, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Fourier-space magnetic field componentsCel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Electronic specific heatd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionality OR dead layer thicknesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Absolute value of electron charge (positive)E(r, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real-space electric fieldE˜(q, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Fourier-space electric field componentsE(ϑ|k2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incomplete elliptic integral of the second kindEg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudogap energy scale (∼ kBT ∗)Ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Quasiparticle dispersion ≡√(ǫk − µ)2 +∆2kf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequencyf˜ ≡ f ′ + if ′′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex frequencyF{ } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier transformF (ϑ|k2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incomplete elliptic integral of the first kindF s1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-symmetric ℓ = 1 Landau Fermi liquid parameterh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planck’s constant OR helicity~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reduced Planck’s constant ( h2π )H(r, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Real-space magnetic H-fieldH˜(q, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Fourier-space magnetic H-field componentsxivList of SymbolsHc1, Hc2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lower, upper critical fieldsHext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External applied magnetic H-fieldI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagonal unit tensorℑ{ } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaginary partJ(r, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real-space electrical current densityJ˜(q, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Complex Fourier-space current density componentsk, k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WavevectorK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Coupling constantkB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Boltzmann’s constantm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carrier massm˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Fourier-space magnetic momentm∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Effective carrier/quasiparticle massme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bare electron massM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetizationn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carrier densitynˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface normal (unit vector)nn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal fluid/quasiparticle densitynp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Density of CuO2 plaquettes (≡ 2Vc for YBCO)ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superfluid densityns/m∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superfluid phase stiffnessNa,Nb,Nc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Demagnetization factors along aˆ, bˆ, cˆN (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positron detector signalp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hole doping (doped holes per CuO2 plaquette)p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Carrier (or particle) momentumP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Microwave powerP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Principal value integrationp1, p2 . . . . . . . . . . . . . . . . . . . . . . . Hole dopings of the lower and upper quantum critical pointspc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Critical hole doping (generic)pCOc . . . . . . . . . . . . . . . . . . . . . . . . . . Critical hole doping for T -linear resistivity (TCO(pCOc ) ≡ 0)pFSc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fermi surface reconstruction critical pointpPGc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudogap critical hole dopingpopt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal hole doping (Tc(popt) ≡ Tc,max)xvList of SymbolsPµ, Pµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Muon spin polarizationq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical charge of carrierQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quality factor of a resonant system (≡ ω0/ωB)Q0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unloaded quality factorq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .WavevectorR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Twin ratio (≡ Amaj/Atot)ℜ{ } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Real partRH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Hall coefficientRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface resistance (≡ ℜ{Zs})S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin angular momentumS˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Complex Poynting vector (≡ E˜× H˜∗)T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TemperatureTc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Superconducting transition temperatureTCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crossover temperature scale for T -linear resistivityTN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ne´el temperatureT ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudogap temperature scaleU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic energy stored in resonatorvF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi velocityv2, v∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Nodal slope of the superconducting gap ∆kV˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Fourier-space voltage componentsVc . . . . . . . . . . . . . . . . . . . . . Effective cavity volume OR conventional unit cell volume (≡ abc)Vs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample volumex . . . . . . . . . . . . . . . . . . . . . . . . CuOx chain layer oxygen occupation fraction for YBa2Cu3O6+xx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Carrier positionxc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical oxygen occupationXs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface reactance (≡ ℑ{Zs})y(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Modified Drude model frequency exponentz . . . . . . . . . . . . . . . . . . . . . . . . . . .Dynamical critical exponent OR depth below sample surfaceZs ≡ Rs + iXs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Surface impedanceα˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex resonator transmission scale factorαi(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal expansivity along iˆ (≡ 1ℓi(T )dℓidT )αsFL . . . . . . . . . . . . . . . . . . . Spin-symmetric Fermi liquid charge current renormalization factorxviList of SymbolsβVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertex correction factorδ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Skin depthδ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirac delta functionδf˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex frequency shift upon sample insertion∆k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Superconducting energy gap∆0, ∆(0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superconducting energy gap maximum amplitude∆γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jump in electronic specific heat coefficient γ across Tc∆λ(T ) . . . . . . Shifts (with T ) in magnetic penetration depth (∆λ(T ) ≡ λ(T )− λ(Tref ≈ 0))∆φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total phase shift around oscillator loopγ . . . . . . . . . . . . . . . . . . . Electronic specific heat coefficient (≡ Cel/T ) OR gyromagnetic ratioΓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Drude linewidth/scattering rate OR resonator constantǫ˜ ≡ ǫ0 − iσ˜/ω . . . . . . . . . . . . . . . . . . . . . . . . . . Complex electric permeability/dielectric constantǫ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Permittivity of free spaceǫi(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . .Integrated thermal expansion along iˆ (≡ e´ T0 αi(T ′) dT ′ − 1)ǫk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Band dispersionζ(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Muon implantation depth probability distributionΘ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heaviside step functionκ˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex propagation constantκ(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal conductivityκ0T . . . . . . . . . . . . . . . . . . . . . . . . . Thermal conductivity low-temperature slope (≡ limT→0κ(T )T )λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Magnetic penetration depthλab . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric average of in-plane penetration depth (≡√λaλb)λL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . London penetration depth (≡√m∗/µ0nse2)µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chemical potentialµ˜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Complex magnetic permeabilityµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic momentµ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Permeability of free spaceν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial critical exponentρ(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical resistivityρs(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Normalized superfluid density (≡ λ2(0)/λ2(T ))ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation length OR superconducting coherence lengthξτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Correlation timescalexviiList of Symbolsσ˜ ≡ σ1 − iσ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical conductivity (general)σ(r, r′; t, t′) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real space (and time) electrical response tensorσ˜(q, ω) . . . . . . . . . . . . . . . . .Complex Fourier-space electrical conductivity tensor componentsσ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Drude conductivity height (≡ nq2τm ) OR limω→0 σ1(ω)σ00 . . . . . . . . . . . . . . . . . . . . . . . . . Universal electrical conductivity limit (≡ limω,T→0 σ1(ω, T )))σ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Real (dissipative) component of conductivity (≡ ℜ{σ})σ2 . . . . . . . . . . . . . . . Imaginary (inductive/screening) component of conductivity (≡ −ℑ{σ})σ1D(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Frequency-independent conductivity offsetσ˜n ≡ σ1n − iσ2n . . . . . . . . . . . . . . . . . . . . . . . . Normal fluid/quasiparticle electrical conductivityσ˜s ≡ σ1s − iσ2s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superfluid electrical conductivityτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Relaxation/scattering lifetimeφext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oscillator phase shift external to resonatorω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Angular frequency (2πf)ω˜ ≡ ω′ + iω′′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Complex angular frequencyω0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Real) resonant angular frequency (≡ ω′0)ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Cutoff angular frequencyωB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bandwidth (full width at half maximum)ωL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Larmor precession frequency (≡ γB)ωp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Plasma frequency (≡√nq2mǫ0 =√σ0τǫ0 )xviiiAcknowledgementsFirst and foremost, I would like to thank my supervisor, Walter Hardy. I have been veryfortunate to have had the opportunity to learn experimental condensed matter physicsunder Walter’s tutelage. The breadth of his experimental expertise – particularly his knackfor always knowing just the right tool or technique for the job – never ceases to amaze me.He is always willing and eager to “get dirty” with hands-on involvement in every part ofthe lab, and despite his emeritus status, remains one of the hardest-working physicists Iknow. He has been very supportive of me throughout my entire degree – especially duringthe hard times on the ill-fated high-pressure susceptometry work – for which I will alwaysbe grateful.Doug Bonn, who has been my (unofficial) co-supervisor over the years, has also playeda huge role in guiding the focus and direction of this thesis work, as well as in supportingmy work financially throughout. Even during his busiest times as department head, hecould always find time to share his insights on my data, and his impressive knowledge ofthe cuprate field.None of the work presented in this thesis (nor, indeed, much of the YBCO research donein the world!) would have been possible without the expert crystal growth skills of RuixingLiang. I am extremely grateful for Ruixing’s patience with my incessant sample growth andannealing requests – particularly the Herculean job of detwinning the giant LEM mosaics!I have greatly enjoyed working with Pinder Dosanjh, who has taught me much of whatI know about fine experimental work, and has helped me out immensely on countless occa-sions. I will always remember his sage advice that the most delicate work should be donefirst thing in the morning – but not until after a coffee.It has been a blast sharing an office with James Day and Shun Chi. I always enjoyedmy lively conversations (and arguments) with James about physics, education, language,and everything else; we usually saw eye-to-eye, but when we didn’t he was most often ableto convince me I was wrong. I have learnt much from my many physics discussions withxixAcknowledgementsShun – especially when he would pose conceptual questions that I had never previouslyconsidered!Over the years, I have had the pleasure of working with many other graduate studentsand postdocs passing through the Supercon lab. In particular, Jake Bobowski showed methe ropes of running a low temperature experiment as a new graduate student. I learnedmuch about low temperature physics (especially running a dilution fridge) from AndreaMorello, whose enthusiasm for research was infectious; I will always fondly remember ourFe8 project, which first piqued my interest in low temperature physics. Many others havehelped make my time in AMPEL 243 memorable, including Brad Ramshaw, Darren Peets,Mahyad Aghigh, Rinat Ofer, and Kevin Musselman.I wish to give special thanks to Rob Kiefl, who brought me into the low-energy µSRteam, and who always made me feel welcome in the foreign land of µSR. I would also like tothank Masrur Hossain and Oren Ofer, with whom I spent many late nights watching muonsdribble in, as well as all of the other people with whom I have had the pleasure of workingwith on the LEM measurements (Zaher Salman, Hassan Saadaoui, Sarah Dunsiger, BastianWojek, Thomas Prokscha, Andreas Suter, and Elvezio Morenzoni).Last, but not least, I have been blessed by much love and support from my entirefamily, both immediate and extended. My sister Kristina has always been there for me,and I am very proud of her accomplishments – even though I had to race to earn my seconddegree (just barely) before she could earn her third! Finally, my parents not only wentthrough the trouble of bringing me into this world, they have been a boundless source ofsupport throughout my entire education and life. I could never have accomplished anythingwithout the love and encouragement (and food!) that they have provided, and I will foreverbe grateful for all they have done for me.xxChapter 1Introduction1.1 The high-Tc cuprate superconductorsSince the 1986 discovery of superconductivity at 35 K in a La–Ba–Cu–O compound (LBCO)[4], soon followed by truly “high” temperature superconductivity in YBa2Cu3O6+x (YBCO,with Tc = 93 K at 6 + x = 6.92) [5], the cuprate (copper oxide) superconductors havebeen the focus of intense and sustained study by theorists, experimentalists, and materialsscientists alike. After nearly thirty years of such efforts, although much has been learned,many of the most fundamental questions about the cuprates remain unanswered, and asuccessful (and complete) theory of the cuprates continues to elude researchers.Nevertheless, the high-Tc field remains very active, as new developments in both theoryand experiment continue to emerge daily. Since the February 2008 discovery of the Fe-basedpnictide superconductors [6], much emphasis has been placed upon both the similarities ofthe two systems and their differences. With this speculation of a common underlyingphysical origin for high temperature superconductivity, there is additional new impetus forunderstanding the physics of the cuprates.The high-Tc cuprates are layered copper oxides based on a perovskite structure. Theprincipal common feature of the cuprates is the presence of one or more CuO2 planes per unitcell, in which the superconductivity is believed to occur. The number of such CuO2 layers,along with the structure and composition of the layers in between, varies between types ofcuprates. The off-plane details not only determine the hole1 doping (carrier concentration)of the CuO2 planes, they have strong effects on the c-axis transport as well. With someimportant exceptions (e.g. YBa2Cu4O8), most of the cuprate compounds being studiedcome from non-stoichiometric families of materials, with fractional cation substitution levelsand/or oxygen concentrations that are variable over some range.1There are also many examples of electron-doped cuprate superconductors; these have similar propertiesto the hole-doped variety, with a similar phase diagram. However, there are significant differences, and inthis thesis one should read “hole-doped cuprate” unless otherwise specified.11.1. The high-Tc cuprate superconductorsTx, pdxt-yt SCAFMT*TcSmall FS(arcs/pockets)PseudogapTCOLarge FS(hole pocket)Strange metalFermi liquid6+x:p:606.320.056.500.09Ortho-II6.670.1251/8th6.920.16Optimal7.000.19N/A0.27TNFigure 1.1: A schematic phase diagram of the high-Tc cuprate superconductors. ImportantYBa2Cu3O6+x oxygen content values 6+x have been labelled here, along with approximatevalues of their associated hole dopings p.Although the dopant chemistry, number of layers, presence/absence of chains, etc., maydiffer between cuprate families, the most important phenomena have been found to fit into auniversal phase diagram of the hole-doped cuprates.1 When the hole doping of each materialis taken into account, each of the cuprates fits into the same characteristic temperature-vs.-doping phase diagram seen in Figure 1.1, within temperature scalings of order unity. Thiscommon phase diagram is strong evidence that for the most part the cuprates share thesame underlying physics, prompting researchers to treat the cuprates as a collective whole.At the zero-doping end of the phase diagram, the relevant physics is that of an antifer-romagnetic insulator, below a Ne´el temperature TN . As the hole doping is increased, theantiferromagnetism weakens, and the Ne´el temperature decreases to zero at p ≈ 0.04; astructural phase transition from tetragonal to orthorhombic crystal symmetry is also foundnear this doping. At slightly higher dopings (p = 0.052) one enters the superconducting re-gion of the phase diagram, in which superconductivity occurs below a critical temperature21.2. The cuprate superconductor YBa2Cu3O6+xFigure 1.2: A unit cell of fully doped YBa2Cu3O7, with equivalent copper and oxygenpositions labelled. For partially oxygenated YBa2Cu3O6+x, only a fraction x of the O(1)oxygen sites in the CuOx chains are occupied. This figure (with some modifications) wasprovided by Darren Peets.Tc whose variation with doping is roughly parabolic;2 this dome extends to a maximumdoping of p = 0.27 [7, 8]. Optimal doping – at which Tc reaches its maximum – is locatedaround popt = 0.16; higher and lower doping levels are referred to as the underdoped andoverdoped regimes, respectively. It should be noted that for YBCO, due to oxygen orderingeffects, hole doping levels are not uniquely determined by oxygen content alone, even forhighly pure samples, and thus some implicit ordering-dependent variations in the x valuesshown in Figure 1.1 must be taken into account [8].31.2. The cuprate superconductor YBa2Cu3O6+x1.2 The cuprate superconductor YBa2Cu3O6+xThe high-Tc cuprate superconductor upon which this thesis will focus is YBa2Cu3O6+x(YBCO). Discovered in late January 1987 [5], its maximum Tc of ∼94 K made it thefirst material displaying superconductivity above the 77 K boiling point of liquid nitrogen,and was thus responsible for igniting a firestorm of activity in the nascent field of high-temperature superconductivity. In the years since, it has remained one of the most popularcuprate materials for study,3 due to both its cleanliness and the availability of data frommany techniques to draw from.Each YBa2Cu3O6+x unit cell (see Figure 1.2) consists of two CuO2 planes separatedby a Y layer, in a closely-spaced bilayer structure. Above and below the planes are BaOlayers, followed by a CuOx “chain” layer at the edge of the cell. The chain layer has oxygensite occupation along only one edge of the cell, forming oriented chain structures of oxygenwhen filled. By varying the fraction x of oxygen sites occupied from 0 to 1 (by adjusting theannealing process used in crystal preparation), one may control the level of hole doping inthe sample.4 Unlike most other cuprate materials, in which variation of doping requires thespecies substitution of cations neighbouring the planes, in YBCO the variable oxygen siteoccupation setting the doping sits a distance away, in the CuOx chain layer. The resultingdisorder potential seen in the planes – where the superconductivity is believed to occur –is thus far less disruptive.Additionally, the chain oxygens may be made to form ordered phases, as shown inFigure 1.3 [10, 11]. An important example is YBa2Cu3O6.5, where half of the oxygensites are occupied, which may form a superstructure of alternating full and empty chainsknown as ortho-II. The ability to minimize the off-plane disorder in this manner is one ofthe main reasons for YBCO’s well-deserved reputation as one of the the cleanest cupratematerials available; however, as we will see later in this thesis, this can present significantcomplications to measurements done over a range of oxygen contents – and ordered CuOx2In many cuprates – including YBCO – a “dip” in Tc relative to this parabolic relationship is foundto occur near 1/8 doping (p = 0.125), associated with the tendency towards charge stripe ordering in thisdoping range and the associated suppression of superconductivity. The magnitude of this suppression variesbetween compounds.3Because YBCO does not cleave well – and when it does cleave, it displays a “polarization catastrophe,”leading to electronic reconstruction which alters the doping of the surface [9] – it is less popular amongstpractitioners of surface measurement techniques such as ARPES and STM, which require a pristine surfacethat resembles the bulk.4Note that, by convention, when one speaks of the hole doping “of the sample”, it is the density of holesin the CuO2 planes alone (where the relevant physics occurs) that is tacitly implied.41.3. Changes from underdoped to overdopedTetragonal Ortho-I Ortho-IIOrtho-III Ortho-V Ortho-VIIIFigure 1.3: A diagram of the oxygen-ordered phases of the CuOx chain layer ofYBa2Cu3O6+x. Shown are the oxygen O(1) site occupation configurations for several possi-ble orthorhombic phases. The red brackets indicate the smallest repeating unit, the lengthof which labels the ortho-# phase. The (non-ordered) tetragonal phase is also shown, whereboth chain-layer oxygen site sublattices may be occupied.chains are not always the desired configuration.1.3 Changes from underdoped to overdopedAn important feature of the cuprate phase diagram, lying on the underdoped side, is thepseudogap region, wherein many different measurement techniques show evidence of a par-tially gapped Fermi surface (with a finite energy gap only along part of the surface) inthe normal state. The pseudogap is associated with a characteristic crossover temperatureT ∗ below which the gapped excitations freeze out. The pseudogap energy Eg ∼ kBT ∗ isobserved to decrease with hole doping, extrapolating to zero energy at a critical dopingpPGc ≈ 0.19 beyond which it disappears. While its presence has been well established exper-imentally [12], the physical origin of the pseudogap state is an active field of research. Mostproposals can be divided into two camps: those that link the pseudogap to superconduct-ing fluctuations, or “preformed pairs” [13–16]; and those that identify the pseudogap withsome distinct form of order [17–20]. This remains a topic of vigorous debate, particularlyin regards to the relationship between the pseudogap and superconductivity.Another well-established feature in the cuprate phase diagram is a region of T -linear51.4. Quantum critical phenomenologyresistivity, wherein the normal state5 electrical resistivity ρ (T ) exhibits a linear temperaturedependence over an extended range of temperatures, from T & 600 K down to a doping-dependent crossover temperature below which the temperature dependence deviates fromlinearity [21, 22]. Although the choice of a criterion to identify the location of this crossoverTCO can be somewhat arbitrary, most studies identify a critical doping pCOc ≈ 0.19 at whichthe linearity extends to zero temperature, with the crossover temperature rising for higherand lower dopings [23, 24]. Since this behaviour stands in contrast to the low-temperatureT 2 dependence of the resistivity expected of a Fermi liquid, the region near pCOc and abovethe crossover temperature is known as the “strange metal” or “non-Fermi liquid” region.Perhaps the most compelling example of a dramatic change in behaviour across thephase diagram comes from recent studies of the normal-state Fermi surface of the thecuprates. All techniques show a large, hole-like Fermi surface for samples on the overdopedside of the phase diagram [25, 26]. On the underdoped side, however, there is significantlymore controversy; some techniques claim a Fermi surface consisting of small hole and/orelectron pockets [27, 28], while others see only Fermi arcs and not enclosed pockets [29].Nevertheless, a general consensus has been reached with the picture of some sort of “small”Fermi surface, in clear contrast with the large Fermi surface of the overdoped side. Sucha topological change in the Fermi surface from the underdoped to the overdoped side ofthe phase diagram requires the existence of a phase transition6 at some critical doping pFScwhere this reconstruction of the Fermi surface occurs [30], with some form of k-space foldingof the Brillouin zone possibly associated with the onset of some form of electronic order.1.4 Quantum critical phenomenologyThe simplest (and most frequently proposed) explanation of the aforementioned featuresof the phase diagram is that they each share a common origin: at T = 0, there is a singlequantum critical point at a critical doping pc ≈ 0.19, at which the pseudogap opens andthe Fermi surface reconstructs. The crossover behaviour and T -linear resistivity seen inthe strange metal regime above the postulated critical point find natural explanations interms of the finite-T behaviour common to quantum phase transitions. The experiments5For the low-temperature normal state measurements referred to here, superconductivity is suppressedwith a large applied magnetic field.6At least one phase transition, that is; see Section 1.7.61.4. Quantum critical phenomenologydescribed in this thesis only concern the presence of the low-temperature phase transitionitself, independent of its quantum critical nature. However, much of the motivation forthe existence of such a transition stems from the natural explanation for many features ofthe cuprate phase diagram in terms of quantum critical phenomenology. Before proceedingfurther, a brief introduction to quantum phase transitions is in order.In a classical system, phase transitions are driven solely by thermal fluctuations; asT → 0, such systems freeze into fluctuationless ground states [31]. However, for quantumsystems (i.e., those for which the relevant order and correlations are of a distinctly quantummechanical nature), quantum fluctuations remain at T = 0, allowing external non-thermalparameters (such as magnetic field or chemical doping) to drive transitions between quantummechanical ground states [32]. The point along the T = 0 line where such a transitionoccurs is known as a quantum critical point (QCP). While a true quantum phase transitiontechnically occurs only at the QCP, signatures of quantum criticality may still be observedat finite temperatures in the vicinity of the QCP; the theory of such quantum criticalphenomena was first developed by Hertz [33] in the late 1970s, and has since been elaboratedupon and adapted by many others [34–36].The basic idea rests upon the deep connection between quantum dynamics and classicalthermodynamics, wherein the statistical mechanics of a d-dimensional quantum systemcan be expressed in terms of the statistical mechanics of an analogous (d+1)-dimensionalclassical system [37], where the extra dimension is temporal. This temporal dimension isfinite in extent, with a length inversely proportional to the original temperature (which thusbecomes infinite as T → 0). The role of temperature in the new system is now assumedby a coupling constant K = p − pc expressing the proximity in parameter space p to thecritical point pc (and thus the strength of the quantum zero-point fluctuations).As a consequence of this intermingling of temporal and spatial dimensions near a QCP,the familiar scaling properties obtained close to a classical phase transition are extendedto the temporal dimension as well. For T = 0, the critical fluctuations possess not only adiverging correlation length scale ξ, but also a diverging correlation time scale ξτ , as thecritical point is approached. Near criticality, the system possesses scale invariance in thetemporal dimension as well as in the spatial dimensions; consequently, the energy ~ω ofcritical fluctuations is also scale invariant.For finite T , the picture becomes more complicated, as the length of the temporal71.4. Quantum critical phenomenologyTx, pAFM6+x:p:60pc~0.19dxt-yt SCTNT*QuantumorderedQuantum criticalQuantum disorderedFigure 1.4: Phase diagram of the cuprates, as in Figure 1.1, but here framed in terms ofthe phenomenology associated with a quantum critical point at pc ≈ 0.19.dimension becomes finite, and the resulting finite-size scaling effects govern the physics.For low temperatures, and for a sufficient distance K from critical coupling, the finitetemperature cutoff timescale ~/kBT remains larger than the scale ξτ ∼ |K|−νz (where ν andz are the spatial and dynamical critical scaling exponents, respectively), and thus quantumfluctuations continue to set the critical behaviour. As T is increased (or K decreased) towhere ξτ ≈ ~/kBT and beyond, the thermal timescale becomes the relevant cutoff, and wecross over to a region – the quantum critical regime – where temperature is the dominantenergy scale.Framing this in terms of the proposed scenario of a QCP under the superconductingdome of the cuprates (see Figure 1.4), one can identify the strange metal regime with thequantum critical regime above critical doping. The T -linear resistivity found in the strangemetal regime arises as a natural consequence of the quantum critical scaling properties nearpc [38, 39]. At T = 0, for p < pc, and in the normal state (produced with applied magneticfield), the system is in an insulating ordered phase, which disappears for p > pc; it is this81.5. A quantum critical point underneath the dome?order which is associated with both the pseudogap and the Fermi surface reconstruction.To either side of the quantum critical point at low temperatures (and below the quantumcritical region) lie the quantum ordered and quantum disordered phases, where quantumfluctuations control the critical phenomena.1.5 A quantum critical point underneath the dome?The characteristic quantum critical phenomenology outlined above describes the observedcuprate phase diagram well in the region above the proposed quantum critical point. Amongthe most compelling “evidence” for a quantum critical point under the cuprate supercon-ducting dome comes not from the cuprates themselves, but from comparison to the heavyfermion materials, whose phase diagrams are quite similar to that of the cuprates [40]. Inthese systems the existence of quantum critical points surrounded by regions of supercon-ductivity has been fairly well established. It is thought that superconductivity in the heavyfermion materials is facilitated by quantum critical fluctuations [41, 42] – a sentiment whichhas often been extended to the cuprates as well [43, 44].Returning to the cuprates, there is additional evidence of a phase transition (quan-tum critical or otherwise) at or near p = 0.19 from other measurements, beyond thosephenomena already mentioned. Some of the earliest evidence comes from specific heat mea-surements [45–47]. In particular, the jump ∆γ in the electronic specific heat coefficientγ ≡ Cel/T as the temperature is raised through Tc – commonly referred to as the “specificheat anomaly” – is found to be roughly flat with doping for samples with p > 0.19; be-low this point, ∆γ drops rapidly with doping, a result attributed to the decreased spectralweight available for superconductivity associated with the opening of the pseudogap belowpc [48].All of the experimental evidence discussed so far has concerned normal state properties,where either the measurements are performed above Tc, or superconductivity is suppressedwith strong magnetic fields. Evidence for a phase transition near optimal doping at zero fieldin the superconducting state is far less abundant. One technique which has produced somesuch evidence is zero-field muon spin relaxation (ZF-µSR), in which the spin precession anddephasing of polarized muons implanted into a sample is used as a probe of the magneticfields within. Measurements with ZF-µSR [49, 50] have shown signatures of “glassy” spin91.6. Early experimental attemptsdynamics for underdoped samples, disappearing for p & 0.20 (consistent with pc = 0.19within their doping resolution). However, the evidence here for a clear phase transition ata well-defined doping is somewhat weak.More dramatic evidence for a sharp phase transition at pc ≈ 0.19 in the superconduct-ing state comes from Raman scattering, an inelastic, polarization-sensitive optical spec-troscopy technique useful for probing the elementary excitations of solids [51]. Measure-ments on (Y,Ca)Ba2Cu3Oy of the intensity of pair-breaking peaks in the A1g and B1gpolarizations [52] show an abrupt change in the doping dependence at p ≈ 0.19. Thesemeasurements were carried out at 10 K and in zero field – deep in the superconductingstate – yet still suggest a change in the electronic state of the system near the same criticaldoping indicated by normal-state techniques.Perhaps due to the relative lack of data (aside from the Raman experiments) showinga zero-temperature phase transition at p ≈ 0.19 that persists in the superconducting state(compared to the normal state, for which ample evidence exists), some theorists have pro-posed scenarios in which such a critical point near optimal doping is absent. Among themost prominent of such proposals is that of an “avoided quantum critical point” promotedby Sachdev and collaborators [53], wherein a spin density wave (SDW) quantum criticalpoint seen in the normal state at pc ≈ 0.19 is shifted to a much lower doping in the super-conducting state, due to the suppression of SDW order by superconductivity. There is someexperimental evidence from transverse-field µSR which supports this picture [54]. However,even assuming the validity of this scenario, a strong possibility remains for the existence ofan additional phase transition to Ising nematic order, which is coincident with the SDWphase transition in the normal state, but unlike the SDW transition is not displaced fromits normal state pc by superconductivity [53]. A phase transition at p ≈ 0.19 could thusstill potentially be experimentally observed in the superconducting state.1.6 Early experimental attemptsWith a lack of conclusive experimental evidence for a QCP that persists in the supercon-ducting state – but with strong reasons to support its existence – a careful study of thelow-energy electrodynamic properties as a function of doping above p = 0.19 was proposedand initiated. A good candidate measurement for such a study is that of the temperature101.6. Early experimental attemptsdependence of the magnetic penetration depth λ(T ) (more accurately, the experimentallyaccessible ∆λ(T ) ≡ λ(T )− λ(T ≈ 0)) at low temperature.7The magnetic penetration depth (also known as the London penetration depth) is thelength scale over which magnetic fields are screened from the bulk of a superconductorvia dissipationless surface currents; it (as well as its temperature dependence) dependssensitively on the geometry of the Fermi surface, as well as the electronic dispersion in itsvicinity, and the details of the superconducting gap [55, 56]. It should therefore be wellsuited to display clear signatures of changes in Fermi surface topology at the proposed QCP,although the exact details remain a subject of debate.8The experiment that was proposed – and attempted – was to be a precision low-temperature ac susceptometry study of ∆λ(T ) as a function of doping, using pressure tovary the doping in a well-controlled manner. The reason for measuring the samples un-der applied pressure was to allow the hole doping of a single sample to be varied in finesteps, without altering the chemistry (which can introduce confounding factors into anystudy examining variations with doping). For measurement at the critical doping in YBCOthere was an additional important reason to use pressure doping, in that some estimatesplaced this just past that of fully-oxygenated YBa2Cu3O7.0 (where p = 0.194 [8]): withoutsubstitution doping, the use of pressure was the only way such a critical point could bereached.Many months of time and effort were invested in this project, particularly in the designand construction of a miniature high-sensitivity ac susceptibility probe suitable for insertioninto the inner bore of a clamp-type pressure cell. The required sensitivity in penetrationdepth needed for detecting the predicted changes in the low temperature slope of the pene-tration depth was to be below an a˚ngstro¨m. Unfortunately, while much progress was madein reducing the noise and drift levels, the final penetration depth sensitivity reached wasslightly better than 1 nm – still an order of magnitude beyond our requirements.7The penetration depth λ is the means by which we obtain the imaginary part of the conductivity σ2(where σ ≡ σ1 − iσ2), from the relation σ2 = 1/µ0ωλ2. One often expresses this in terms of the superfluiddensity ns instead, since in the low frequency limit σ2 = e2ns/m∗ω, and thus ρs(T ) ≡ ns(T )/ns(0) =λ2(0)/λ2(T ). Perhaps more fundamentally, the quantity nsm∗ is proportional to the superfluid phase stiffness,the energy cost for spatial variations in the phase of the superconducting order parameter.8A calculation of Carbotte et al. [57], using a modified Yang-Rice-Zhang model [15] for the pseudogap,shows there to be a step in λ(0) at the QCP, but no change in the T = 0 value of dλ/dT . However, thereis no known a priori reason for such an equality in dλ/dT after reconstruction, and this could well be anartifact of the YRZ model used.111.7. Two quantum critical points underneath the dome?1.7 Two quantum critical points underneath the dome?Tx, pAFM6+x:p:60Spinorderp2~0.19p1~0.08dxt-yt SCTcTNChargeorderFigure 1.5: A proposed phase diagram scenario for the cuprates with two quantum criticalpoints – a point p1 ≈ 0.08 where spin density wave (SDW) order gives way to charge densitywave (CDW) order, and a point p2 ≈ 0.19 where CDW disappears. In this scenario, thedome of superconductivity is two domes (associated with p1 and p2) which have mergedtogether.Over the period while the high-pressure susceptibility project was still ongoing, newevidence emerged suggesting a different picture of the phase diagram. Quantum oscillationmeasurements on YBa2Cu3O6+x between 6 + x = 6.49 and 6.54 (near p1 = 0.09) demon-strated a divergence of the effective quasiparticle mass m∗ with decreasing oxygen x as oneapproaches 6+xc ≈ 6.46 [58]. Such a divergence, corresponding to 1m∗ → 0, is characteristicof a metal-to-insulator quantum critical point, which the authors proposed to exist at xc.Here a picture of two quantum critical points was put forth – one near p1 ≈ 0.09 and theother at p2 ≈ 0.19 – with the suggestion that the superconducting Tc dome might in realitybe two merged domes of superconductivity associated with each of these quantum criticalpoints, a scenario previously encountered in some heavy fermion systems such as CeCu2Si2121.8. The conductivity doping scan[40, 59, 60]. Alternatively, the enhancement of Tc near p1 could be only a local enhancement(due to quantum critical fluctuations) of a single dome of superconductivity associated withthe quantum critical point at p2 [61].Additionally, Hall effect measurements at high magnetic field (up to 60 T) demonstrateda change in the behaviour of the Hall coefficient temperature dependence at a criticalhole doping of p = 0.08 [62]. Above this doping, the Hall coefficient RH(T ) (measuredat high field to suppress superconductivity) was observed to change sign as a functionof temperature, from positive at high temperature to negative as T → 0, a behaviourascribed to the presence of a high-mobility electron pocket of the Fermi surface. Below thisdoping, however, RH(T ) remained positive for all temperatures – a sign of the disappearanceof the electron pocket upon decreasing hole doping below 0.08. This was interpreted interms of a Lifshitz transition, a change in the topology of the Fermi surface; this wasproposed to be due to a change in the k-space folding of the Brillouin zone resulting froma change in some underlying electronic order (such as spin or charge density waves) orthe incommensurability thereof. This transition coincided with observed changes in the a–banisotropy of the resistivity, providing supporting evidence for the involvement of some formof unidirectional order. The natural proposal was a scenario with spin order for p < p1 beingreplaced by charge density wave order in the range p1 < p < p2, with order disappearingaltogether for p > p2 (see Figure 1.5).More recently, measurements of the upper critical field Hc1 as a function of doping [63]have provided extremely compelling evidence for the existence of two quantum critical points(at p1 ≈ 0.08 and p2 ≈ 0.18) underneath the superconducting dome. At high magneticfields of 30 T the superconducting Tc dome is observed to contract down to two separatesuperconducting domes centred on these points, with Tc dropping to zero in between: thedomes fully separate. This measurement also suggested that the higher-doping quantumcritical point p2 might lie in a region below the doping of fully-oxygenated YBCO – andthus accessible through oxygen content variation without the need for pressure doping.1.8 The conductivity doping scanIn light of this new information, there was renewed interest in undertaking a doping de-pendence study of the electrodynamics of YBa2Cu3O6+x – but this time over a much wider131.8. The conductivity doping scanrange of hole dopings, accessible through oxygen content variation alone. While previouswork had been done in this doping range, it had almost exclusively focused on only threeoxygen concentrations (ortho-II YBa2Cu3O6.5, optimally-doped YBa2Cu3O6.92, and fullyoxygenated YBa2Cu3O7.0), with few exceptions. In particular, no detailed data of the dop-ing dependence of the microwave conductivity of YBa2Cu3O6+x between 6 + x = 6.5 and7.0 was known to exist. Even neglecting the possibility of high-impact discoveries in thisrange, the availability of this data would be of intrinsic value for the field.Seeking to fill this void in an important region of the phase diagram – and to search forevidence for unexpected changes in conductivity properties with doping that might revealthe presence of a quantum critical point – the main thrust of the thesis work to be describedwas a detailed doping dependence study of the microwave conductivity – both the real andimaginary components – for YBa2Cu3O6+x, at several oxygen contents (and thus dopings)between 6+x = 6.49 and 6.998. The imaginary (inductive) part σ2 of the conductivity wasobtained using a 940 MHz microwave cavity perturbation technique to measure shifts in themicrowave penetration depth ∆λ. Measurements of the real (resistive) part σ1 of the con-ductivity were carried out with a bolometric broadband microwave spectroscopy techniqueoptimized for measuring the frequency-dependent surface resistance of superconductors atlow temperatures.For the microwave penetration depth measurements, a very considerable effort was putinto characterizing and eliminating systematic effects that could otherwise overwhelm thesubtle intrinsic doping dependence displayed by the penetration depth. In particular, var-ious doping-dependent corrections to the data (for effects such as thermal expansion andc-axis contributions from the sample edges) can potentially be confounding factors, anddifferent choices of data sources used for these corrections can change both the quantitativeand qualitative conclusions regarding the doping dependence of a physical quantity. Thesensitivity of the results to these different choices will be explored and documented in thisthesis, and the final results should provide a useful reference for future studies examiningthis doping range.Much of the resulting penetration depth data was unsurprising – over most of the dop-ing range, parameters interpolate fairly smoothly with doping, and parameter values arecomparable with previous studies (where comparable data exists). However, at the higherdopings examined, at and above optimal doping, surprising trends in behaviour were ob-141.8. The conductivity doping scanserved – particularly visible in changes in the anisotropy between a- and b-axis temperaturederivatives of superfluid density. While remaining uncertainties regarding systematic ef-fects prevent any strong claims from being made here, the data does not appear to show apeak in penetration depth as one approaches p =0.18–0.19 – the predicted location of theputative “p2” quantum critical point – similar to that seen in the pnictide superconductorBaFe2(As1−xPx)2 near its quantum critical point [64]. However, the data does provide someevidence for a change in anisotropy near optimal doping (p = 0.16), the nature of whichremains unclear.For the surface resistance measurements, many of the same considerations apply – sys-tematics and data fitting considerations must be carefully accounted for in order to extractthe true doping dependence, and these details have been documented in this thesis. Justas with the penetration depth data, the resulting conductivity data carry intrinsic valuefor future reference. Data were consistent with previous results (where available), and overmost of the range, conductivity properties varied smoothly with doping. However, unex-pectedly large scattering rates were observed near p = 0.12–0.13, along the a axis only –this was particularly surprising, since the samples in question were (nominally) well-orderedortho-VIII and ortho-III samples. Such samples were expected to be better ordered thanthe higher doping samples, and thus to have lower loss.In an attempt to unravel this mystery, an ortho-VIII YBCO 6.67 sample that had alreadybeen measured was subsequently remeasured after disordering its chain layer oxygens (bya low-temperature quench following a high-temperature anneal). The b-axis conductivityof the sample was barely affected – but the a-axis scattering rate of the disordered samplewas observed to be much lower than that of the ordered sample. This was a completelyunexpected result, and appeared to contradict previous work on ortho-II YBCO (6 + x ≈6.5) in which disordering was shown to clearly increase the scattering rate [65, 66]. Anexplanation proposed in this thesis concerns the domain walls formed in the CuOx chainlayer between ordered regions of different phase: the scattering potential produced in theCuO2 planes from such extended scatterers may potentially be more effective at scatteringthan that produced by more numerous (but homogeneously distributed) chain ends in thedisordered phase. The appearance of this peak near p = 0.125 – where charge density waveorder has been observed to be strongest – may not be coincidental, and there may be acomplex interplay at work between chain oxygen order and charge order.151.8. The conductivity doping scanWhatever the mechanism, this serendipitous discovery that chain-oxygen-disorderedYBa2Cu3O6+x samples can display lower electronic scattering rates than ordered samples insome regions of the phase diagram has already had an important impact on the field. Recentquantum oscillation studies – informed by this result – have begun to employ chain oxy-gen disordered YBa2Cu3O6+x samples in their measurements, up to oxygen contents muchhigher than 6.67, and have found that the scattering rate is indeed decreased substan-tially. This decreased scattering rate has made possible quantum oscillation measurementsat much higher dopings than were previously accessible, opening entire new regions of thephase diagram for study. As a direct result of this work, quantum oscillation measurements– submitted for publication, but as yet unpublished – have now shown a clear divergenceof effective mass m∗ as one approaches p ≈ 0.18 [67] – striking evidence for the quantumcritical point that this study initially sought to confirm.A separate part of the doping scan project – yet very closely related – was the use oflow-energy µSR measurements of the absolute value of the magnetic penetration depth.For practical reasons, the microwave methods described in this thesis are only suitable forextracting shifts ∆λ(T ) in the microwave penetration depth as a function of temperature;to extract the absolute penetration depth, λ(T ), one needs a separate technique to obtainan absolute value of λ(Tmeas) at some measurement temperature Tmeas, to which these shiftscan be referenced. For these purposes, the technique of low-energy µSR was employed.While somewhat distinct from the main project, the penetration depth values obtainedin this way played an important role in the analysis, and are thus presented in this thesisas well. In addition, low-energy µSR measurements of the iron pnictide superconductorBa(Co0.074Fe0.926)2As2 were also taken, and were combined with microwave measurementsof the author to produce a complete picture of the temperature dependence of the superfluiddensity in this material; this is a good demonstration of the power of combining microwavespectroscopy and low-energy µSR, and will be documented here as well for posterity.16Chapter 2Microwave electrodynamics ofsuperconductors2.1 Electrical conductivityBefore proceeding further, it is important to note the conventions that we will be follow-ing: In addition to the consistent use of “MKS” SI units throughout this thesis, a phaseconvention is used9 such that harmonic time dependence is expressed as e+iωt, with Fouriertransforms defined accordingly; this convention leads to Zs = Rs + iXs and σ = σ1 − iσ2.2.1.1 Linear response theoryWithin linear response theory, the electric current J(r, t) at position r and time t is deter-mined from the electric field E(r′, t′) at positions r′ and times t′ via the integral equationJ(r, t) =ˆ ∞−∞dt′ˆd3r′ σ(r, r′; t, t′)E(r′, t′), (2.1)where the kernel σ(r, r′; t, t′) is the conductivity tensor (which need not be diagonal ingeneral). Although the integral in this form extends over all r′ and t′, in all physicalsituations causality must be imposed; that is, the “stimulus” E(r′, t′) at time t′ must notaffect the response J(r, t) at an earlier time t < t′.10 Thus σ(r, r′; t, t′) can be nonzero onlyfor t > t′.Additionally, we may require translational invariance in space and time; we consider theconductivity to depend only on differences in time and position (τ ≡ t− t′ and r ≡ r− r′,respectively). This assumption relies on the system being homogeneous on the spatial andtemporal scales being probed – conditions which will break down at the atomic scale and9This convention is common in microwave spectroscopy – perhaps due to historical close contact withthe world of electrical engineering, where this convention is used exclusively.10More rigorously, |r− r′| ≤ c(t− t′); the events must have null or time-like separation.172.1. Electrical conductivityacross material boundaries. Our expression (2.1) now becomesJ(r, t) =ˆ t−∞dt′ˆd3r′ σ(r− r′; t− t′)E(r′, t′) (2.2)≡ [σ ∗E] (r, t) , (2.3)which is the convolution of σ and E over space and (past) time.For typical electromagnetic measurements, such a real-space formulation will typicallybe less useful than the corresponding description in terms of frequency and wavevector,obtained by the appropriate Fourier transforms. By virtue of the Faltung theorem, theFourier transform of the convolution reduces to a product of Fourier-transformed quantities;that is,F{σ ∗E} = F{σ}F{E} , (2.4)yielding the simple resultJ˜(q, ω) = σ˜(q, ω)E˜(q, ω), (2.5)where E˜(q, ω) and J˜(q, ω) are the complex (q, ω)-space Fourier components of the electricfield and current, respectively. Note that σ˜(q, ω) remains a tensor.In certain situations – such as for the unconventional superconductors encountered inthis thesis, for which the coherence length ξ is shorter than all other relevant length scalesin the superconducting state – we may make the approximation of local electrodynamics,wherein we assume that the current at a point depends only on the field at that point. Theq-dependence then drops out, yieldingJ˜(ω) = σ˜(ω)E˜(ω). (2.6)Finally, for all work in this thesis, measurement geometries have been chosen to probethe conductivity tensor σ˜ along crystallographic axes of the sample, which also coincide withthe principal axes of σ˜; therefore, we seldom need to consider the general tensor characterof σ˜, working only with the three diagonal components σ˜xx, σ˜yy, and σ˜zz as appropriate.1111For the crystallographic aˆ, bˆ, and cˆ axes, these σ˜ii will be respectively denoted σ˜a, σ˜b, and σ˜c through-out, without the redundant double indices.182.1. Electrical conductivity2.1.2 Response function causality, analyticity, and the Kramers-KronigrelationsAs mentioned in Section 2.1.1, any physically meaningful response function σ(t−t′) must becausal. It can be shown [68] that such a requirement is equivalent to demanding analyticityof its Fourier transform σ˜(ω) in the lower12 half-plane ℑ (ω) < 0. As a result of thisanalyticity, the Cauchy residue theorem states that˛ σ˜(ω′)ω′ − ω dω′ = 0, (2.7)where the integration contour here skirts the ℑ (ω) < 0 side of the ℜ (ω) axis, closed by asemicircle at infinity. The integral over the semicircle will vanish, leaving the contour alongthe real axis. After careful treatment of the integration near the poles, followed by somemanipulation making use of the parity of σ1 and σ2 [68, 69], one arrives at the Kramers-Kronig relations, the paired expressions for the real and imaginary parts of σ˜ = σ1 − iσ2 interms of each other:σ1(ω) = +2πPˆ ∞0ω′σ2(ω′)ω′2 − ω2 dω′, (2.8)σ2(ω) = −2πPˆ ∞0ωσ1(ω′)ω′2 − ω2 dω′. (2.9)Here P denotes principal value integration. While the utility of these expressions – gettingone response function component “for free” from its partner – is obvious, the caveat is thatcomplete knowledge of a response function for all ω ∈ [0,∞) is required to make use of this,at least in principle. In practice, however, extrapolations of a response function beyond themeasurement window can sometimes be sufficient to allow a useful approximate solution.In this manner, we will be making use of Equation 2.9 in Section 5.2.2 to allow the effectsof the quasiparticle screening conductivity σ2n to be accounted for in our measurements ofσ1n.At this juncture it is also important to note a powerful optical sum rule – known as thef -sum rule – which relates the integrated real conductivity to the density and charge of the12Recall our phase convention with e+iωt harmonic dependence of fields.192.1. Electrical conductivitycarriers, through a quantity known as the spectral weight A [69, 70]:A ≡ 2πˆ ∞0σ1(ω) dω =∑rq2rnrmr(2.10)where the index r specifies the carrier type, and qr, nr, and mr are the charge, numberdensity, and mass of carrier r. This can be motivated from the Kramers-Kronig transformsabove, using the fact that at very high energy, absorption (and thus σ1) must vanish [68, 71].The above sum rule extracts all of the bare masses and charges in the system – includingthe core electrons – but does so by integrating σ1(ω) well past the energies of all interactions.A limited form of this sum rule, integrated up to a cutoff frequency ωc, may instead be used(here considering only one carrier type, of bare charge q, which is not renormalized) [72, 73]:A(ωc) ≡2πˆ ωc0σ1(ω) dω =q2neffmeff. (2.11)Here A(ωc) is an effective spectral weight up to frequency ωc, and neff and meff are cutoff-dependent parameters expressing the effective density and mass of the carrier; with theappropriate choice of cutoff (e.g. integrating well into the tail of a Drude peak, but stoppingbefore interband transitions), one can extract the spectral weight of the relevant free carriersparticipating in conductivity at that energy scale.2.1.3 The Drude modelThe simplest natural starting point for a description of the low-energy electrodynamicsof normal metals is the Drude model, wherein we consider “nearly free” charged carriersaccelerated by a time-dependent (but locally spatially uniform) electric field E(t), alongwith an additional damping mechanism acting to restore momentum to equilibrium withina relaxation time τ . This relaxation time may be taken generically to represent an effectivecarrier scattering lifetime, and will in general depend on frequency; however, for now, wetake τ to be frequency-independent.The time-domain dynamics of a particle of charge q and mass m described by thissituation may be expressed asp˙(t) = mx¨(t) = qE(t)− mx˙(t)τ , (2.12)202.1. Electrical conductivityb where x is the position of the particle. Given a density n of such (noninteracting) chargecarriers, we may then express this in terms of the current density J = qnx˙ asJ˙(t) = nq2m E(t)−1τ J(t). (2.13)It is more useful to Fourier transform this expression and work in the frequency domain;here we use the convention E(t) = ℜ{E˜e+iωt}and J(t) = ℜ{J˜e+iωt}. This yieldsiωJ(ω) = nq2m E(ω)−1τ J(ω), (2.14)which may be simply rewritten asJ(ω) = nq2τm11 + iωτE(ω); (2.15)using our familiar definition for conductivity from Equation 2.6 (here assuming our conduc-tivity tensor to be diagonal, i.e. σ(ω) = σ(ω)I), we thus arrive atσ˜(ω) = nq2τm11 + iωτ , (2.16)or in terms of σ0 ≡ nq2τm ,σ˜(ω) = σ01 + iωτ . (2.17)Using the sign convention σ˜(ω) = σ1(ω)−iσ2(ω), we may extract the real and imaginaryparts of the conductivity, yieldingσ1(ω) =σ01 + (ωτ)2 , (2.18a)σ2(ω) =ωτσ01 + (ωτ)2 . (2.18b)These Drude-model σ1(ω) and σ2(ω) are shown in Figure 2.1 with their important featureshighlighted.Particularly in the context of optics, the conductivity is often expressed in terms of the212.1. Electrical conductivity0-12-13-14-15-100/20 = 1/~~-2Conductivity 1,2()Angular frequency  1( )  2( )~-1Figure 2.1: The real (σ1) and imaginary (σ2) components of conductivity for the Drudemodel, from Equations 2.18a and 2.18b, respectively.plasma frequency ωp ≡√nq2mǫ0 =√σ0τǫ0 asσ˜(ω) = ǫ0ω2pτ1 + iωτ ; (2.19)the utility of this parametrization becomes apparent when we consider the real part of thedielectric constant, defined here13 to be ǫ˜ = ǫ0 − iσ/ω. Taking the real part ǫ1 ≡ ℜ{σ˜}, wefindǫ1 = ǫ0 −σ2ω = ǫ0 −1ωǫ0ω2pτωτ1 + (ωτ)2= ǫ0(1− (ωpτ)21 + (ωτ)2). (2.20)For the typical case where ωpτ ≫ 1, we see that ǫ1 crosses zero and changes sign atω = ωp√1− (ωpτ)−2 ≈ ωp (the so-called plasma edge), above which the material becomes13Since there is no meaningful distinction between free and bound charges and currents at finite frequency– at least from a macroscopic electrodynamics standpoint – there is freedom here as to how to partitiontheir contributions. The choice we make here is a common one.222.1. Electrical conductivitytransparent as a result [68, 74].A test application of the optical sum rule (Equation 2.10) to the Drude σ1(ω) (Equation2.18a) yieldsA = 2πˆ ∞0σ1(ω) dω =2πˆ ∞0σ0 dω1 + (ωτ)2= σ0τ =nq2m (2.21)= ǫ0ω2p; (2.22)Equation 2.21 affirms what was claimed in Section 2.1.2, while Equation 2.22 is a commonstatement of the sum rule in optics contexts.Finally, we note that the applicability of the Drude model transcends the limited scenarioin which we have derived it. Allowing for a tensorial inverse effective mass (i.e., 1/m →(1/m∗)ij), the model extends readily to real, anisotropic materials. In the appropriate limits,it emerges nearly unscathed from more rigorous treatments of conductivity, including bothquasiclassical Boltzmann dynamics and the fully quantum Kubo formalism [68, 69].2.1.4 The two-fluid modelIn order to extend this Drude model to describe the electrodynamics of a superconductor,we adopt what is known as the two-fluid model, wherein we divide our charge carriersinto two components: a conventional “normal fluid” (associated with quasiparticles excitedout of the condensate) and a dissipationless “superfluid” (associated with paired electronsin the condensate). The nomenclature of this model betrays its origins in the study ofsuperfluid helium, for which it has seen much use [75]. While such a division into normaland superconducting components is not rigorously valid, this model nonetheless provides anexcellent phenomenological description of the electrodynamics at low energies ~ω ≪ 2∆ thatare well within the superconducting energy gap.14 Throughout this thesis, we shall adoptsubscript labels n and s to denote quantities pertaining to the normal and superconductingcomponents, respectively.The total conductivity is given by the sum σ˜ = σ˜n + σ˜s of these two channels. For nowwe consider only the superfluid conductivity, which we assume to take the form of a Drude14We can safely assume the limit ~ω ≪ 2∆ for what follows; the highest microwave frequency measuredin this thesis is 26.5 GHz, whereas all YBa2Cu3O6+x samples measured had Tc > 50 K, corresponding to2∆h & 4.46 THz: more than two orders of magnitude greater than ω.232.1. Electrical conductivityconductivity as in Equation 2.15, but here with τs →∞:σ˜s(ω) = limτs→∞nsq2sτsm∗s11 + iωτs= nsq2sm∗slimτs→∞τs1 + (ωτs)2︸ ︷︷ ︸σ1s−i nsq2sm∗slimτs→∞ωτ2s1 + (ωτs)2︸ ︷︷ ︸σ2s, (2.23)since the limits may be taken separately here for the real and imaginary components. Firstexamining the simpler limit, for the imaginary conductivity component σ2s, we findσ2s(ω) =nsq2sm∗s1ω limτs→∞✟✟✟✟✟✟✯1(ωτs)21 + (ωτs)2= nsq2sm∗sω. (2.24)For the case of the limit in σ1s, we encounter the situation of a Lorentzian centred at ω = 0whose width 1/τs is decreasing in inverse proportion to its height τs – preserving total area– towards the limiting case of a delta function of the same total area. This can be seen byconsidering the Kramers-Kronig relation for σ2 from σ1 (Equation 2.9), which yieldsnsq2sm∗sω= σ2s(ω) = −2πˆ ∞0ωω′2 − ω2σ1s(ω′) dω′, (2.25)thusπnsq2s2m∗sω=ˆ ∞0ω2ω2 − ω′2σ1s(ω′) dω′, (2.26)which admits the solution15σ1s(ω) =πnsq2sm∗sδ(ω). (2.27)Given the Ansatz of a delta function form, one could also have derived this same prefactormore easily from the optical sum rule (Equation 2.10).Gathered in one place, and setting q2s = q2n = e2 and m∗s = m∗n = m∗, the final superfluidconductivity is thenσ˜s(ω) =nse2m∗(πδ(ω)− iω), (2.28)15The normalization here relies on the convention´∞0 δ(ω) dω =12 ; another commonly observed convention(which must tacitly assume lower integration limits of 0−, i.e.´∞0−) carries an extra factor of 2 in thedenominator.242.1. Electrical conductivityand the total conductivity σ˜ = σ˜s + σ˜n isσ˜(ω) =[πnse2m∗ δ(ω) + σ1n(ω)]︸ ︷︷ ︸σ˜1(ω)−i[nse2m∗ω + σ2n(ω)]︸ ︷︷ ︸σ˜2(ω). (2.29)Rewritten in terms of the London penetration depth λL ≡√m∗µ0nse2 , this becomesσ˜(ω) =[ πµ0λ2Lδ(ω) + σ1n(ω)]− i[ 1µ0ωλ2L+ σ2n(ω)]. (2.30)None of the conductivity expressions in this section depend on how we choose to countcarriers, with respect to considering Cooper pairs of charges as units versus individual(albeit paired) charges; the combined “pairing” substitutions (n → n2 ,m → 2m, q → 2q)leave the conductivity invariant. For our purposes, we will consider ns to be the density ofsuperconducting electrons, and not pairs, with the corresponding choices for ms and qs; thiswill be more convenient when dealing with sum rules, where the sum of “normal electrons”and “superconducting electrons” remains invariant.If we apply the optical sum rule (Equation 2.10) to the total two-fluid conductivityabove (Equation 2.29) – and noting that the same sum rule will apply for σ1n, but involvingonly the normal fluid – we have162πˆ ∞0σ1(ω) dω =2πˆ ∞0σ1n(ω) dω +2πˆ ∞0σ1s(ω) dω,∴ ne2m∗ =nse2m∗ +nne2m∗ (2.31)This expression, also known as the Ferrell-Tinkham-Glover sum rule [76, 77], shows thatthe superfluid and normal fluid spectral weights (As and An, respectively) must sum to thetotal spectral weight of carriers A, a quantity independent of temperature; thus any changesin the superfluid spectral weight must show up in that of the normal fluid. Expressing Asin terms of λL and An in terms of conductivity, we see that spectral weight changes in thesystem due to changes in temperature must satisfy2π∆[ˆ ∞0σ1n(ω) dω]= −∆( 1µ0λ2L); (2.32)16In the case where multiple bands are contributing to the normal and superconducting components,appropriate sums over bands should be taken.252.1. Electrical conductivitytaking the shift from T = 0 to finite T then gives2πˆ ∞0[σ1n (ω, T )− σ1n (ω, T = 0)] dω =1µ0λ2L(T = 0)− 1µ0λ2L(T ). (2.33)Finally, one assumption which can be tested is whether or not the total spectral weightenters the condensate and disappears from the normal fluid by T = 0; this would requirelimT→0 σ1n(ω, T ) = 0, or in other words:2πˆ ∞0σ1n (ω, T ) dω ?=1µ0λ2L(T = 0)− 1µ0λ2L(T ). (2.34)We will examine this question later, in Section 5.3.6.2.1.5 Quasiclassical formalism for penetration depthWhile the simple free electron picture presented above will be sufficient for describing muchof the phenomenology described in this thesis, we have provided no connection betweenthese phenomenological quantities and the physical properties of the material. One partic-ularly useful formalism capable of providing such insights is the so-called “quasi-classical”approach, most famously described by Chandrasekhar et al. [55]; here we will follow thelead of Prozorov and Giannetta [56] and Prozorov [78].In the London limit, and when one neglects impurity scattering, the superfluid densityns,i relevant for the i direction is given byns,i =kF4π3˛FSdSk ×[(vF ⊗ vFv2F)ii(1 + 2ˆ ∞∆kdE ∂f∂EN(E)N(0))], (2.35)where kF is the magnitude of the Fermi wavevector, the integration is over the Fermi surface,vF is the Fermi velocity, ⊗ represents the dyadic tensor product, ∆k is the gap function atwavevector k, E ≡√(ǫ− µ)2 +∆2 is the quasiparticle energy, ǫ is the normal metal banddispersion, µ is the chemical potential, N(E) is the density of quasiparticle states at energyE, and f is the Fermi-Dirac distribution function.From this expression, it is clear that the full details of the band dispersion, gap function,and Fermi surface topology will influence the superfluid density (and thus the penetrationdepth) – and we still have yet to include the effects of scattering and Fermi liquid interac-tions. While we shall not delve further here into the specifics, it is clear that a complete262.2. Surface impedancetreatment of the superfluid properties must take all of this into account.2.2 Surface impedance2.2.1 General expressionsIn the limit where the response currents are confined to a skin depth δ much smaller thanother relevant sample dimensions, they are most appropriately described in terms of aneffective surface current Keff [74]. One may then define a complex surface impedance Z˜s(ω)by the relation between the electric and magnetic fields at the surface of the conductor,given the surface normal nˆ pointing out from the bulk:E˜c(ω) = Z˜s(ω)(H˜c(ω)× nˆ). (2.36)In terms of the conductivity, Z˜s can be shown to be expressed as [74]:Z˜s(ω) =√iµ˜(ω)ωσ˜(ω) . (2.37)The quantities Z˜s, µ˜, and σ˜ will each be complex functions of frequency in general. For thematerials and frequencies to be discussed in this thesis, µ˜(ω) ≈ µ˜(0) ≈ µ0 to an excellentapproximation,17 one which will be used for what follows. Expanded in terms of their realand imaginary parts, this may be written – as per our phase convention – asZ˜s(ω) ≡ Rs(ω) + iXs(ω) =√iµ0ωσ1(ω)− iσ2(ω), (2.38)where Rs, Xs, σ1, and σ2 are real-valued. The quantities Rs and Xs are known as thesurface resistance and surface reactance of the material.Omitting the explicit ω dependence of these quantities for the moment, we may rewritethis asRs + iXs =√µ0ω (−σ2 + iσ1)σ21 + σ22. (2.39)17At the lowest temperatures (T . 5 K), paramagnetic impurities can contribute to give a stronglytemperature-dependent susceptibility; this could contribute to the curvature sometimes observed in the lowtemperature penetration depth, something typically ascribed to impurity scattering [79] but which mayinstead be due (at least in part) to paramagnetic impurities.272.2. Surface impedanceFurther expansion after squaring both sides yields the two coupled equationsR2s −X2s = −µ0ωσ2|σ˜|22RsXs =µ0ωσ1|σ˜|2, (2.40)where |σ˜| ≡√σ21 + σ22. Substituting the second line into the first to eliminate Xs, solvingthe resulting quadratic equation for R2s , and finally returning to the first line to obtain X2s ,we obtain:Rs =√µ0ω2|σ˜|√1− σ2|σ˜| (2.41a)Xs =√µ0ω2|σ˜|√1 + σ2|σ˜| . (2.41b)As will be the case for the bolometry measurements described in Chapter 5, one oftenknows σ2(ω) (in our case, approximately) over the relevant frequency range, and wishes toextract σ1(ω) from measurements of Rs(ω). Defining σr ≡ µ0ω2R2s (such that Rs =√µ0ω/2σr),Equation 2.41a may be rewritten in in a form quadratic in |σ˜|, from which a solution forσ1 =√|σ˜|2 − σ22 may be extracted:σ1 =(σr2 ±[(σr2)2− σrσ2] 12)2− σ2212, (2.42)where the signs + (−) give the appropriate solutions for the cases σ1 > (<)√3σ2, respec-tively. While this rather opaque expression provides us with the general solution for σ1from known values of Rs and σ2, it provides no immediate insight into how the measuredquantities Rs and Xs are related to the material properties. Fortunately, for the case ofa superconductor at temperatures T ≪ Tc, one can make use of approximations (to bedescribed in the following section) which are far more transparent.Finally, we note that one may calculate the time-averaged power absorbed by the sample,using the complex Poynting vector S˜ ≡ E˜× H˜∗; for an outward-pointing surface normal nˆ,the time-averaged areal power density entering (and thus dissipated as heat in) a sampleis given by 12ℜ{S˜ · nˆ}[74]. Given the field configuration being considered – with uniform,mutually perpendicular H and E parallel to the surface S of the sample – we may calculate282.2. Surface impedancethe power P absorbed by the sample to beP ="S12ℜ{S˜ · nˆ}da = 12"Sℜ{− (Rs + iXs)[(H˜× nˆ)× H˜∗]· nˆ}da= 12"SRs|H˜|2da =12RsH2AS , (2.43)where for the last assignment we assumed uniform H and Rs on the sample surface with areaAS ; sample faces with different Rs values may be treated with the appropriate weighted sum.This expression will be used later for interpreting our broadband bolometric spectroscopymeasurements, wherein Rs(ω) will be extracted from measurements of the absorbed powerP (ω).Note that the area AS used here represents the full sample area on which currents arebeing induced (e.g., two ab faces and two bc faces, for H ‖ bˆ) – thus approximately twicethe area of one flat side for a thin platelet. However, unless otherwise stated, sample areasquoted throughout this thesis will ordinarily refer only to the area of one of the large plateletfaces, as is conventional.2.2.2 Surface impedance of a superconductorFor a superconductor well below Tc (see Equation 2.29), |σ˜s| ≫ |σ˜n|, and for ω > 0 we seethat |σ˜(ω)| ≈ σ2s(ω) =(µ0ωλ2L)−1. Using this in Equation 2.41b, we obtainXs(ω) ≃ µ0ωλL. (2.44)Now, inserting this into the second line of Equation 2.40, we obtainRs(ω) ≃12µ20ω2λ3Lσ1n(ω). (2.45)These approximations are very accurate for T ≪ Tc, and make the rough behaviour of thesurface impedance components readily apparent: Xs is primarily dependent on the super-fluid screening λL, while Rs is proportional to σ1n – but with a large λ3L dependence on thesuperfluid screening. While measurements of Xs alone should provide clean approximationsof the behaviour of λL (and thus nsm∗ ), measurements of Rs also require accurate knowledgeof λL to extract σ1n.292.3. Microwave cavity perturbation2.3 Microwave cavity perturbation2.3.1 Introduction and phenomenologyThe basic premise of cavity perturbation measurement is a simple one: a small sample of amaterial to be studied is inserted into a resonant electromagnetic cavity, and the resultantchanges in the cavity resonance modes – both from the initial insertion into the cavity, andfrom subsequent changes in temperature, magnetic field, etc. – yield information on thepermittivity, permeability, and conductivity of the sample.Although the term “cavity” will be used loosely and liberally throughout this thesis, thetechnique is by no means limited to traditional enclosed metallic volumes, and is applicableto a far wider variety of systems for which there is a well-defined resonance involving only a(quasi-)local electromagnetic field configuration. In particular, all of the cavity perturbationmeasurements to be shown in this thesis were performed in a loop-gap resonator (to bedescribed in Section 3.1.1) – not particularly cavity-like, and far more reminiscent of anLCR circuit whose (complex) inductance is perturbed by the sample in the bore of theloop; yet the same considerations to be discussed (including the derivation in Section 2.3.2)will apply.Most real resonant systems have multiple (and often an infinite spectrum of) resonantmodes, and the framework to be discussed can be adapted – with some additional compli-cation – to this general case. However, in what follows, we will only discuss the case of asingle, well-defined resonant mode that is well-separated in frequency from (and, at most,very weakly coupled to) other resonant modes. In the cavity perturbation measurementsto be presented, only the single lowest-frequency resonant mode was probed, and highermodes are well separated and quite justifiably neglected.For a two-port resonant microwave cavity, and in the limit of weak coupling, the complextransmitted microwave voltage V˜out as a function of frequency ω takes the form of a complexLorentzian [80]:V˜out(ω) =α˜V˜in1 + iQ(ωω0 −ω0ω) , (2.46)where V˜in is the complex input microwave voltage, ω0 is the resonant frequency, Q ≡ω0ωB is the quality factor associated with the full width at half-maximum power (FWHM)bandwidth ωB, and α˜ is a complex scale factor depending on coupling (amongst other302.3. Microwave cavity perturbation0-B/2   0   0+B/20P0/2P0a)Transmitted power PDrive frequency BB = 0/Qd     -2Q0-B/2   0   0+B/2- /2- /40/4/2Drive frequency d       0=Phase shift b)Figure 2.2: The transmitted power (a) and voltage phase shift (b) through a microwavecavity resonator.factors). We will absorb all losses and phase shifts external to the cavity into α, assumedhere to be (approximately) frequency independent near the resonance. Direct couplingbetween input and output – which would add a complex background offset to V˜out, presumedto vary weakly with frequency over the narrow resonance window – is not accounted forhere; through transmission measurements away from resonance, such coupling was seen tobe negligible for the experiments we will be discussing, and for simplicity will therefore beomitted from our treatment here.In practice, for a high-Q resonator one is almost always working sufficiently close to thecavity resonance (i.e., |ω − ω0| ≪ ω0) to make the approximationωω0− ω0ω =(1 + ω0ω)( ωω0− 1)≈ 2( ωω0− 1), (2.47)which, when applied to Equation 2.46, yieldsV˜out(ω) =αV˜in1 + 2iQ(ω−ω0ω0) = α˜V˜in1 + 2i(ω−ω0ωB) . (2.48)The phase shift of the microwave voltage across such a system would be∆φ(ω) ≡ arg V˜out(ω)V˜in(ω)= φext + tan−1[2Q(1− ωω0)], (2.49)where φext ≡ arg α˜ is the phase shift external to the cavity, and thus for small frequency312.3. Microwave cavity perturbationshifts δω near ω = ω0, one finds a corresponding phase shift (see Figure 2.2b)δ∆φ ≈ −2Qω0δω. (2.50)To work in terms of real microwave power, we take the complex squared norm V˜ V˜ ∗ ofvoltages, yieldingPout(ω) =P01 + 4Q2(ω−ω0ω0)2 =P01 +(ω−ω0ωB/2)2 , (2.51)where α˜, V˜in, and any relevant impedances have been combined into P0; see Figure 2.2a.Here we may confirm by inspection that the transmitted power is indeed a Lorenztiancentred at ω0, with FWHM bandwidth ωB and with maximum value P0.Notes on complex frequency and the time domainIt is useful at this point to describe the relationship of the Lorentzian voltage response above(Equation 2.48) to equivalent expressions in terms of complex frequency, and to relate theseto the time-domain behaviour of the fields and field energy (equivalently, power) of a freedissipative resonator. Most expressions herein are quite basic and are most likely alreadyfamiliar to the reader, but these definitions and relations are collected here as a convenientreference and reminder.We will work in terms of complex frequencies f˜ ≡ f ′ + if ′′, or equivalently, 2πf˜ = ω˜ ≡ω′ + iω′′. This can be seen as an analytic continuation of ω off of the real line and into thecomplex plane, and arises naturally in the spectra of systems with dissipation.(Note that all frequencies considered to be complex in this thesis will be denoted bytildes (f˜ , ω˜). All quantities not written as such are to be taken as either as purely real,or as the real part of the complex quantity represented by the same symbol. We also notethat expressions in terms of both f and ω will be used throughout this thesis, with themost appropriate or convenient choice used for the expressions being considered, typicallyto minimize the appearance of 2π. In all cases one may substitute ω ↔ 2πf , transferring allappropriate subscripts and adornments. In addition, we shall often omit the word “angular”when describing the angular frequency ω, in such cases where no ambiguity in meaning mayarise.)Here, the real part (ω′) has the usual meaning of an oscillatory frequency (ω0 above).322.3. Microwave cavity perturbationThe imaginary part (ω′′) of the frequency, representing the rate of decay of the oscillatorysolution with time, is equivalent to half of the bandwidth ωB defined above. Additionallyusing Q ≡ ω0ωB =ω′2ω′′ , we find the following equivalent relations for ω˜:ω˜ ≡ ω′ + iω′′ = ω′ + iωB2 = ω′(1 + i 12Q)(2.52)We may rewrite Equation 2.48 in terms of the real and imaginary parts ω′0 and ω′′0 ofthe complex resonant frequency ω˜0 (here setting α˜V˜in → V˜0) to obtainV˜ (ω) = −iω′′0 V˜0ω − (ω′0 + iω′′0)= −iω′′0 V˜0ω − ω˜0(2.53)This expression can be verified to be (within prefactors) the Fourier transform ofV˜ (t) = V˜0e+iω˜0t Θ(t), (2.54)a simple complex exponential time dependence, which is “turned on” for t ≥ 0 by a Heav-iside step function Θ(t). This models the behaviour of a resonator starting at t = 0 withamplitude V˜0 in its resonant mode ω0, after which it is left to oscillate freely. Consideringonly t ≥ 0 for now (dropping Θ(t)), we rewrite Equation 2.54 asV˜ (t) = V˜0e+iω′0te−ω′′0 t = V˜0e+iω′0te−ω′0t2Q . (2.55)Here we clearly see that our solution is oscillatory at frequency ω′0, with an exponentiallydecaying envelope of e−t/τV , where the exponential decay time constant of the voltage at acoupling loop (or equivalently, the cavity fields) isτV =1ω′′0= 2Qω′0= Qπf ′0. (2.56)From this we may interpret Qπ as the number of oscillation periods required for the voltage(and fields) to decay to a fraction e−1 (≈ 36.8%).We can also rewrite this in terms of the real output power (∝ V˜ V˜ ∗):P (t) = P0e−2ω′′0 t = P0e−ω′0tQ . (2.57)332.3. Microwave cavity perturbationThe output power is therefore a simple exponential decay e−t/τP , with time constantτP =τV2 =12ω′′0= Qω′0= Q2πf ′0; (2.58)thus we can also identify Q2π with the number of oscillation periods required for the power(or stored energy) to decay to a fraction e−1.Finally, if we consider the stored energy U(t) = U0e−t/τP , noting that the total powerPtot leaving or dissipated in the resonator will be Ptot(t) = −dUdt =ω′0Q U(t), we find additionaluseful interpretations of Q:Q = ω′0U(t)P (t) = 2πf′0Energy storedPower dissipated (2.59a)= 2π U(t)T0P (t)= 2π Energy storedEnergy lost per cycle , (2.59b)where T0 ≡ f ′0 is the oscillation period. While derived for an undriven resonator in freedecay, the same considerations apply for a continuously driven resonator at equilibrium;that is, the external driving power required to maintain a given stored energy must matchthe power dissipation specified by Equations 2.59a-b.2.3.2 Derivation of the cavity perturbation equationWe consider a resonant electromagnetic system, into which a small perturbation is intro-duced. We write subscripts i = 1, 2 to indicate two different configurations of the cavityand its contents, along with the resultant fields. As a reminder, the notational conventionused (as introduced in Section 2.1.3, and used throughout) is that the physical, real-valuedexpression for a field Ei(r, t) (where the index i denotes the particular cavity configuration)is expressed as the real part Ei(r, t) = ℜ{E˜i(r, t)}= ℜ{E˜i(r)e+iω˜it}(and similarly for B,H, D, and J). In the text below, explicit r, ω˜i, and t dependences are omitted for simplicity;E˜ and H˜ are to be read as E˜(r, t) and H˜(r, t) (and not merely as their time-independentprefactors).Before proceeding, we work out expressions for the curls of E˜ and H˜, using Maxwell’sequations; we shall assume linear media, described by response tensors defined as B˜i = µ˜iH˜i,342.3. Microwave cavity perturbationD˜i = ǫ˜iE˜i, and J˜i = σ˜iE˜i. We then find:∇×E˜i = −∂B˜i∂t =− iω˜iµ˜iH˜i (2.60a)∇×H˜i = J˜i +∂D˜i∂t = (σ˜i + iǫ˜iω˜i) E˜i (2.60b)Motivated by the derivations of Ormeno [81] and Broun [82], we form the productW ≡ H˜1×E˜2 − H˜2×E˜1, a complex quantity which depends on the fields of both cavityconfigurations 1 and 2. We start by taking the divergence of this quantity, using the vectoridentity ∇ · (A×B) = B · (∇×A)−A · (∇×B):∇ ·W = E˜2 ·(∇×H˜1)− H˜1 ·(∇×E˜2)− E˜1 ·(∇×H˜2)+ H˜2 ·(∇×E˜1)= E˜2 · (σ˜1 + iǫ˜1ω˜1) E˜1 − H˜1 · (−iω˜2µ˜2) H˜2− E˜1 · (σ˜2 + iǫ˜2ω˜2) E˜2 + H˜2 · (−iω˜1µ˜1) H˜1. (2.61)With some tedious rearrangement, this can be reduced to∇ ·W = −i{∆ω˜[ǫ˜2(E˜1 · E˜2)− µ˜2(H˜1 · H˜2)]+ω˜1[∆ǫ˜(E˜1 · E˜2)−∆µ˜(H˜1 · H˜2)]− i∆σ˜(E˜1 · E˜2)}, (2.62)where we have used the notation ∆Xi ≡ X2 − X1. Here we have also assumed that theresponse tensors are symmetric, and that the fields are aligned along common principalaxes of the tensor; additionally, we drop the tensor notation, and now assume that thesequantities are to be read as the appropriate component for the local field directions.18 Wenow integrate this quantity over an arbitrary volume V , followed by further rearrangementto separate out ∆ω˜, yielding∆ω˜ = 14U12{−i˚V∇·W dV + i˚V∆σ˜(E˜1 ·E˜2)dV+ ω˜1˚V[∆µ˜(H˜1 ·H˜2)−∆ǫ˜(E˜1 ·E˜2)]dV}, (2.63)18When either of these assumptions is not true, then one can instead proceed by making the substitution∆(σ˜)(E˜1 · E˜2) → E˜1σ˜2E˜2 − E˜2σ˜1E˜1 (and similarly for µ˜ and ǫ˜) in what follows.352.3. Microwave cavity perturbationFigure 2.3: A schematic of the microwave cavity geometry, including sign conventions fornormal vectors, shown here for the case of two samples S1 and S2.where we have definedU12 ≡14˚V[µ˜2(H˜1 ·H˜2)− ǫ˜2(E˜1 ·E˜2)]dV. (2.64)We now consider a cavity volume C which encloses (and includes) an arbitrary numberof sample volumes Si (see Figure 2.3). For all surfaces we define the normal vector nˆ to beoutward from the surface; for positively-oriented surface integration, however, we see thatda = +nˆ da for each sample surface ∂Si, but da = −nˆ da for the enclosing cavity surface∂C. Employing the divergence theorem, we integrate over the volume V ≡ C −∑i Si, thespace inside the cavity but outside each sample:˚V∇·W dV ="∂CW · da−∑i"∂SiW · da="∂CW · (−nˆ da)−∑i"∂SiW · (+nˆ da)= −"∂VW · nˆ da, (2.65)where we have defined ∂V ≡ ∂C+∑i ∂Si to be the sum of all surfaces bordering the interiorspace V of the cavity.To proceed further, we assume that each of the surfaces in the system is a suffi-362.3. Microwave cavity perturbationciently good conductor, and thus may be treated in the surface impedance approximationE˜t = Z˜snˆ×H˜t (Equation 2.36). This will certainly be the case for the metallic and super-conducting cavities and samples under consideration; in the “thin limit” where skin depthsbecome comparable to sample dimensions, effective surface impedances may still be usedhere. For this “good conductor” limit, we may take the fields near the surface of the sampleto be transverse to the sample; that is, E = Et and H = Ht. It then follows from somesimple vector algebra manipulation thatW =(Z˜s1 − Z˜s1)(H˜1t ·H˜2t)= ∆Z˜s(H˜1t ·H˜2t), (2.66)and consequently˚V∇·W dV = −"∂V∆Z˜s(H˜1t ·H˜2t)da. (2.67)We may now cancel the factor of ei(ω˜1+ω˜2)t which appears in every term in the numeratorand denominator; i.e., E˜ and H˜ are now to be read as only the time-independent prefactorsE˜(r) and E˜(r), respectively. We assume that the cavity Q is sufficiently large (i.e., ω′ ≫ ω′′)that the phase of E˜ differs from that of H˜ by ±π2 everywhere in the cavity, and furthermorefix the phase of H˜ to be zero (without loss of generality). Finally we note that over almostthe entire cavity the fields remain nearly the same in this perturbative limit, allowing us torewrite U12 (Equation 2.67), to an excellent approximation, to beU12 ≈14˚V(µ˜2 |H2|2 + ǫ˜2 |E2|2)dV ≡ U, (2.68)where the change in relative sign between the terms comes from the aforementioned e±iπ2phase factor on E˜, and U is the total electromagnetic energy stored in V . We are finallyleft with∆ω˜ = 14U{i"∂V∆Z˜s(H˜1t ·H˜2t)da+ i˚V∆σ˜(E˜1 ·E˜2)dV+ ω˜1˚V[∆µ˜(H˜1 ·H˜2)−∆ǫ˜(E˜1 ·E˜2)]dV}. (2.69)One important point to notice is that perturbative frequency shifts due to separate pertur-bations (e.g. changing Z˜s on multiple surfaces) are additive; in particular, shifts ∆ω′′ inthe imaginary frequency of the cavity – and thus in 1/Q – from separate loss mechanisms372.3. Microwave cavity perturbationare additive. Starting from a lossless cavity (Q = ∞), we add the loss contributions of theresonator surface(s) (1/Q0) and those from samples (1/QSi) to get the total Q:1Q =1Q0+∑i1QSi. (2.70)We shall restrict the remainder of this discussion to the case of a single sample S withsurface ∂S placed in the resonator.Since the cavity volume V outside the sample is only vacuum, µ˜ = µ0, ǫ˜ = ǫ0, and σ˜ = 0throughout V , allowing these terms to be removed from consideration.19 If we consider onlythe situation where a perfectly-conducting sample has already been inserted into the cavityfrom which further frequency shifts are measured (a good approximation to the situationdescribed, for which a superconducting sample is held at a fixed position in the resonator),then the remaining volume integral drops out as well. For the final simplification, we notethat only the sample temperature is changed during measurements, while ∆Z˜s ≈ 0 for thecavity surface ∂C, allowing us to restrict our surface integration to ∂S.Assuming that both ∆Z˜s and H˜1t ·H˜2t are constant over the sample surface20 – validfor a sufficiently small sample in the “thin” field geometry – and using our definitionsZ˜s = Rs + iXs and ω˜ = 2π(f0 + ifB2), we have∆ω˜ = 2π(∆f0 + i∆fB2)= (i∆Rs −∆Xs)2As〈H˜1t ·H˜2t〉∂S4U , (2.71)where As here is the full surface area of the sample on which screening currents are carried,and not just the area of the largest face on one side. (Here we have switched from workingwith ω to f , the more convenient parameter for direct comparison to experiment.) Wedefine the resonator constant Γ to beΓ ≡〈H˜1t ·H˜2t〉∂S4πU ; (2.72)19These terms would become relevant if we had instead (and equivalently) defined V to include the sample,and not terminate at its surface (in which case its contribution to the ∆Z˜s surface integral would vanishinstead).20Where contributions from different crystal faces with different Z˜s are considered, appropriate averagesmust be taken; this will typically involve simply substituting ∑iAi∆Z˜s,i in Equation 2.71 and in whatfollows.382.3. Microwave cavity perturbationFor small samples placed in the same position in the same resonator, this will remain verynearly unchanged from run to run. In terms of this constant Γ, the real and imaginaryparts of Equation 2.71, expressed separately, become∆f0 = −ΓAs∆Xs (2.73a)∆fB = 2ΓAs∆Rs. (2.73b)We note that ∆fB = ∆(f0Q)≈ f0∆(1Q)for large Q and small shifts in f0; from this,and rewriting Equations 2.73a,b in terms of Rs and Xs, we obtain∆Xs = −1ΓAs∆f0 (2.74a)∆Rs =f02ΓAs∆( 1Q). (2.74b)Rewritten together in terms of the relative shift in complex frequency ∆f˜0/f0, we get∆f˜0f0= ∆f0f0+ i2∆( 1Q)= iΓAsf0(∆Rs + i∆Xs) (2.75a)= iΓAsf0∆Z˜s (2.75b)(Note that other common definitions of Γ seen in the published literature may alreadyinclude As and/or f0.)Finally, except very near Tc, σ2 ≫ σ1, and thus Xs ≃ µ0ωλ(T );21 application to Equa-tion 2.74a yields∆λ = − 12πµ0ΓAs∆f0f0. (2.76)2.3.3 Thin platelet solutionBefore discussing the solution for shifts in the cavity for a thin platelet, it is importantto define our measurement geometry, along with the notation we will use for the relevantquantities; see Figure 2.4 for an overview. Shown here is the “a-axis orientation,” in whichthe external H-field is applied along the b axis, resulting in screening currents flowing in the21Note that here we switch away from using λL for the penetration depth in favour of λ; we reserve theformer for the ideal London penetration depth λL ≡√m∗µ0nse2, whereas the true penetration depth λ maybe subject to modifications from scattering and nonlocality. In this thesis, λL will be written in place of λwherever the approximation λ ≈ λL is explicitly intended.392.3. Microwave cavity perturbationλaλc caH || bJcabFigure 2.4: A schematic of the sample measurement geometry, showing the relative ori-entations of fields, currents, and penetration depths and their labelling convention. Theorientation shown here (with H‖bˆ) is what we denote the “a-axis orientation,” used formeasuring λa (or σa); to obtain measurements in the b-axis orientation (for λb or σb), onesimply rotates the sample 90◦ in the ab-plane. Figure provided courtesy of Jake Bobowski[71] (with slight modifications).402.3. Microwave cavity perturbationaˆ direction. It is the a axis which will label quantities pertaining to these currents, such asthe conductivity σa, surface impedance Za, and penetration depth λa; one should take careto note that the latter quantity expresses the depth in the cˆ direction that the field in the bˆdirection penetrates into the sample! As shown, there will also be some screening currentsalong the cˆ direction which will contaminate the results; the use of thin samples, combinedwith appropriate corrections, can minimize such effects. To obtain measurements in theb-axis orientation, one simply rotates the sample 90◦ in the ab-plane; for the measurementsdone in this thesis, samples were typically measured in a given probe in the a-axis orientationfirst, followed by a rotation of the sample to measure the corresponding b-axis quantities.For a thin superconducting platelet (c ≪ a, b) in a low-demagnetization geometry withH⊥ cˆ, carrying out the analysis above for δf˜0, the full complex frequency shift upon insertionof the sample,22 yields [83, 84]23δf˜0f0= Vs2Vc[1− tanh (κ˜c/2)κ˜c/2], (2.77)where Vs is the sample volume, Vc is the effective cavity volume, and κ˜ ≡√iωµ0σ is thecomplex propagation constant; κ˜ = (1+ i)/δ for a metal of skin depth δ, and κ˜ ≈ 1/λ for asuperconductor with penetration depth λ well below Tc. Considering the superconductingcase at low temperature, where λ ≪ c/2 (the “thick limit”), the tanh term goes to 1, andEquation 2.77 simplifies toδf0f0≃ Vs2Vc[1− 2λc], (2.78)where we have ignored the small σ1n, and δf0 is now manifestly real in this limit. UsingVs = abc = 12cAs, we obtain the changes in fractional frequency shift due to changes in λ:∆(δf0f0)≃ − As2Vc∆λ; (2.79)comparison with Equation 2.76 allows us to relate the effective cavity volume Vc to theresonator constant Γ used previously, via Γ = (4πµ0Vc)−1.22A remark on notation: we use δ to denote changes from an empty cavity upon sample insertion,as opposed to our use of ∆ to denote further changes upon variation of sample properties (such as withtemperature).23In Ref. [83] this expression is given for real κ only, whereas in Ref. [84] the complex magnetic magneticmoment m˜ of such a platelet sample is shown to have the exact same functional form; by examining Equation2.69, we see that by treating the sample as part of the volume rather than a boundary, then ∆ω˜ ∝ ∆|m˜|,and thus this simple κ → κ˜ generalization to complex κ˜ holds.412.3. Microwave cavity perturbationThis treatment, however, neglects the field penetration of length λc into the bc faces. Forsamples of non-negligible thickness – and particularly for highly anisotropic superconductorssuch as the cuprates, for which λcλa can exceed 100 – this must be accounted for; as we shall seein Section 4.5.2, such corrections are typically small relative to ∆λa, b at low temperatures(¡ 1% ), but can exceed 20% by T ≈ 0.5Tc for the lowest dopings studied. Neglecting theimaginary component of the screening length,24 we find the relative frequency shifts in theH‖bˆ and H‖aˆ directions (respectively) upon sample insertion [82, 85, 86]:25δff∣∣∣∣aˆ= Vs2Vc1− 8π2∞∑oddn≥11n2[ 2γnctanh γnc2 +2κnatanh κna2] (2.80a)δff∣∣∣∣bˆ= Vs2Vc1− 8π2∞∑oddn≥11n2[ 2γ′nctanh γ′nc2 +2κ′nbtanh κ′nb2] , (2.80b)where we defineγn ≡1λa√1 +(nπλca)2(2.81a)γ′n ≡1λb√1 +(nπλcb)2(2.81b)κn ≡1λc√1 +(nπλac)2(2.81c)κ′n ≡1λc√1 +(nπλbc)2. (2.81d)If we consider the case of a partially twinned sample, with twin ratio R ≡ Amaj/Atotbeing the fraction of the sample in the “majority” crystallographic orientation, we canapproximate the change in frequency with the appropriate weighted averages over orien-tations for penetration depths in the above quantities, yielding for the (nominal) aˆ and bˆ24Imaginary screening lengths δ˜ can be accommodated here with the replacements λi → δ˜i; their contri-bution to estimates of ∆λ will be negligible except very near Tc, and without reliable high temperature Rsmeasurements available to account for them, we shall forego this complication here.25Note the notation: the H‖bˆ orientation measures mostly ∆λa, and the H‖aˆ orientation measuresmostly ∆λb. We therefore refer to H‖bˆ as the “aˆ orientation” and vice versa.422.3. Microwave cavity perturbationorientationsδff∣∣∣∣aˆ= Vs2Vc1−8π2∞∑oddn≥11n2[R( 2γnctanh γnc2 +2κnatanh κna2)+(1−R)( 2γ¯nctanh γ¯nc2 +2κ¯natanh κ¯na2)] (2.82a)δff∣∣∣∣bˆ= Vs2Vc1−8π2∞∑oddn≥11n2[R( 2γ′nctanh γ′nc2 +2κ′nbtanh κ′nb2)+(1−R)( 2γ¯′nctanh γ¯′nc2 +2κ¯′nbtanh κ¯′nb2)] , (2.82b)where in addition to Eqs. 2.81a–d we define their “overbarred” counterparts,γ¯n ≡1λb√1 +(nπλca)2(2.83a)γ¯′n ≡1λa√1 +(nπλcb)2(2.83b)κ¯n ≡1λc√1 +(nπλbc)2(2.83c)κ¯′n ≡1λc√1 +(nπλac)2. (2.83d)We note in passing that κ¯n = κ′n and κ¯′n = κn, but we maintain the new symbols for sym-metry and notational consistency. Also note that the sample dimensions a and b must alsobe considered to be subject to R-weighted averages of the appropriate thermal expansionsalong the a and b axes.2.3.4 Demagnetization effectsAlthough all of the measurements to be discussed in this thesis are taken in an H⊥ cˆ “edge-on” orientation with a low demagnetizing factor (such that the field just outside the surfaceof the sample is very close to the applied field), demagnetization effects are not necessarilynegligible and must be accounted for.26Calculation of the demagnetizing fields for samples of arbitrary geometry can be diffi-26For general microwave measurements in finite E one must also consider depolarization effects; however,measurements in this thesis are done exclusively in the E ≈ 0 limit.432.3. Microwave cavity perturbationcult, and in general must be solved numerically with electromagnetic finite-element mod-elling software. This can be particularly problematic with superconducting samples, forwhich most of the magnetic field variation occurs within a penetration depth λ of the sam-ple surface – a length scale which is poorly matched to the typically much larger sampledimensions.Fortunately, for the sole case of an ellipsoid (and appropriate limits thereof, such asflat disks or infinitely long cylinders), this demagnetization field is uniform throughout thesample; even better, its value has an analytic solution, given by Osborn [87]. For an externalfield Hext applied to such an ellipsoid along a principal axis with demagnetization factor Nand magnetization M , the H field seen throughout the sample becomes H = Hext −NM .In the case of a superconductor – where B = 0 is enforced – M is negative, and the effectof demagnetization (giving N > 0) is to increase the effective magnetization – and thussurface current, surface field, and the cavity perturbation frequency shift – for a givenapplied field. (One may visualize this as the superconductor expelling lines of magneticflux, thus increasing the effective density of flux lines at the surface and the effective field tobe screened.) In our case, this results in division of the left-hand sides of Equations 2.82a,bby a factor of (1−N), giving our final expressions:δff∣∣∣∣aˆ= Vs2Vc(1−Na)1−8π2∞∑oddn≥11n2[R( 2γnctanh γnc2 +2κnatanh κna2)+(1−R)( 2γ¯nctanh γ¯nc2 +2κ¯natanh κ¯na2)](2.84a)δff∣∣∣∣bˆ= Vs2Vc(1−Nb)1−8π2∞∑oddn≥11n2[R( 2γ′nctanh γ′nc2 +2κ′nbtanh κ′nb2)+(1−R)( 2γ¯′nctanh γ¯′nc2 +2κ¯′nbtanh κ¯′nb2)] ,(2.84b)where here Na and Nb are the appropriate demagnetization factors along the a and b axesof the sample, respectively.While the thin, nearly-rectangular platelets studied in this thesis are certainly not per-442.3. Microwave cavity perturbationfectly modelled by ellipsoids – particularly near corners, where the concentrated fields canbe much larger than the applied field, a potentially problematic scenario – a rough esti-mate of demagnetization effects may still be obtained by modelling the samples as very flatellipsoids of equivalent dimensions, which we will do here.Consider a sample of thickness c and nominal in-plane dimensions a and b, with a ≥b ≥ c ≥ 0, which we will approximate by an ellipsoid of major27 axes (a, b, c). Labelling thedemagnetization factors for these three axes as Na, Nb, and Nc, respectively, Osborn finds(expressed here in SI) [87]:Na =cosφ cosϑsin3 ϑ sin2 α[F (ϑ|k2)− E(ϑ|k2)](2.85a)Nb =cosφ cosϑsin3 ϑ sin2 α cos2 α[E(ϑ|k2)− F (ϑ|k2) cos2 α− sin2 α sinϑ cosϑcosφ](2.85b)Nc =cosφ cosϑsin3 ϑ cos2 α[sinϑ cosφcosϑ − E(ϑ|k2)](2.85c)where we define the angles ϑ, φ, and α, as well as the elliptic modulus k, bycosϑ ≡ c/a, (2.86a)cosφ ≡ b/a, (2.86b)k ≡ sinα ≡ sinφsinϑ =√1− (b/a)21− (c/a)2 . (2.86c)Here F (ϑ|k2) and E(ϑ|k2) are the incomplete elliptic integrals of the first and second kinds,respectively,28 defined byF (ϑ|k2) ≡ˆ ϑ0dφ√1− k2 sin2 φ, (2.87a)E(ϑ|k2) ≡ˆ ϑ0√1− k2 sin2 φ dφ. (2.87b)For the case where a = b, Na and Nb can be reduced to a simpler expression, given byNa = Nb =12(m2 − 1)[m2√m2 − 1sin−1(√m2 − 1m)− 1], (2.88)27Osborn [87] used semi-major axes, but a rescaling of all dimensions here by a factor of two is of noconsequence.28We also differ from Osborn here in our choice of elliptic integral parameter convention; our choice ismore common for computer algebra systems such as Mathematica.452.3. Microwave cavity perturbationwhere m ≡ a/c; in fact, this limiting expression must be used for a = b, where the denomi-nators of Equations 2.85a and 2.85b vanish.46Chapter 3Microwave spectroscopy tools andtechniques3.1 The cavity perturbation probe3.1.1 The 940 MHz loop-gap resonatorMany of the cavity perturbation experiments done in the UBC Superconductivity lab havebeen performed with conventional right cylindrical cavities made of copper electroplatedwith a Pb:Sn alloy. Cooled to liquid helium temperatures (4.2 K at 1 atm, ∼1.1 K whenpumped down to typical vapour pressures of ∼0.3 Torr), below the 7 K superconducting Tcof the Pb:Sn coating, such cavities can have enormously high Q factors on the order of 108.These cavities are typically operated in their TE011 modes, and the sample is placed inthe very centre of the cavity, which is the location of an electric field node and a magneticfield antinode, where the magnetic field gradients are small. The sample is positioned in thecavity on a high-purity sapphire plate or rod, running along the resonator axis (where E = 0)for minimal disturbance. In most systems, means are provided for moving the sample intoand out of the resonator during a run to enable measurement of the unloaded cavity qualityfactor Q0, from which sample loss contributions are measured via 1/Qs = 1/Qloaded−1/Q0.Coupling of the microwave power into and out of the cavity is accomplished through twosmall coupling holes (with D ≪ λ/10) on either side of the cavity, next to each of whichis placed a current loop at the end of a semi-rigid coaxial cable (the coupling loops). Thecoupling strengths may be adjusted in situ with micrometer-driven translation stages atthe top of the probe controlling the loop heights.The minimum dimensions required for such a geometry are on the order of half anelectromagnetic wavelength; while this is feasible above 10 GHz, it becomes impractical(and, most importantly, larger than the inner dimensions of our liquid helium experiment473.1. The cavity perturbation probedewars) in the low GHz regime and below. Nevertheless, for measurements of the magneticpenetration depth, one wishes to work at the lowest frequencies possible to minimize thecontribution of σ2n (which varies as ∼ ω) to Xs. This necessitates the use of a differentgeometry: the loop-gap resonator.A loop-gap resonator [88–91] (often called a split-ring resonator) – is constructed froma conducting open-bore solenoidal structure severed by one or more length-wise slits, intowhich dielectric material may be inserted; the entire structure sits in a low-loss metallicenclosure to prevent radiation losses. Instead of standing-wave modes of an enclosed cavity,the loop-gap resonator acts as a LC-circuit resonator, with the loop providing the inductanceand the gaps providing the capacitance. Thus, rather than being determined by a cavitydimension ℓ setting free-space standing wave modes at f ∼ c/2ℓ (as with conventional cavityresonators), the resonant frequency of the loop-gap is set by the geometry of the loop andthe width and dielectric constant associated with the gap. This frequency may be manytimes lower than that of a conventional resonant cavity of equivalent size. An additionaladvantage is that the field configuration of the resonant mode involves solenoidal magneticfields in and around the loop (which may be quite uniform at its centre), with electric fieldsconfined to the gap region, with minimal fringing fields – thus a sample placed in the centreof the bore of the loop satisfies the assumptions made in Section 2.3.2. As mentioned inSection 2.3.2, the cavity perturbation expressions derived therein are just as valid for theloop-gap resonator as for more conventional cavity geometries.The cavity perturbation measurements performed in this thesis were focused on penetra-tion depth measurements, since Rs measurements were to be performed with a bolometricprobe better suited to the task (see Section 3.5). For these purposes, we used a single-gaploop-gap resonator with a resonant frequency of ∼940 MHz. The cavity (shown in Figures3.1 and 3.2) is constructed of a copper half-loop bolted tightly on one side (with blind screwholes from underneath) to a copper support block, with a sapphire plate held firmly in thegap between the other side of the loop and the block.All metal parts of the resonator and its surroundings (including the outer enclosuresurrounding the loop-gap) are electroplated with Pb0.95Sn0.05 before assembly to eliminateresistive losses (when at liquid helium temperatures) from exposed normal metal, enablingvery high quality factors after initial plating ofQ > 106. Over a timescale of months to years,this quality factor has been found to slowly decrease as the plating and superconducting483.1. The cavity perturbation probe1GapLoopFigure 3.1: A photograph of the loop of the loop-gap resonator, with a side view schematicof the assembled resonator. Dimensions displayed are in millimetres; the length of the loop(into the page; not shown) is 11.43 mm. Figure provided by Jake Bobowski [71] based onoriginal material by Saeid Kamal [86].493.1. The cavity perturbation probeFigure 3.2: Schematic view of the loop-gap assembled and in operation; the resonatorhousing is not shown for clarity. Figure provided by Jake Bobowski [71].joints degrade with exposure, and at the time of writing the current Q factor at 1.2 K isapproximately 3× 105 – quite adequate for the penetration depth measurements on whichwe will focus, but somewhat too low for accurate Rs measurements of superconductors.As with the cylindrical cavities, coupling to the loop-gap resonator is accomplishedthrough two adjustable-height coupling loops on the outside of the resonator enclosure, inthe vicinity of small coupling holes. In addition to vertical positioning, the angular positionof the loops relative to their coupling holes may also be varied to optimize coupling.The resonance frequency has been found to vary over a range of ∼937–945 MHz betweendifferent cooldowns, a variation believed to be due to different thermal contraction historiesfor each cooldown and different pressures on the sapphire plate in the gap. Such variationwill affect the capacitive factor in the resonant frequency, but should negligibly impact theinductance and therefore the proportionality between fractional frequency shifts δf˜f andshifts in the sample surface impedance ∆Z˜s, as these are determined by the essentiallyimmutable H-field geometry of the loop.503.1. The cavity perturbation probe3.1.2 The sample stage and environmentAs discussed above, the sample is held in the loop-gap resonator at the end of a sapphirehot-finger, in this case a thin sapphire plate (∼0.13 mm thick, ∼1 mm wide, ∼25 mm long),extending into the loop parallel to its axis. In contrast to the cylindrical cavities, wheresamples are typically positioned in the very centre of the cylinder along the axial nodal lineof the electric field, here the sample is centred along the length of the loop, but offset inthe transverse direction away from the gap, to minimize the exposure of the sample to thefringing electric field of the gap. The sample is held on the plate with a very thin layer ofDow Corning High Vacuum Grease (a clear silicone-based grease), which provides adhesionand good sample–plate thermal contact while presenting additional microwave loss that hasbeen found to be measurably negligible.The sapphire sample plate extends outside of the loop and then outside of the resonatorenclosure through yet another small hole, beyond which it is attached to a sapphire blockwith a thin layer of vacuum grease followed by encapsulation in GE-7031 Varnish, an in-sulating modified phenolic resin produced by General Electric that has good mechanicalstrength and thermal conductivity at cryogenic temperatures.29 On the reverse side of thesapphire block a CernoxTM zirconium oxynitride resistance thermometer from Lake ShoreCryotronics [92] and a surface-mount heater resistor are attached (with GE Varnish), usedfor temperature measurement and control of the sapphire block and thereby the attachedsample plate. Due to the high thermal conductivity of sapphire, the combined plate andblock system may be maintained at uniform temperature, with minimal thermal gradients.The block itself is affixed (with Stycast 1266 clear epoxy) to a thin-walled quartz tube,itself anchored to a platform directly attached30 to the bottom of the sample pot, held atthe temperature of the outside liquid helium bath through immersion. The quartz tube –a poor thermal conductor at low temperature – acts as a large thermal resistance to thehelium bath, enabling the sapphire block and plate to be temperature regulated well abovethe helium bath temperature without the use substantial heating power, while still allowingthe sample to cool to (nearly) the bath temperature in a reasonable time (on the orderof several minutes from 100 K to 1.2 K). The leads from the heater and thermometer are29This unusual configuration allows the plate to be easily removed for cleaning – not easily accomplishedif varnish penetrates the gap under the plate – while still fixing the plate in a manner less prone to accidentaladjustment than grease alone.30Except for the 3He pot version, to be described shortly.513.1. The cavity perturbation probevarnished down to the tube to provide good thermal heat sinking, ensuring that the leadsare near the temperature of the block upon reaching the thermometry. The heater andthermometer leads exit the pot via epoxy feedthroughs, and are routed up the probe viaflexible wiring (directly immersed in the bath) to a 12-pin circular connector at the topof the probe; from here, shielded cables provide the connection to a Conductus LTC-20temperature controller. Because the LTC-20 is designed for sourcing much larger heatingpowers than desired (up to 50 W), a ÷275 resistive voltage divider is placed between theheater and the LTC-20 to rescale its output range to more appropriate levels.The sample pot, containing the sample stage and its accoutrements, is bolted to thebottom face of the cavity enclosure and sealed with an indium O-ring compressed by saidbolts. Such seals are immersed in superfluid helium during an experiment, and thus mustbe made rigorously leak-tight. The entire inside of the sample pot, cavity enclosure, andthe thin-walled stainless steel tubes extending to the top flange of the probe (conduits forthe coaxial coupling loops and the central pumping line) comprise a single enclosed vacuumspace; this is evacuated before cooldown by turbomolecular vacuum pumping through thecentral pumping line at the top of the probe.Loading of the sample (and removal, via the inverse of what follows) is accomplishedby sliding the sample pot into place along alignment rails, carefully inserting the sapphiresample plate into the cavity. We note here that this sample mounting system offers nocapability for removing the sample from the resonator in situ to extract the unloadedquality factor Q0. While this is indeed a hindrance for accurate measurements of thesample loss contribution 1/Qs (and thus Rs), it was a deliberate design choice; the fixedsample stage provides increased mechanical stability relative to the movable stages, allowingfor more accurate and repeatable measurements of the resonant frequency (and even of theQ, for a given cooldown). As this low-frequency resonator probe is optimized and primarilyused for penetration depth measurements, this fixed sample position configuration is mostlyadvantageous.The loop-gap resonator is thermally well-anchored to its enclosure, which is directlyimmersed in the liquid helium bath, and any variations of the bath temperature will quicklypropagate to the loop-gap. As both the Q and the resonance frequency of the cavity will varywith temperature, this would be problematic. The solution is to regulate the temperatureof the helium bath, with an Allen-Bradley 100Ω carbon resistor as a thermometer, and523.1. The cavity perturbation probean ordinary metal-film resistor as a heater. The resistance of the bath thermometer isread out by an ac resistance bridge with a setpoint-difference output, which provides theerror signal for a PID controller driving the bath heater (both built by the UBC Physics &Astronomy Department electronics shop). Due to the extremely large thermal conductivityof superfluid helium, the thermal gradients associated with the temperature regulation arenegligible, very effectively maintaining a spatially and temporally uniform temperature inthe bath surrounding the probe. The bath temperature is typically also monitored andrecorded by computer, to allow for subsequent screening of data: as the helium bath leveldrops below the level required for stable regulation, or when spurious “glitches” in thebath temperature occur, the experiment data for the corresponding time interval can beidentified and further scrutinized or vetoed. This same bath temperature control scheme isused on most of the cryogenic probes in this lab, providing a stable cryogenic environmentfor measurement.The 3He-pot refrigeration systemIn addition to what has been described, a second sample pot configuration has also beenused with this probe, one which includes a one-shot charcoal-pumped liquid 3He pot toextend the measurement base temperature from ∼1.1 K to below ∼0.4 K. For this pot, theplatform on which the quartz tube sits contains a reservoir of liquid 3He, both pumped andrefilled through an auxiliary pumping line. This pumping line contains baffles to dampenthermoacoustic (also known as Taconis) oscillations [93], which transport heat down thepumping line, and were observed in the form of pressure oscillations before insertion of thebaffles. The 3He pot is only weakly connected to the bath through a thin-walled stainlesssteel tube standoff, allowing its temperature to be brought well below the outside 4He bath,while still allowing the pot to be sufficiently cooled by the bath to allow the liquid 3He tocondense inside initially.As the price of 3He has become astronomical in present times, its pumping and storageis necessarily done in a closed system to minimize loss, with the pot connected to a stor-age volume and an activated charcoal cryogenic sorption pump designed for helium dewarinsertion. When the sorption pump dip probe – essentially just a long tube packed withchips of activated charcoal – is warm (T ≫ 20 K), the 3He condenses as liquid in the 3Hepot at the bottom of the probe (TB.P. = 3.191 K at 1 atm). When the sorption pump is533.2. Swept-frequency cavity transmission measurementdipped into a liquid helium storage dewar, the enormous surface area of the charcoal heldnear 4 K becomes an extremely effective substrate for gas adsorption, quickly removing gasfrom the pumping line and presenting a high pumping speed for the liquid 3He in the pot.While temperatures below 300 mK can be reached in principle, the lowest base temperatureobtained with the current system is 400 mK, with a typical pot lifetime of several hours.After the pot empties, the sorption pump dip probe is removed from the storage dewar tobe warmed, freeing the 3He to recondense in the probe’s 3He pot for another refrigerationcycle.While this 3He refrigeration system has been used quite successfully in the recent past forpnictide penetration depth measurements [94], all of the measurements to be presented inthis thesis were performed above 1 K, without a need to extend to lower temperatures. Theadditional complications presented by the use of the 3He refrigeration (most importantly, thefar slower thermal time constants of the 3He system, limiting the amount of data that maybe collected in a given run) were therefore not found to be justified; however, the existenceof this important low-temperature capability is documented here for future reference.3.2 Swept-frequency cavity transmission measurementThe simplest mode of operation for the resonant cavities used in our laboratory involvesmeasurement of the transmission31 of the microwave cavity as a function of frequency. Forthe limit of weak coupling to the cavity, the transmitted microwave voltage Vout as a functionof frequency ω takes the form of a complex Lorentzian (repeated from Equation 2.46):V˜out(ω) =α˜V˜in1 + iQ(ωω0 −ω0ω) . (3.1)The simplest microwave measurements are not sensitive to phase, measuring only the totaltime-averaged power reaching a detector (repeated from Equation 2.51):Pout(ω) =P01 + 4Q2(ω−ω0ω0)2 . (3.2)31The cavity resonance may also be measured via reflection, although this mode of operation is moredifficult to carry out – particularly without the use of a vector network analyzer – and has not been muchutilized in this laboratory.543.2. Swept-frequency cavity transmission measurementPower (µW)2.61997E-6-1.80028E-77.6721E-232E-74E-76E-78E-71E-61.2E-61.4E-61.6E-61.8E-62E-62.2E-62.41997E-6Frequency (MHz)947.92778940.57329 942 943 944 945 946 947Figure 3.3: An example of swept-frequency mode raw cavity transmission data with aLorentzian fit. The plot and data are extracted directly from the experiment control/datacollection LabVIEW VI.This is true for the swept-frequency and time-domain cavity transmission measurementsperformed for the work in this thesis; a simple “square-law” diode detector was used tomeasure the power transmitted through the cavity, with no phase information recorded.The hardware setup in both cases is very simple (see Figure 3.4: an HP 83620A microwavesynthesized sweeper (with a frequency range of 10 MHz to 20 GHz) is connected via flexiblecoaxial cable to the input coupling loop of the resonator, which for this work was always the940 MHz loop-gap resonator described above. The output coupling loop is connected viaflexible coax to an HP 423A coaxial crystal detector, whose near-DC output is proportionalto the incident microwave power (and thus proportional to the square of the rms microwavevoltage). See Figure 3.3 for an example of real data and the corresponding Lorentzian fit.For noise and interference suppression, the detector is tightly shielded by a metal en-closure, out of which only its input SMA connector protrudes; inside the enclosure, thelow-frequency side of the detector drives the input of an amplifier, typically operated ata gain of 103. The output of this amplifier is connected directly to an input channel ofa Tektronix TDS 520B digital oscilloscope; a “blanking” signal output from the synthe-553.2. Swept-frequency cavity transmission measurementHP 83620ASynthesizedSweeperHP 423ACrystal DetectorAmplifierPC(LabVIEW)TriggerInputGPIB (IEEE-488)TektronixTDS 520BOscilloscopeFigure 3.4: The electronics setup for swept-frequency cavity perturbation measurement.sizer, synchronized to the start of each sweep, provides the requisite trigger signal for theoscilloscope.The experiment control and data acquisition is performed through a GPIB (IEEE-488)bus interface by a computer running the graphical programming environment LabVIEW,developed by National Instruments. We employ a LabVIEW “Virtual Instrument” – orVI – which automates the temperature control, synthesizer control, oscilloscope controland readout, and performs the Lorentzian fits to obtain f0 and Q. Given a set of desiredmeasurement temperatures and tolerances, this VI will handle the entire process of collect-ing and fitting resonance curves at the programmed temperatures, including tracking theappropriate centre frequency, span, and amplitude of the synthesizer sweeps as they changewith temperature. The end result is a data file containing thermometry data, along withthe Lorentzian fit parameters, for each temperature setpoint.It should be mentioned that some experimental groups employ setups which additionallyrecord the phase (such as with the use of a vector network analyzer, or VNA), which allowsboth components to be fit to a complex Lorentzian, which is equivalent in a sense to doublingthe resolution, and which often improves fit robustness in the presence of noise [80]. For thepurposes of the work presented here, however, our scalar detection setup was found to bemore than adequate for Rs and Xs measurements near and above Tc, with other techniquesavailable for lower temperatures.563.3. The Robinson oscillatorWhile future measurements could make use of VNA-based detection, more immediategains could be available through a full cavity Pb replating to restore its initial Q of > 106,which (as noted in Section 3.1.1) has slowly degraded over many years to its current valueof ∼ 3×105. It should be noted that such a procedure carries substantial risk of irreparabledamage to the components, and thus (perhaps unsurprisingly) has not been attempted forthis particular resonator in more than a decade.3.3 The Robinson oscillator3.3.1 ElectronicsFor the most precise measurements of the resonance frequency f0 – essential for achievingsub-˚angstro¨m resolution for changes in the magnetic penetration depth as a function oftemperature – the resonator is incorporated in a Robinson oscillator configuration. Devel-oped by Neville Robinson for use in NMR [95], the defining characteristic of the Robinsonoscillator is the use of a dedicated limiter to stabilize the amplitude of oscillation, ratherthan the marginal oscillator’s reliance on the saturation of the amplification system for thispurpose. A schematic diagram of the electronics setup used for the Robinson oscillator isshown in Figure 3.5.In the main loop of the oscillator, the signal from the output of the loop-gap resonatoris sent through two microwave amplifiers, then through an adjustable bandpass filter. Thefiltered signal enters a -20 dB directional coupler, at which 1% of the signal is diverted forfrequency measurement. The remainder of the signal passes through a limiter – essentiallya unity-gain amplifier designed to operate cleanly in its output saturation regime – followedby a “trombone”-style (adjustable path length) phase shifter, and finally through a second-20 dB directional coupler, before returning to the input of the resonator and completingthe oscillator loop.The -20 dB signal from the first directional coupler is first amplified and then inputto the IF (intermediate frequency) terminal of a double-balanced mixer; an HP 83620Asynthesizer provides the stable fixed-frequency LO (local oscillator) reference to the mixer,typically set to fLO ≈ 100 kHz above the resonator frequency at the base temperature.The mixed-down output signal – whose fundamental frequency is the difference |fosc − fLO|between those of the oscillator and reference signals – is sent to a final RF amplifier before573.3. The Robinson oscillatorPC(LabVIEW) GPIBHP 83620ASynthesizedSweeperMixerRFAmplifierLimiterµWAmplifierDirectionalCouplerDirectionalCouplerPhase ShifterHP 8473BDiodeDetectorHP 5345ACounter / TimerµWAmp.µWAmp.Loop-gap resonatorLORFIFTektronix TDS 520BDigital OscilloscopeBandpassFilter-20 dB-20 dB~940 MHzFigure 3.5: The Robinson oscillator circuitry, used for precision cavity resonance frequencydeterminations for magnetic penetration depth measurements.583.3. The Robinson oscillatorbeing connected to an HP 5345A counter/timer. This “difference signal” may also beobserved on an oscilloscope for diagnostic purposes. Additionally, the -20 dB output ofthe second directional coupler is connected to an HP 8473B diode detector to monitor theoscillator loop power.As will be discussed in Section 3.3.2, a properly tuned oscillator will track the resonantfrequency f0 of the cavity resonator. By accurately measuring the difference frequency of themixer output, and given a stable synthesizer frequency fLO, we can perform measurementsof ∆f0 with sub-hertz precision, equivalent to sub-˚angstro¨m resolution in ∆λ for our typical∼mm2 sample areas. Note that the accuracy in the absolute frequency f0 will depend onthat of the synthesizer frequency fLO, which will be somewhat greater than 10 Hz. However,as only the relative shift ∆ff will enter here, even an uncertainty of 1 kHz in fLO – so longas it remains stable in time – would contribute a negligible uncertainty (relative to othercontributions) of only one part in 106 to ∆λ.Just as with the swept-frequency mode of operation, the experiment control is handledby computer control of the instruments over the GPIB bus, using a custom LabVIEWVI. In this case, the VI simply reads out the counter/timer, in addition to setting thetemperature control according to a user-provided list of temperature setpoints. Unlike forthe swept-frequency mode, some additional manual adjustment of the phase shifter andfilter (to tune the oscillator on resonance) is also required.3.3.2 Principles of oscillator operationIn order to sustain stable oscillation, an oscillator loop must satisfy the condition that thesignal voltage component at the oscillation frequency ω0 returns to the same phase (modulo2π) upon traversing the loop; furthermore, the total loop gain must be unity in steady-stateoperation.32 In practice, in order to allow for the initial buildup of oscillations, there mustbe a (necessarily nonlinear) gain greater than one for amplitudes smaller than the steady-state value. Under such conditions, the oscillator will start itself from any finite voltagecomponent at ω0 which may be present, including thermal noise.As previously discussed, in the case of a high-Q resonator such as the loop-gap oscillatordescribed above, the transmission of a microwave cavity is sharply peaked near ω0 (Equation32These conditions – which are necessary (but not sufficient) for stable oscillations – are together knownas the Barkhausen stability criterion [96].593.3. The Robinson oscillator3.2); additionally, the phase shift across the oscillator changes sharply near ω0, as given by(repeated from Equation 2.49):∆φ ≡ arg VoutVin= φext + tan−1[2Q(1− ωω0)], (3.3)where φext encompasses all phase shifts external to the cavity. The oscillator loop will beresonant at frequency ωr when ∆φ(ω = ωr) = 0; introducing this requirement into 3.3 yieldsωr = ω0(1 + tanφext2Q)(3.4)Note that while the phase of the signal transmitted through the cavity changes sharply nearω0, the loop may only oscillate at a independent frequency ωr, differing from ω0 by an offsetwhich depends on both the external phase shift φext and the cavity Q. By adjusting thephase (performed here by adjustment of the trombone) such that the total external phaseshift of the system is zero (modulo 2π) at ω0 – where the cavity phase shift is also zero – wefind ωr = ω0, maximizing the transmission of the cavity. Furthermore, this position is alsowhere the slope of ∆φ(ω) is greatest, and thus where ωr is least sensitive to shifts in externalphase. Assuming an external phase that depends weakly on frequency on the scale of theresonance width, the oscillator frequency ωr will exactly track ω0, with no dependence onQ and no further phase adjustments required.When tuned far off resonance, the transmission of the cavity will be insufficient for theunity loop gain required to sustain oscillations. However, for smaller shifts of the externalphase away from zero, the cavity transmission may remain sufficiently large, wherein theoscillator will operate off-resonance. In this case, as can be seen from Equation 3.4, theoscillator frequency will have an additional Q dependence, with shifts in ωr (such as withsample temperature) produced by both shifts in ω0 and shifts in Q. This illustrates theimportance of tuning the oscillator to be as close to the natural cavity resonant frequencyas possible.It is apparent that the performance of the oscillator for the purposes of accuratelytracking the cavity resonance will depend strongly on the stability of the external phaseφext, both in terms of drifts in the time-averaged phase and the phase noise of the limiterand amplifiers. For a system initially tuned to ωr = ω0, we see that in the limit of small603.4. Time-domain measurementshifts δφext, the resulting frequency shifts will beδωr ≈ω02Qδφext; (3.5)therefore, the higher the cavity Q, the smaller the sensitivity of the oscillator to shifts(or noise) in the external phase. For the oscillator configuration that was used with theloop-gap resonator in this thesis, the effects of short term phase noise could be readilyaveraged out with longer averaging times for the frequency counting. The dominant sourceof measurement uncertainty was instead found to be due to a shifting oscillator frequency.One contribution to these frequency shifts may be associated with temperature changesor mechanical drift of the resonator itself, due to changes in helium bath level and settlingwith time. However, the primary culprit is believed to be shifts in phase of the externaloscillator setup. This includes shifts in electrical path lengths due to thermal expansionand contraction of the coaxial cables, due to changes in both the helium bath level andthe room temperature;33 temperature dependences of the active electronics may also playa role. To counter the effects of slow drift in the phase, a base drift subtraction scheme (tobe described in Section 4.3.2) is employed, which is effective for removing a continuous slowdrift in frequency, but less so for sudden jumps.3.4 Time-domain measurementThe advantage of using an oscillator scheme rather than swept-frequency transmission mea-surements is the ability to measure the resonant frequency (and shifts thereof) almost di-rectly, since frequency measurements can be made with much greater precision and accuracythan can be obtained from Lorentzian fits to transmission as a function of frequency. How-ever, one obvious disadvantage of the oscillator method is that it provides no informationabout the Q of the resonator and thus cannot be used for measurements of Rs.In practice, this does not prove much of a limitation, since other methods have beendeveloped for this purpose. A particularly accurate method for measuring the cavity Q canbe made in the time domain, where the cavity is driven near its resonance frequency, andthen the input microwaves are switched off; the power exiting the cavity is then measured33The room temperature in the laboratory where these measurements are performed can vary by severaldegrees Celsius during the course of a typical day.613.4. Time-domain measurementas it rings down. The exponential decay can be fit quite accurately (to Equation 2.57) fora direct determination of Q.The apparatus and setup for this measurement modality is identical to that of the swept-frequency measurements described in Section 3.2 above, but with the microwave synthesizerset for repeated pulses of continuous-wave rf on resonance, and with the oscilloscope setto measure the appropriate time window after the applied rf is stopped. Additionally, thecomputerized data collection and analysis here involves fits to signals that are exponentialin time, rather than Lorentzian in frequency. Care must also be taken to choose an amplifierof sufficiently high bandwidth; since the timescale of cavity decay can be far shorter (.50µs for the loop-gap resonator at present) than the period of standard 100-ms sweeps, thisis a more stringent requirement for time-domain measurement.A significant advantage of this mode of operation over Lorentzian fits to swept-frequencytransmission measurements is its insensitivity to fluctuations in the cavity resonance fre-quency ω0. For many reasons – including acoustic vibrations of the sample position in thecavity – the resonance frequency of the cavity may fluctuate about its mean on a timescalecomparable with the measurement time. While such fluctuations will average out over thegate time of the oscillator technique (typically 10 or 100 seconds), they can add significantvariability to the shape of the averaged frequency-sweep waveform, introducing substantialnoise to measurements of ω0 and Q extracted from fits. For time-domain measurement,the cavity energy (and output power) ringdown occurs over a timescale τP = Q/ω0 (thatis much shorter than most such fluctuations (particularly for lower Q), and which is onlyweakly affected by small shifts in ω0 during the decay interval.Unfortunately, while this mode of operation can offer significant improvements in theaccuracy of Q measurements, one is still limited by the ability to subtract the contribution1/Q0 due to losses in the unloaded cavity. The very low Ohmic losses present in thesuperconducting samples being studied (particularly at the lowest temperatures) can bemuch smaller than the intrinsic losses of the resonator (particularly in its current condition),with a correspondingly small fractional contribution to the total resonance linewidth.In order to resolve this small sample contribution 1/Qs to the total measured 1/Q =1/Q0 + 1/Qs, linewidths must be measured very accurately, and the unloaded cavity Q0must remain stable over a course of such measurements. For the loop-gap resonator probein question, the latter constraint is the more onerous, as there are no means of removing the623.5. Bolometric broadband spectroscopysample from the loop in situ, and in recent measurements Q0 has been observed to vary onthe order of 10% between cooldowns – and sometimes even between liquid helium transfersduring the same cooldown.Consequently, we find that the extraction of small, low-temperature Rs values of super-conducting samples from Q measurements – with any measurement mode – is not particu-larly reliable with this probe at present. The use of a higher-Q resonator could amelioratethis situation in the future, although as mentioned at the end of Section 3.2, accomplishingthis by replating the existing loop-gap resonator components is not without risk.3.5 Bolometric broadband spectroscopy3.5.1 IntroductionFor all of the reasons outlined above, the use of microwave cavity perturbation for Rsmeasurements of small superconducting samples can be less than optimal. One additionalproblem is the fixed-frequency nature of the resonators we use; the resonators used in ourlab each operate at a particular fixed unloaded resonator TE011 mode frequency ω0. Inorder to measure Rs or Xs at multiple frequencies via cavity perturbation, an equivalentnumber of resonators would be required.34 A detailed study of the frequency dependenceof surface impedance becomes a daunting – if not infeasible – prospect.For the surface reactance, the situation is not so dire, at least at low temperature; forsuperconducting samples well below Tc, Xs is dominated by the imaginary component ofthe superfluid conductivity σ2s(ω) = 1/µ0ωλ2L, giving Xs(ω) ≃ µ0ωλL(T ) (Equation 2.44)to a very good approximation. As λL is frequency independent by definition, measurementof Xs (or equivalently λL) at a single frequency will suffice. On the other hand, in this lowtemperature regime, the surface resistance Rs is dominated by σ1n(ω, T ), which possessesa strong and nontrivial frequency dependence containing important information about thequasiparticle dynamics. A method for obtaining detailed Rs(ω) curves is therefore critical.Fortunately, our group35 has developed a powerful non-resonant method for this purpose,using the Ohmic heating of the sample in an rf field as a measure of its Rs. The sample is34Resonators operated at multiple frequencies certainly do exist, and have been used quite successfully forcavity perturbation by the Broun group at Simon Fraser University [97] amongst others; the use of multipleresonances does carry its own additional set of complications.35The current apparatus (used here) was designed and built by Jake Bobowski [71, 98], and is an improvedversion of the pioneering work of Patrick Turner, Saeid Kamal, and others [99–101].633.5. Bolometric broadband spectroscopySample Stage Quartz tubeReference Stage CuGa AlloySapphire PlateCutoff HoleRectangular CavityCentre Conductor(Septum)Figure 3.6: A cutaway view of the assembled bolometric spectrometer. Samples (not shown)are mounted on the sapphire fingers, centred on either side of the septum. Microwave currenttravels into the septum in the “into page” direction, as shown here; the resulting magneticfields circulate around the septum, parallel to the sapphire plates. The plates are situatednear the cavity endwall, where E → 0. Not shown are the heaters and thermometers, whichare located on the sapphire plates outside the cavity, past the cutoff holes. Figure providedcourtesy of Jake Bobowski [71].mounted to a thermal stage which is weakly connected to the low-temperature bath, and thepower absorbed by the sample is measured bolometrically ; that is, through the temperaturerise of the thermal stage. Through the use of a non-resonant cavity configuration, thistechnique allows continuous-frequency low-temperature measurement of Rs(ω, T ) over abroad frequency range (from ∼100 MHz to 26.5 GHz) with great sensitivity.3.5.2 Technique overviewAs mentioned, the use of Ohmic heating as a measurement of surface resistance is the heartof the bolometric broadband spectroscopy method. (For brevity, we will often refer tobroadband bolometric spectroscopy as “bolometry” throughout this thesis.)The sample is placed on a thin sapphire finger inside a nonresonant36 microwave “cavity”composed of a rectangular septum (acting as a centre conductor of a coax, and connected36While any such structure will indeed have resonant modes, here we are operating well below their cutofffrequencies, which is why we characterize it as nonresonant for our purposes.643.5. Bolometric broadband spectroscopyto the center conductor of the input coax) shorted to the end-wall of a rectangular cavity.The sample is placed very close to the end-wall of the cavity, midway along the width ofthe septum. In this geometry, the sample is very near an electric field node and a magneticfield antinode, and can be considered to be in an applied magnetic field only.Microwaves are driven down a length of semirigid coaxial line to the bottom of the probe,where its center conductor connects to the septum, and its outer conductor is grounded tothe cavity body; microwave currents thus flow through the septum into the cavity endwall,returning along the outer conductor of the coax. All inner surfaces of the Cu cavity (thoseexposed to the fields, at least) are coated in Pb0.5Sn0.5 solder, which is superconductingat the cavity operation temperature (Tc ≈ 7.0 K, Tbath ≈ 1.2 K), in order to minimizemicrowave-induced heating of the cavity body itself, which could contribute a backgroundmodulation to the bolometry signal.In the presence of an applied rf magnetic fieldHrf, a sample with a finite real (dissipative)component of conductivity σ1 will absorb power from the fields, dissipated in the form ofOhmic heating. The sample will therefore increase in temperature according to its heatcapacity and the energy absorbed. However, the sample is thermally well-connected to asapphire plate, into which it quickly comes to thermal equilibrium at the same temperature.The sapphire plate has a high thermal conductivity, such that the temperature of the entireplate-sample system should be very nearly uniform at any given point in time, within therelevant time scales. The sapphire plate is glued into a fused quartz tube, which acts asa thermal weak link to the bath; the quartz tube is anchored to a copper block which isconnected via copper cold-fingers to the liquid helium bath outside the probe.The purpose of the weak link is to allow the power absorbed by the sample to resultin a corresponding measurable temperature rise of the bolometry thermal stage. A square-wave-modulated rf signal is sent to the cavity, resulting in a corresponding modulated powerabsorption by the sample, and thus a modulated temperature of the bolometry stage. Thisis read out by a sensitive thermometer and detection system which amplifies the modulationof the thermometer signal.This thermometry signal may have a complicated waveform in general, whose shape andscale is dictated by both the low- and high-pass filtration of the amplification system aswell as the thermal time constants of the bolometer stage and its thermal links. The latterwill vary significantly as a function of helium bath and bolometry stage temperatures, as653.5. Bolometric broadband spectroscopywell as with variations in various thermal contact parameters which are neither predictablenor repeatable between cooldowns.Life is simplified through the use of a known resistance heater, here a miniature (butotherwise conventional) chip resistor attached to the same sapphire plate as the sample, butplaced some distance away, outside of the cavity and away from the rf fields. By applyinga square-wave-modulated voltage across this heater, we can generate a precisely knownOhmic heating signal, acting as a proxy for the same power of microwave absorption froma real sample; this “standard candle” allows accurate calibration of the absolute microwavepower absorbed by the sample.Measurement of the sample power absorption (and sample dimensions) alone is notsufficient to extract the surface resistance – knowledge of the H field is necessary. It mightseem a trivial exercise to calculate the field at the sample position for a given current intothe septum – perhaps just a messy foray into an early chapter of Jackson [74]. In reality, thissetup (in large part due to the unavoidable impedance mismatch resulting from the ∼0 Ωshort to ground where the septum connects to the cavity endwall) supports a complicated setof reflections, the resulting standing waves of which combine to give a strongly frequency-dependent field for a given nominal microwave power driven into the coax at the top ofthe probe. This pattern of standing waves is highly variable and nonrepeatable betweencooldowns and runs, precluding the use of a single reference dataset for correcting thisfrequency dependent power.Without a reliable and repeatable way to predict the field at the sample, we resort tothe Gordian solution of simply measuring it. More precisely, we measure the field at aposition in the cavity – the other side of the septum – which is nearly electromagneticallyequivalent by symmetry. To do so, we introduce a second “reference” bolometry stagenominally identical37 to the first, but with a sample of known surface resistance. In thismanner, it can be seen that for the purposes of obtaining the sample Rs we no longer needto consider the value of the field if we simply take the ratio of powers Psamp/Pref absorbedby the two stages; since P = 12ARsH2 (Equation 2.43), the fields cancel in the ratio, andgiven known sample dimensions (and known reference sample Rs) we can readily extractthe unknown sample surface resistance.37Some thermal properties are in fact chosen to be different, in particular to accommodate the usually-greater heating signal from the lossier reference sample.663.5. Bolometric broadband spectroscopyHowever (as hinted at in the preceding paragraph), the fields are only nearly equivalentin the two positions. A sloping background is observed in the frequency dependence of theRsratio for identical AgAu test samples placed on the reference and sample bolometry stages.This is quite possibly due to small but differing E-field contributions at higher frequenciesdue to the finite distance of samples from the endwall, which differs very slightly betweenthe two stages. Additionally, TE01 and TE10 modes can be excited (with appropriatesymmetry breaking) above 24.4 GHz, and are observed to generate sharp anomalies in theratio between the sample and reference signals at and above a frequency of ∼29.45 GHz.In this thesis, this measured background frequency-dependent ratio is always divided outfrom the data, and no measurements were taken above 26.5 GHz.3.5.3 Thermal stage considerationsAs a bolometric method, the importance of the thermal properties of the bolometry stagesare paramount. The bolometric detection scheme employed measures the time-modulatedrf-induced Ohmic heating of the sample through the resulting small temperature oscillationsinduced on the thermal stage to which the sample is thermally anchored. For such a schemeto be effective, there are many conditions which must be satisfied:1. The sample stage must be thermally connected to the bath strongly enough to dis-sipate the average sample heating at the desired temperature, but weakly enough toallow a sufficient oscillatory sample stage temperature signal, and to prevent bathtemperature variation from interfering with this signal.2. The specific heat of the sample stage must be small enough to allow such a signal.3. The sample, thermometer, and calibration heater must be thermally anchored to thestage well enough such that thermal lag between each is much shorter than the heatingmodulation period.4. The thermal diffusivity of the sample stage must be sufficiently large to allow its tem-perature to become uniform on a much shorter timescale than the heating modulationperiod.These conditions must be satisfied for the entire temperature range over which one wishesto measure; fortunately, the bolometry apparatus in use was designed by Jake Bobowski to673.5. Bolometric broadband spectroscopysatisfy all such requirements over the temperature ranges being considered. Since it sufficesfor our purposes simply that these requirements are met, we shall not delve too deeplyinto the specifics of the thermal properties of the system and its design, which has beenthoroughly documented in the Ph.D. thesis of Dr. Bobowski [71]; we will, however, brieflyprovide enough details here to explain the basic operation of the bolometry apparatus.The sample sits on a high purity sapphire plate made from the highest available purityHEMEX grade sapphire from Crystal Systems, Inc. Similar to the sample stages of thecavity perturbation probes, this sapphire is glued to a thermal weak link – for the sample-side bolometry stage, this is a thin fused-quartz tube (possessing low thermal conductivity atlow temperatures), whereas for the reference stage this is a solid rod of higher-conductivityCu:Ga alloy designed for the typically greater heating of the normal metallic AgAu referencesample. These thermal weak links act as low-pass filters, and allow the bolometry stages tobe modulated above the bath temperature while the average microwave heating power canstill be removed.At the end of the thermal weak links is a copper base, supported by tubes of VespelSP-22, a 40% graphite-filled polyimide-based plastic with low thermal conductivity at lowtemperature. These tubes mechanically anchor the bolometer stages to the bolometry cavitybody while providing mechanical isolation. For thermal contact to the bath, continuouscopper cold fingers, running from the outside helium bath and into the vacuum can, arefastened to the copper base of the bolometer as well as to a second copper block anchoredto the tube part way along its length.Dilution refrigerator bolometry configurationOne of the major design goals of the current bolometry apparatus was to make the mainbolometry body modular. The primary motivation for this was so that the body and mea-surement stages of the bolometer could be removed from its 4He immersion cryostat probeand mounted on a 3He/4He dilution refigerator, extending its measurement temperaturerange down to .100 mK.In this mode of operation, the body of the bolometer is attached to the bottom of thecopper radiation shield, bolted to the still of the dilution refrigerator held near 700 mK.The bolometry stages – which are thermally isolated from the bolometer body by theirVespel tubes – are connected via copper braid to a specially designed thermal filter which683.5. Bolometric broadband spectroscopyis anchored to (and thus held at the ∼100 mK temperature of) the dilution fridge mixingchamber. Careful and rigorous heat sinking of electronics and rf lines were employed tominimize the heat load.3.5.4 Electronic detection and temperature controlThe most important components of the bolometry stages are the thermometers. In thesetup used here, they are especially important, since the thermometers play three criticalroles: the bias current through the resistive thermometer provides Ohmic heating to setthe sample stage temperature; the time-averaged dc resistance of the thermometer gives ameasurement of the average sample stage temperature; and the fast resistance oscillationsof the thermometer provide a measurement of the temperature modulation (and thus rfpower absorption) of the sample stage.While previous iterations of the bolometric broadband spectrometer have used Cer-nox resistance thermometers for this component [99, 100], the current probe uses neutron-transmutation-doped (NTD) germanium thermistors from Haller-Beeman Associates. TheseNTD thermometers have been measured to exhibit substantially lower current noise thantheir Cernox equivalents (no measurable excess noise under bias, compared to 40 dB excesswith the Cernox) along with greater “dimensionless sensitivity” (TR |dR/dT |) over most ofthe relevant measurement temperature range [71, 101]. To accommodate the use of thebolometer from dilution refrigerator temperatures up to 20 K, both the sample and thereference bolometry stages contain two different NTD thermistors, with R(T ) calibrationcurves optimized for measurement either above or below 1 K.693.5.BolometricbroadbandspectroscopyFigure 3.7: Schematic of the amplification and bias circuitry for the bolometers, as described in the text. Figure provided courtesy ofJake Bobowski [71].703.5. Bolometric broadband spectroscopyAccomplishing thermometry, temperature control, and signal detection with a singlecomponent simultaneously does indeed add some level of complication to the electronics, asone needs a way to decouple these various functions such that they may be performed sepa-rately and without mutual interference. This is performed using the “bias box” electronicsshown in Figure 3.7. A bias current supplied by an alkaline battery is connected acrossthe thermometer and a series ballast resistor; measurements of the bias current are takenfrom a voltage measurement across the known bias resistor. The resulting voltage acrossthe thermometer is amplified by a gain of 10 (partly to serve as a buffer) before measure-ment; from these two voltages and the known bias resistance, the thermometer resistancecan be inferred; from the known calibration curve, this is used to determine the sampletemperature.This temperature is set by the Ohmic heating power generated by the thermometeritself, which is set by the bias current; this current (and thus the temperature) is setby adjusting the bias resistor, with the fixed bias voltage from the battery. Setting thetemperature is typically an iterative process, where bias resistors are adjusted until thedesired thermometer resistance (corresponding to the desired temperature) is obtained.For the signal measurement, the voltage across the thermometer is input into an amplifierchain containing a 15 Hz low pass filter for noise reduction, followed by two 0.1 Hz highpass filter stages (as we are not interested in the dc offset signal, only the modulation). Afinal amplification stage with adjustable gain (up to 106) is used, before input to the dataacquisition system for measurement.3.5.5 Computer control and data acquisitionMost of the initial setup for a bolometric broadband spectroscopy run – such as the config-uration of microwave amplifiers/filters, dropping and bias resistors, signal amplifier gains,and temperature measurements – involves physical configuration changes in the appara-tus which must be done manually at present. However, just as for the cavity perturbationmeasurements, the main bolometry experiment control and data acquisition is subsequentlyperformed via computer control using the LabVIEW graphical programming environment.Control of the microwave synthesizer power and frequency – along with auxiliary HP3478A digital multimeter measurements used for bath temperature monitoring – is carriedout over a GPIB (IEEE-488) instrumentation bus, using a GPIB-USB peripheral connected713.5. Bolometric broadband spectroscopyFilters**optionalMicrowave*amplifierFigure 3.8: Schematic of the electronics system configuration for the bolometry experiment,described in detail in the text. Figure provided courtesy of Jake Bobowski [71].723.5. Bolometric broadband spectroscopythrough a USB fibre-optic isolation system to the experiment computer (see Figure 3.8) Thebolometry signal measurement, heater calibration voltage output, and microwave modula-tion signals are handled by a National Instruments NI USB-6218 data acquisition system(DAQ), providing both analog and digital inputs and outputs suitable for this purpose.While the DAQ has a direct USB electrical connection to the computer, optical isolationis placed between the DAQ and the experiment. The bolometry signal inputs and calibrationheater voltage outputs use AFL-300 analog fibre-optic links from A.A. Lab Systems Ltd.The digital output controlling the microwave synthesizer’s AM modulation is connected tothe synthesizer through a UBC-built fibre-optic link, after first being inverted and rescaledfrom standard TTL levels (+5 V high/0 V low) to appropriate levels specified for thesynthesizer’s AM input (-1 V “high”/0 V low), using a simple -1/5–gain amplifier circuit.The DAQ is used to drive a known amplitude square wave voltage across the series com-bination of the bolometry calibration heater and an adjustable ballast resistor; by measuringthese resistances, DAQ square wave output voltages can be readily converted into knownsquare wave Ohmic heating power applied to the calibration heaters (and thus the thermalstages). By measuring the response of the detection system to this known square-wave heat-ing signal, we then have a calibration for a known square-wave heating signal which maybe compared with the thermal response to square-wave microwave heating, the signal wewish to measure. All bolometric signal measurements are performed using a digital lock-intechnique, described in [98].73Chapter 4YBa2Cu3O6+x conductivity dopingdependence survey I:Penetration depth4.1 The YBa2Cu3O6+x samples4.1.1 Sample preparationAll of the YBa2Cu3O6+x samples used in this thesis were grown, annealed, and detwinnedby Dr. Ruixing Liang at UBC. The samples were grown in specially prepared BaZrO3crucibles by a self-flux method, in which the crystals grow by precipitation out of a melt.A starting mixture of YO1.5, CuO, and BaCO3 powders is placed in the crucible and in afurnace, heated up to 1015 ◦C for 12–16 hours to melt, and then slowly cooled to 950 ◦C.Growth is stopped by tipping the crucible over in situ at 950 ◦C, pouring the melt out of thecrucible onto porous ceramic and partially decanting the mixture, allowing easier extractionof high-quality crystals.After samples have been grown and extracted from the melt, the chain-layer oxygencontent x of the samples must be set by annealing in an oxygen atmosphere at 400 to860 ◦C, with the temperature and oxygen partial pressure specified by the desired oxygencontent. The process can take days or weeks, depending on the temperature, the sizeof the samples, and the oxygen content desired. This must be followed by an oxygenhomogenization process carried out by annealing samples for one week at ∼ 570 ◦C in asealed quartz capsule containing pieces of YBCO ceramic of the same oxygen content. Foroxygen contents where the sample is orthorhombic at ∼ 570 ◦C (x > 0.56), this procedureis instead performed after detwinning, as described below.As-grown samples – and samples that have spent any time in the tetragonal crystal-744.1. The YBa2Cu3O6+x sampleslographic phase, by exceeding the temperature of the orthorhombic-to-tetragonal phasetransition – are usually in a “twinned” condition. Upon cooling into the orthorhombicphase, the four-fold C4 symmetry of the tetragonal phase (where the aˆ and bˆ axes areequivalent) is broken into a two-fold C2 symmetry, with the unit cell distorting to become∼2% longer in the b direction than the a direction. This symmetry is spontaneously brokendifferently in different parts of the sample, dividing the sample into a- and b-axis domainswhich did so in opposite fashions – a phenomenon known as twinning.To correct this, samples are compressed between two gold-lined pressure anvils, undera uniaxial pressure of ∼100 MPa along an in-plane direction, at a temperature of ∼ 180–250 ◦C where O(1) chain oxygen atoms (see Figure 1.2) are sufficiently mobile to changelocations within the unit cell, but where oxygen exchange with the atmosphere is negligible.The mechanical strain generated in the sample realigns the twins such that the a axis(which has the shorter lattice constant) is aligned with the direction of compression. Thisprocedure is not always fully successful at converting the sample into a single twin domain,but we shall see later that a small degree of residual twinning may be accounted for. Afterdetwinning, some samples must undergo the homogenization procedure described above.38Twin domain boundaries precipitate oxygen segregation in their vicinity, which remainsbehind after detwinning; oxygen homogenization is carried out to remove this.Finally, as discussed in Section 1.2, YBa2Cu3O6+x may be made to form chain-oxygen-ordered phases. While samples will order spontaneously (but slowly) at room temperature –as the ordered phase is the thermodynamically stable one – good, longer-range oxygen chainorder can be produced by appropriate annealing procedures. The temperature program ofannealing is set by the oxygen content and particular order desired, and sometimes involvesa fast ice-bath quench of the sample (sealed in a quartz tube) from a high temperature to0 ◦C, to quickly bring the sample below a temperature region where an undesired order isformed. Such procedures have been used to produce the appropriate order for many of thesamples discussed in this thesis.All of these growth and preparation procedures have been described extensively in areview article by Liang et al. [102], from which most of the above information was obtained;the reader is invited to refer to this paper for more details.38For samples with x ≤ 0.56, homogenization is performed before detwinning; see [102] for more details.754.1. The YBa2Cu3O6+x samples4.1.2 Doping determinationThe concentration p of doped electron holes – most commonly referred to as simply the dop-ing – is the primary parameter controlling the physics of the (hole-doped39) high-Tc cupratesuperconductors. For certain cuprates such as La2−xSrxCuO4 (LSCO) the doping can besimply identified with the cation substitution fraction x, to an excellent approximation;almost exactly one hole is doped into the plane per substituted ion, giving p = x.For many other cuprates, including YBa2Cu3O6+x, the doping has no such simple,exact relationship. This is particularly true for YBCO and its isostructural RBa2Cu3O6+xbrethren (with R here being a rare earth element40), wherein only consecutive pairs ofchain-layer oxygen atoms contribute to the planar hole doping [103, 104]; in a simplifiedpicture of this process [105], a chain segment of length ℓ dopes ℓ− 1 holes into the plane.41Consequently, the hole doping of a YBCO sample depends not only on the total oxygencontent 6+x (where x is the average chain site occupancy), but also on the particularconfiguration of the oxygen atoms in the chains.For most cuprates yet studied, the superconducting transition temperature Tc has beenfound to approximately obey a “universal” parabolic doping dependence, given byTc(p) = Tc,max[1−(p− popt∆p)2], (4.1)where popt ≈ 0.16 and ∆p ≈ 0.11 for all cuprates, with only Tc,max varying between com-pounds [7]. Such a relationship naturally lends itself to use for doping determination frommeasurements of Tc, for which it has seen much use.Unfortunately, although this universal Tc(p) relationship finds a remarkably wide rangeof applicability, for most cuprates it is found to break down near 1/8-doping (p ≈ 0.125),where the actual Tc is suppressed to some degree below the parabola. This phenomenonwas associated with charge order, and has been observed to be strongest in systems knownto have static stripe ordering.Quite recently, as discussed in Section 1.7, compelling evidence has emerged whichsuggests that this paradigm of a single parabolic dome interrupted by a dip is inaccurate,39For electron-doped cuprates, this role is instead played by the electron concentration n.40For our purposes, the rare earth elements are defined to be scandium, yttrium, and the lanthanides.41This picture is not entirely accurate, as (for example) some holes reside in the chain bands – but itcaptures the basic idea.764.1. The YBa2Cu3O6+x samples0.00 0.05 0.10 0.15 0.200102030405060708090100 Measured Tc Parabolic model Cubic interpolationTc (K)Hole doping p0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20-15-10-50Tc (K)Hole doping pFigure 4.1: The relationship between Tc and hole doping p in YBa2Cu3O6+x. Data pointsare from Liang et al. [8], with the line being the p(Tc) interpolation used for this thesis.and should be replaced by a scenario involving two separate domes of superconductivitywhich have merged together. High-field measurements by Grissonnanche et al. [63] ofthe doping dependence of the upper critical field Hc2 of YBa2Cu3O6+x, YBa2Cu4O8, andTl2Ba2CuO6+δ at low temperature, combined with Tc measurements of YBa2Cu3O6+x atfixed high field, reveal a picture of domes of superconductivity centred on quantum criticalpoints at p1 ≈ 0.08 and p2 ≈ 0.18, and which contract towards these points and eventuallyseparate from each other with increased applied field. While this new interpretation of theform of the Tc dome is an important revelation, it will minimally impact our interpolationof the Tc versus hole doping data which follows.The most reliable and accurate determination of the hole doping in YBCO has beenfound to come from the c-axis lattice constant c, as measured by x-ray diffractometry [8].774.1. The YBa2Cu3O6+x samplesWhile neither Tc nor p is uniquely determined by x, it was shown that both can be uniquelydetermined by c. This measured Tc(p) relationship fills the void left by Equation 4.1,allowing p to be determined reliably from Tc measurements. This is the method which willbe used for the determination of YBa2Cu3O6+x doping throughout this thesis.For the determination of the hole doping p of each YBCO sample used in this thesis, theraw Tc(p) data of Liang et al. [8] have been used (see Figure 4.1), along with measurementsof the Tc of each sample. For most samples (when the option was available), the magne-tization curves were taken with the Quantum Design MPMS DC SQUID magnetometer.Magnetization curves were taken on zero-field cooled samples using a 1 Oe applied field inthe H ‖ cˆ orientation; the midpoint temperature of the transition (where M(T ) reaches 50%of its low temperature limit) is taken to be the Tc for these purposes. To create a functionalform of Tc(p) suitable for interpolation, the data was modelled with an overall parabolicrelationship of Equation 4.1 used with Tc,max = 94.3 K, along with a suppression term thatis modelled by a cubic polynomial interpolant to the Tc deviation data near p = 0.125; thisis numerically solved to determine p(Tc). While a “suppressed parabolic dome” is indeedused for this interpolation scheme, it fits quite well away from the “dip,” and the cubicinterpolation to real data near p = 0.125 does not depend on our interpretation.We emphasize that the use of a uniform doping scale is critical for the self-consistency ofa doping dependence survey such as this. This same hole-doping determination scheme hastherefore been used for all YBCO data shown in this thesis or used for analysis, includinginterpolations of absolute λ values and thermal expansion data from other groups andtechniques. Where both Tc and p values are reported, the Tc takes precedence and is usedto extract p with the aforementioned interpolation scheme.Anecdotally, hole doping determination schemes have been found to vary between (andsometimes within) laboratories; while discrepancies may still exist between different de-terminations of Tc – both between techniques and between operational definitions of thethreshold level defining Tc – they are believed to represent a small fraction of the discrep-ancies between reported doping scales, at least for quality samples with narrow transitions.784.2. Experimental procedure4.2 Experimental procedure4.2.1 Cryogenic proceduresAll penetration depth measurements presented in this thesis were carried out with a 940MHz loop gap resonator probe, as described in Section 3.1. Samples were mounted on thesapphire plate with a minimal quantity of vacuum grease, both for adhesion and for thermalcontact. After mounting the sample pot and sealing its indium O-ring, the interior space ofthe probe is evacuated with a turbomolecular pump to a pressure below 10−5 torr at roomtemperature before cooling down; preliminary evacuation beyond this level is unnecessary,as gas adsorption onto probe surfaces at 4 K and below will act as an extremely effective“cryopump” for residual gas.Most experiments performed in the UBC Superconductivity laboratory (including allthose presented in this thesis) use a quite traditional cryogenic approach, in which theexperiment probe extends down into an inner glass dewar filled with liquid helium, itselfsurrounded by an insulating annulus of liquid nitrogen held in an outer glass dewar. Tominimize heating from external thermal radiation, the inner vacuum surfaces of both dewarsare silvered, except for two narrow (∼1–2 cm) vertical slits extending the full height at thefront and rear of the dewars, providing windows for direct visual monitoring of cryogenlevels.While the vacuum space of the outer nitrogen dewar is permanently sealed, the innerhelium dewar has a connection to an external gas manifold to allow it to be thoroughlyflushed out with dry nitrogen (or air) and re-evacuated before each cooldown. This isnecessary because glass is permeable to helium when warm; warm helium gas left in thedewar after an experiment (or even the ambient helium present in a typical low temperaturephysics lab) will slowly gather in the vacuum space of the dewar; because helium does notfreeze out at low temperature, this will result in a serious heat leak – a condition known asa “soft dewar.”In light of this, the reader may wonder why one would choose to use such a glass dewarsystem, particularly with the availability of modern superinsulated metal dewars whichare more durable and less susceptible to “going soft.” The answer lies in our commonuse of a pumped liquid helium bath to cool our probes from 4.2 K to base temperaturesof 1.1–1.2 K. A large, high-pumping-speed Stokes pump (situated two floors beneath the794.2. Experimental procedurelaboratory, connected by large diameter pipe) is used to pump on the liquid helium bath,achieving pressures as low as 0.26 torr (35 Pa). At these temperatures, the liquid heliumwill be well below its ∼2.17 K lambda transition, and will be in the superfluid state. A wellknown property of superfluid helium is its tendency to form a film which may climb walls,very effectively transporting heat back down into the bulk of the fluid below. Althoughthe helium will form such a film in both glass and metal dewars, the thickness is greateron metal surfaces, which are rougher than glass; the film creep flow rate – and thus itsheat transfer – has been found to be three times greater for clean metal surfaces than forglass [106].The probe is always precooled to nearly 77 K with liquid nitrogen to remove most ofits specific heat before attempting to cool with less efficient (and more expensive) liquidhelium. For initial cooldown starting from a warm dewar station, the probe is typicallyleft to cool overnight inside the experiment dewar with a filled liquid nitrogen “jacket” inthe outer dewar; once the experiment dewar is cold, a warm probe outside that dewar canbe cooled by direct immersion in liquid nitrogen for faster turnaround if desired. Liquidhelium is then transferred slowly into the experiment dewar, with the goal of utilizing asmuch of the the enthalpy of the cold helium gas rising past the warm probe as possible untilthe probe is near 4 K; once the probe is immersed in liquid helium, only the latent heat ofevaporation is extracted, making far less efficient use of the cooling potential of the helium.After the liquid helium is transferred and the thermometer on the sample plate indicatesthat the inside of the probe has reached 4.2 K, we begin slowly pumping on the liquid heliumbath. After 20–30 minutes, the bath reaches a base temperature of ∼1.1 K, with the actualnumber varying between cooldowns and drifting during the lifetime of a helium bath. Oncethe bath temperature has settled to near its base, bath temperature regulation is established,as described at the end of Section 3.1.2. The bath temperature regulation setpoint istypically set to ∼10–20 mK above the initial bath temperature, found to be a good rangewhere regulation heating power remains positive throughout the run without generatingexcessive additional boiloff. Once the temperature regulation has reached equilibrium, theexperiment can begin.The pumped liquid helium bath remains at a level sufficient to maintain stable temper-ature regulation for a period of 8–14 hours.42 At this time, the liquid helium is replenished42This highly variable bath lifetime depends on numerous factors, including initial pumpdown speed,804.2. Experimental procedureto begin a new cooldown cycle. To optimize the usage of equipment time and cryogens, ex-periments are typically run on a 24-hour basis while the dewar is cold, with helium transferstwice daily in the mornings and late evenings. Throughout the experiment, the outer liquidnitrogen “jacket” dewar needs to be replenished; because filling the nitrogen jacket disturbsthe experiment (due to vibration from initial boiloff and a shift of the external heat loadinto the helium dewar), this is typically done during helium transfers.4.2.2 Measurement procedure: swept-frequency operationFor typical penetration depth runs, the experiment is first run in the swept-frequency mode,for several reasons: the cavity resonance frequency – which may vary by several MHzbetween cooldowns – must first be located; the coupling loops must be optimized for near-optimum transmission; the resonance must be characterized, including checks for powerdependence; and measurements near Tc (where the oscillator mode cannot operate) mustbe taken. The experiment is configured as described in Section 3.2 and shown in Figure3.4, with the resonance lineshape continuously monitored on the oscilloscope.After locating the resonance and adjusting the synthesizer sweep settings to appropri-ately display it, the input and output coupling loop positions are adjusted, initially tomaximize the transmission amplitude. This should correspond to a configuration with bothloops immediately adjacent to the coupling holes of the resonator enclosure, with the loopareas rotated to face the holes. After maximization, both loops are then pulled back equallyuntil the transmission amplitude is approximately 50% of the maximum. In many cavityperturbation experiments, such steps are performed because strong coupling will perturbthe cavity resonance. For this probe, the coupling is relatively weak even at its maximum,and loop positions have not been observed to significantly impact Q; however, the sensi-tivity of the transmitted signal to loop vibrations and disturbances has been found to bedecreased when the loops are better separated from the coupling holes.At this stage, preliminary Lorentzian fits to the line are performed at base temperatureto determine the maximum resonance Q – anomalously low Q values outside the typicalrange of (3.5–4.5)×105 can be symptomatic of serious problems (such as a superconducting“weak link” between the loop and its mounting) requiring a warmup. The Q and lineshapebath regulation temperature, dewar vacuum conditions, and dewar inner surface conditions; during a typicalweek of experiments, frost gathering on the inside dewar walls will steadily decrease dewar hold times as theweek progresses, motivating a warmup to room temperature after several days of continuous measurement.814.2. Experimental procedureof the resonance are then investigated as a function of applied microwave power, to check forany power dependence. Such power dependence – usually ascribed to weak superconductinglinks between the loop and its mounting block – reveals itself in the form of steps anddistortions of the Lorentzian lineshape as the applied power crosses a critical threshold.Such power dependence is nearly always seen for powers above +5 dBm, well above ourtypical maximum operating powers of . −10 dBm; however, problems occasionally arise forcertain cooldowns where such power dependence is observed at lower values. In these cases,measurement is performed with special care to stay below the onset of power dependence.Early in this characterization process, and before any real measurement, the sampletemperature is increased to 100 K for several minutes; this allows evaporation of any residualprobe gases which may have condensed onto the sample and sample plate during probecooldown, which has been observed in the past to give anomalous temperature-dependentbackground contributions to the desired signal. This “sample bake” procedure is repeatedafter every new transfer cycle to minimize the possibility of such signal contamination.After this preparatory work, we are ready for measurement. The LabVIEW VI dis-cussed in Section 3.2 provides automatic control of the measurement process, given thedesired temperature program and power settings. Through Lorentzian fits to the transmit-ted power signal, the VI extracts the f0 and Q of the resonance as a function of sampletemperature. For the measurements done for this thesis, emphasis was placed upon col-lecting denser data near Tc, where the oscillator is ineffective and the frequency shifts arewell within the resolution of swept-frequency operation. Sparse data was collected in thismode farther below Tc, where the oscillator has far greater resolution. In most cases, mea-surements were repeated several times (nonconsecutively) at each temperature setpoint toallow for averaging and estimation of statistical uncertainties.4.2.3 Measurement procedure: oscillator operationAfter sufficient data has been collected in the swept-frequency mode of operation, theexperiment is switched to the oscillator configuration, as discussed in Section 3.3.1 andshown in Figure 3.5. The microwave synthesizer – now providing the local oscillator signalto the mixer – is set to continuous-wave operation at a frequency ∼100 kHz above themeasured resonance frequency, giving a ∼100 kHz difference signal from the mixer thatis well-optimized for the timer-counter. The narrow-band filter is set to the resonator824.3. Data preprocessingfrequency, and the trombone phase shifter is adjusted to tune the loop phase until theoscillator starts operation. Because oscillation is possible over a range of phase shifts,but (as discussed in Section 3.3.2) we wish to operate as close as possible to the cavityresonance, the input coupling loop is pulled back (thus the transmission amplitude reduced)until oscillations are only sustained over a narrow window of phase shifter settings. Thislocates the peak of the cavity transmission curve, and sets the oscillator frequency to thecavity resonance frequency; after this, the input coupling loop is relowered to set the desiredpower, as measured by the diode detector.At this point, the oscillator is in operation, and measurement may begin. The rawdata output of the oscillator experiment consists of frequency readings of the ∼ 100 kHzdown-converted signal taken with the timer-counter, along with temperature measurementsfrom the start and end of the counting period. Similar to the swept-frequency measure-ment, desired temperature programs are provided to a LabVIEW VI, which automatesthe temperature control and data collection for the experiment. Depending on the mea-surement and desired resolution, several such experimental runs may be necessary; mostoscillator measurements in this thesis required two or three 8–12 hour runs to allow sufficientaveraging.4.3 Data preprocessing4.3.1 Data selectionFor any cavity perturbation experiment, the base frequency stability of the resonator iscritical for the accuracy of measurements. This is especially true for measurements ofsmall superconducting samples at low temperatures, where one seeks to resolve the effectivecavity volume change associated with a˚ngstro¨m-scale changes in sample penetration depth– fractional changes on the order of parts in 1010. This places stringent requirements on thestability of all properties of the resonator and of the microwave circuitry which can affectthe resonance to this level – requirements which aren’t always met during all points of anexperiment.The first step of the data analysis process involves removing from the dataset pointswhich occur during such times of instability, particularly near the start and end of a heliumbath lifetime. Points near sudden jumps in the base frequency are also removed. Care is834.3. Data preprocessingtaken not to “cherry-pick” the data in such a way as to bias the results; only blatant outliersmany standard deviations from the mean of other (nominally identical) measurements areconsidered for removal.On the other hand, for the swept-frequency measurements, data seldom need to befurther screened or processed, with the exception of the occasional observation of an errantfit (usually characterized by single points with fit parameters off by orders of magnitude fromneighbours). Drift and instabilities in the cavity frequency are usually much smaller than thefrequency resolution available from Lorentzian fits, and thus no corrections are necessary(nevertheless, the drift subtraction presented in the following section is still carried outduring the extraction procedure for calculating ∆f(T )/f0 from raw f(T ) data). However,data taken at intermediate and low temperatures – where oscillator data with much higherresolution are available – are usually discarded for the purposes of averaging, after theagreement of the methods is verified in the overlapping range; the much greater uncertaintiesin the swept-frequency data would lead to a negligible contribution to a weighted average,and a greatly increased stochastic spread for an unweighted average.4.3.2 Oscillator data drift correctionEven during periods of relative stability in the base frequency – typically present fromabout one hour after pumpdown until the helium level reaches the top flange of the vacuumcan – there is usually a slow continuous drift of the base frequency. This is believed to beassociated primarily with the variation of the losses and dimensions of the resonator andespecially the coaxial microwave path as the helium bath level decreases during the run;variations in pumping line pressure as well as ambient temperature and pressure for theroom-temperature electronics also contribute to this drift.Fortunately, over short periods of time such drift has been found to be fairly uniform,reasonably well-approximated by a linear interpolation between base frequency measure-ments. There are occasional “jumps” in the base frequency that this cannot account for;such jumps, when sufficiently large, are usually easy to identify, and the surrounding datacan be removed accordingly.While the perturbative shift ∆ω˜ due to the sample at any given temperature should notchange during the experiment, the various parameters of the resonator and the oscillatorloop components do change with time. Any differences measured in frequency for the same844.3. Data preprocessingsample temperature can thus be ascribed to an equivalent shift in the baseline frequency ofthe oscillator loop alone.The drift subtraction scheme that was employed makes use of the selection of a “basetemperature” for the sample from which all frequency shifts ∆f(T ) are measured. Duringthe measurement process, we periodically return to this base temperature for measurement,allowing this “base frequency drift” to be measured as a function of time. All frequencyshifts are measured with respect to this baseline, which is linearly interpolated in betweenbase measurements. The uncertainty in this interpolation is the single largest source ofuncertainty for the measured oscillator frequency shifts; during any given interval betweenbase points, the baseline frequency may jump or otherwise deviate from linearity. As anestimate of this uncertainty, we take one half of the total frequency shift between adjacentbase points; this may be used for the purpose of weighting measurements which occurredduring periods of greater drift or instability less heavily.We emphasize the fact that all of the frequency shifts (and ∆λ) are relative here, andwe are free to choose any reference point from which to measure. However, most of themeasurements were taken with a base temperature of 5 or 10 K, rather than the using thelowest measurement temperature for this purpose. Due to the substantially greater timerequired to cool to 1.5 or 2 K rather than 5 K from a high temperature, this is found tosignificantly improve measurement speed. At the lowest temperatures where accuracy isimportant (.10 K), base measurements were typically taken every one or two points; athigher temperatures, where frequency shifts become significantly larger than the base drift(and drops to base temperature take significantly longer) the interval between base pointscan be 5–10 measurement points.As previously mentioned, such drift effects tend to be worse near the start and end ofthe lifetime of a helium bath. It was often found to be most efficient to arrange for the firstset of points to be taken at higher temperature, where the frequency shifts are much largerand the fractional changes associated with drift are tolerable.4.3.3 AveragingIn most cases, measurements of the frequency shift from base temperature at each temper-ature are repeated several times, both in the same run and across several runs – and at hightemperatures, with two techniques. This allows for stochastic noise to be averaged out,854.4. The penetration depth extraction codeand an estimate of uncertainty to be extracted. After the processing described above, weconcatenate the data from all runs from both techniques, in the form of relative shifts ∆f(T )f .In rare cases where different base temperatures were used for different runs, frequency datawere shifted accordingly.For the oscillator data points, we have the additional option to average the data weightedby drift, rather than perform an unweighted average. Such a weighted average is oftenfound to substantially reduce the noise in the resulting average; however, rare unfortunatemeasurements with adjacent base points of nearly identical frequency can disproportionately“pull” the average, giving spurious results. In such cases, a baseline uncertainty value isestimated from nearby drift rates and used to set a minimum value for weighting purposes.4.4 The penetration depth extraction codeThe lowest-order approximation for the conversion of temperature-dependent frequencyshifts ∆f to penetration depths ∆λ is the simple linear relationship (Equation 2.79):∆f(T )f0≃ −As∆λ(T )2Vc, (4.2)where f0 is the unloaded cavity frequency, As is the full area of the sample, and Vc is theeffective volume of the cavity.This basic expression, while quite adequate in many cases for T ≪ Tc, neglects poten-tially important corrections, including c-axis contributions, thermal expansion, demagneti-zation, and sample twinning. A proper accounting for all such effects is required when oneis attempting to interpret small doping-dependent changes in the electrodynamic propertiesin isolation from other doping- and sample-dependent properties.To do so, we use the thin-limit expression of Equations 2.84a and 2.84b, which we rewrite864.4. The penetration depth extraction codehere for convenience:δff∣∣∣∣aˆ= Vs2Vc(1−Na)1−8π2∞∑oddn≥11n2[R( 2γnctanh γnc2 +2κnatanh κna2)+(1−R)( 2γ¯nctanh γ¯nc2 +2κ¯natanh κ¯na2)] (4.3a)δff∣∣∣∣bˆ= Vs2Vc(1−Nb)1−8π2∞∑oddn≥11n2[R( 2γ′nctanh γ′nc2 +2κ′nbtanh κ′nb2)+(1−R)( 2γ¯′nctanh γ¯′nc2 +2κ¯′nbtanh κ¯′nb2)] . (4.3b)In using these expressions, we take into consideration doping-dependent thermal expansionof sample dimensions, absolute values of in-plane penetration depths λa(0 K) and λb(0 K),c-axis penetration depths λc(T ), sample-dependent detwinning ratios, as well as estimatesof the demagnetization factors.These equations do not account for corrections due to imaginary skin depth, nor dothey allow us to separate the normal fluid screening contributions of σ2n from the superfluidscreening of σ2s in our interpretation of our screening length measurements. These effectswill primarily become important near Tc (although σ2n contributions can sometimes alsobe important for intermediate temperatures43), whereas the focus of our measurements isthe low temperature behaviour.The coupled Equations 4.3a–b are numerically solved in tandem for λa(T ) and λb(T )given the measured ∆ff data in conjunction with all of the other required physical parame-ters. Each of these has associated uncertainty which would be difficult to propagate throughsuch a numerical solver in any direct fashion. To circumvent this difficulty, a Monte Carloframework was adopted instead, allowing parameter values to be drawn (pseudo)randomlyfrom their appropriate probability distributions, accounting for correlations where appro-priate.Used extensively by high energy physics experimentalists, Monte Carlo techniques arefar less commonly seen in condensed matter experiment, where the nature of the measure-43A comparison of the relative sizes of such contributions in YBCO for 1.1 and 22.7 GHz is provided in[101], and a particularly nice demonstration of where such effects become important in the d-wave super-conducting heavy fermion compound CeCoIn5 is given in [107].874.4. The penetration depth extraction codements – in particular, the presence of sample variability – typically results in less emphasison absolute quantitative results and their uncertainties, and more emphasis on the overallqualitative behaviour. Where quantitative comparisons are both meaningful and necessary– such as in the work here – such techniques allow for a simple treatment of error propaga-tion through complicated analysis routines (such as numerical root-finding, deconvolution,Fourier transforms, etc.), generating “Bayesian-style” probability density functions for pa-rameters automatically without any assumptions of being Gaussian.One obvious downside is the need to iterate such routines many times in order to samplea sufficient range of parameter space to adequately determine the probability density func-tions of the unknown parameters. Fortunately, Monte Carlo routines are typically of theso-called “embarrassingly parallelizable” variety, with Monte Carlo instances fully separableand thus readily distributed over multiple threads and processors – an important practicalconsideration, especially given that future advances in computation are expected to comevia parallelism rather than clock-speed.For each penetration depth dataset collected for this thesis, and for each choice ofinput parameters (to be discussed below), the Mathematica-based [108] Monte Carloextraction code was run for 1000 iterations (plus one separate “sanity check” iteration withall parameters evaluated at their mean values). For separate choices of input parameters,the correction routine was repeated accordingly. This was found to be quite adequate fordetermining the probability density functions of the extracted penetration depths; moreover,the statistical means and standard deviations were found to be sufficient (and preferred) formost purposes, placing even less stringent requirements on the degree of sampling required.Each thousand-iteration run required approximately one hour of computation time on a2012-era mid-range laptop personal computer with a quad-core CPU (Intel Core i7-2630QM;“Sandy Bridge” series). Throughput was improved by executing four such Mathematicanotebooks in parallel in separate “kernels” running on separate CPU cores. The performanceof this code could be improved by coding in a lower-level language such as C++; however,the diagnostic tools and flexibility provided by Mathematica and the consequent greaterspeed of implementation were found to outweigh any execution performance issues, at leastgiven the quantity of data to be analyzed.884.5. Input parameters for ∆λ extraction4.5 Input parameters for ∆λ extraction4.5.1 Absolute λa and λbWhile relative shifts in the penetration depth ∆λ(T ) = λ(T ) − λ(Tbase) can be measuredquite accurately by the cavity perturbation methods we employ, the situation is far lessfavourable for the determination of an absolute value λ(Tbase) from which such shifts mustbe referenced [83]. Since the absolute values λa(T ) and λb(T ) enter the full expressions forfrequency shifts nonlinearly (Equations 2.82a and 2.82b), we need to settle on values forthe absolute penetration depths at this stage of the analysis.There are several choices of absolute penetration depth measurements from a variety ofexperimental methods to choose from in the published literature (as well as some partiallyunpublished low-energy µSR measurements to be detailed in Chapter 6). While values forsome techniques and at some dopings exhibit good agreement, substantial discrepanciesexist elsewhere. Given that superfluid density ns ∝ λ−2 and real conductivity σ1 ∼ λ−3Rs,even small differences in λ can potentially be of great consequence for the interpretation ofour measurements.Values from the published literature of λ(0) obtained from various methods are presentedin Section 6.4. For measurements where λa and λb are resolved (rather than an in-planeaverage λab =√λaλb for a sufficient range of doping, we are limited to the low-energyµSR (LE-µSR) measurements taken as part of this work, as well as previous electron spinresonance (ESR) measurements of ∼1% Gd-doped YBCO (Gd-ESR) by Tami Pereg-Barnea,Patrick Turner, and collaborators here in the Superconductivity group at UBC [109].While in essential agreement at higher dopings, measurements from the two techniquesappear to disagree at the lowest dopings, where LE-µSR shows a substantial suppression ofλ−2 below p ≈ 0.12, while the lowest Gd-ESR point is consistent with a continued lineardecrease in λ−2. Therefore the correction calculations for λ(T ) have been run assumingboth sets of λab values to examine the resulting differences in extracted values; however, forillustrative purposes, results assuming the LE-µSR values will be used by default for whatfollows unless otherwise stated. Where appropriate, we will also show comparisons betweenresults assuming absolute λ measurements from both techniques.894.5. Input parameters for ∆λ extraction0.08 0.10 0.12 0.14 0.16 0.18 0.20020406080100120 Low-energy SR Gd-doped ESRa-2(0 K) (m-2)Hole doping pa axis0.08 0.10 0.12 0.14 0.16 0.18 0.20020406080100120140160180200b axis Low-energy SR Gd-doped ESRb-2(0 K) (m-2)Hole doping pFigure 4.2: Measurements of the extrapolated T = 0 values of the in-plane magnetic pen-etration depths for each axis, plotted as λ−2a (0 K) (top) and λ−2b (0 K) (bottom). Theblack data were taken from low-energy µSR measurements at the Swiss Muon Source atthe Paul Scherrer Institute (see Chapter 6), and the red data were collected via microwaveESR measurements of Gd-doped YBCO by Pereg-Barnea et al. [109].904.5. Input parameters for ∆λ extraction0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.001020304050607080901001101/a2 (m-2)T/Tc 6.998  6.80 6.49 6.998 Gd-ESR a,b(0) 6.80 Gd-ESR a,b(0) 6.49 Gd-ESR a,b(0)Figure 4.3: A demonstration of the sensitivity of extracted λ(T ) (plotted as λ−2a (T )) tochoices of absolute penetration depth λa,b(0 K); a-axis data shown for three representativedopings. The solid, darker data values use the low-energy µSR values of λa,b(0 K) foranalysis, while the dashed, lighter data use the Gd-ESR data. For clarity, only selectedrepresentative error bars are shown.914.5. Input parameters for ∆λ extraction0.08 0.10 0.12 0.14 0.16 0.18 0.200123456789 Gd-ESR (Pereg-Barnea et al.) Far IR (Homes et al.) Chosen interpolationc(0 K) (m)Hole doping pFigure 4.4: The zero-temperature c-axis penetration depth λc(0 K) for YBa2Cu3O6+x as afunction of hole doping p, from Gd-ESR (Pereg-Barnea et al. [109]) and far IR (Homes etal. [110]) measurements, and the linear interpolation used for analysis. The uncertaintieson the infrared data are unknown.4.5.2 c-axis penetration depthSince our samples are neither infinitesimally thin nor infinitely wide, we must consider theflux penetration into the side faces of the sample (with length scale λc), changes in whichwill present as c-axis “contamination” to our ∆λa,b(T ) measurements. In order to correctfor this, we must know λc(T ) at each of the measured dopings.Measurements of λc(0) as a function of doping have been previously obtained from theaforementioned Gd-ESR studies [109], as well as far-infrared spectroscopy measurementsby Homes et al. [110] (see Figure 4.4). As was the case for λa,b data, here too we find somedisagreement between measurements, in this case near the intermediate value of p ≈ 0.13,where the Gd-ESR value for λc(0) is ∼60% of the interpolated infrared value. Given theuncertain reliability of the extraction procedure by which λc was inferred from the Gd-ESR924.5. Input parameters for ∆λ extractionmeasurements, we will employ the infrared λc data over most of the doping range, alongwith the highest-doping λc data point from Gd-ESR (to which the infrared data appears toextrapolate). Note that the linear interpolation was performed here for λc(p) rather thanλ−2c (p), and the data is represented in Figure 4.4 accordingly.For the temperature dependence of λc in YBa2Cu3O6+x, two doping-dependent sets of∆λ(T ) data were considered (see Figure 4.5a): those of Bonn et al. [112] for x = 6.60,6.95, and 6.99 (Tc = 61.1, 91.9, and 90.6 K, respectively) using samples grown in yttria-stabilized zirconia (YSZ) crucibles; and the measurements of Hosseini et al. [111] on aheavily-underdoped x ≈ 6.35 sample grown in a barium zirconate (BaZrO3) crucible. Thelatter sample was measured with a hole doping which was varied by oxygen ordering withtime at room temperature; here only the Tc = 19.94 K measurement is used. For eachsample, λc(0 K) was interpolated (from the data discussed above) using the hole dopinginferred from Tc, except for the optimally-doped x = 6.95 sample, to which popt = 0.163was assigned.From Figure 4.5a we see that, as a function of reduced temperature T/Tc, the overallform of λ2c(0)/λ2c(T ) remains similar between dopings, despite very large changes in λc(0)itself over this range; however, the curves are still quite different quantitatively, particularlyat high temperature. The doping dependence appears to be without clearly discernibletrends, with the optimally-doped (green) curve being well-separated at high temperaturefrom the curves at higher and lower doping. One issue to consider is the difference inTc(p) for YSZ-grown samples relative to cleaner BaZrO3-grown samples44 – the maximumTc found at optimal doping for YSZ-grown samples can be 1–2 K below the ∼94.3 Kmaximum for the BaZrO3 case, and thus using a YSZ Tc with the BaZrO3-based dopingscale of Section 4.1.2 is problematic, particularly near optimal doping. This is compoundedby the great sensitivity of the shape of λ2c(0)/λ2c(T ) to λc(0), which varies strongly withdoping. For Figure 4.5b, the YSZ-grown sample data has been plotted with the values ofλc(0) corresponding to slightly different sample Tc values (shown in legend) on the BaZrO3doping scale, chosen to maximize their overlap with the Hosseini BaZrO3 data. Only slightshifts of Tc are required to give decent agreement between curves for most temperatures,44The cleanliness situation is not quite so clear-cut in this case: while the YSZ-grown samples are or-dinarily less pure, the BaZrO3-grown samples here are ∼1% Gd-doped – which one might not consider a“dilute” impurity. While this concentration of Gd doping has been shown to minimally affect ∆λ(T ), exceptfor a slight upturn below ∼5 K that is accounted for [109, 113], it is not known how large of an effect thismay have on the absolute penetration depth λ(T = 0 K), though it is believed to be small.934.5. Input parameters for ∆λ extraction0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.70.80.91.0 Tc 19.9 K 61.1 K 91.9 K 90.6 Kc2(0 K)/c2(T)T/Tca)0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.400.900.920.940.960.981.000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.70.80.91.0b)c2(0 K)/c2(T)T/Tc Eff. Tc 19.9 K 63.5 K 92.3 K 90.0 KFigure 4.5: a) The normalized inverse-squared c-axis penetration depth λ2c(0)/λ2c(T ) ofYBa2Cu3O6+x from several sources. The black curve with Tc = 19.9 K is a measurementon a BaZrO3-grown sample (Hosseini et al. [111]), while the coloured curves are from olderYSZ-grown samples (Bonn et al. [112]). Inset: The low temperature data. b) The samedata plotted with λc(0) taken at slightly different dopings (see Tc in legend) to illustratethe sensitivity to such changes.944.5. Input parameters for ∆λ extraction0 10 20 30 40 500102030405060708090100a (nm)Temperature (K) 6.49 6.80 6.998 6.49, YSZ c(T) 6.80, YSZ c(T) 6.998, YSZ c(T) 6.49, c=0 6.80, c=0 6.998, c=01 10 100110100100010000a (nm)Temperature (K) 6.49 6.80 6.998 6.49, YSZ c(T) 6.80, YSZ c(T) 6.998, YSZ c(T) 6.49, c=0 6.80, c=0 6.998, c=0Figure 4.6: (Top) A demonstration of the sensitivity of extracted ∆λa to contributionsfrom ∆λc(T ), with corrections using ∆λc data from BaZrO3-grown (solid) and YSZ-grown(wide-dashed) samples, and without corrections (short-dashed); a-axis data shown for threerepresentative dopings. (Bottom) The same data in log-log scale, shown up through Tc;error bars have been omitted here to allow the lines to be distinguished.954.5. Input parameters for ∆λ extractionsuggesting a weak doping dependence for the form of λ2c(0)/λ2c( TTc ).Examining the low temperature behaviour (see inset of Fig. 4.5a), the lowest-dopingYSZ data (red) appears to have a substantial T -linear component rather than the expectedT 2 quadratic temperature dependence, unlike the other curves and previous experimentalevidence. This is believed to be an artifact of the subtraction procedure used to generatethe curves, which can quite possibly leave behind a residual from the ∆λa and ∆λb data,which have a strong T -linear term.In light of these issues with the YSZ data, our primary choice for λc(T ) was thus touse the BaZrO3 data alone, taking the λ2c(0)/λ2c(T ) curve measured for Tc = 19.9 K to beuniversal and scaling with the appropriate λc(0 K, p) accordingly. However, an additionalset of λc data was generated using just the YSZ data: here, the form of λ2c(0)/λ2c( TTc )was interpolated linearly in doping, pointwise in T/Tc; this interpolated function was thenlikewise scaled by the appropriate λc(0 K, p) value.Penetration depth correction calculations were run using both choices of λc(T ) for thesake of completeness. A comparison of the data corrected using the BaZrO3-based ∆λcdata versus data corrected with the YSZ-grown ∆λc data, along with uncorrected data, areshown in Figure 4.6 for ∆λa data at three representative dopings. Corrections for c-axispenetration depth can be significant, and the choice of data used for corrections can beimportant. However, only the λa,b data corrected with the BaZrO3-based λc data will beused for further analysis.4.5.3 Thermal expansionBy the very nature of the “excluded volume” microwave techniques used for this thesis work,variations in the sample dimensions with temperature are indistinguishable from equivalentshifts in the microwave penetration depth. While low-temperature thermal expansion co-efficients are small, the resulting expansion is proportional to the full sample dimensions,and can be comparable to shifts in penetration depth for sufficiently thick samples. Theseeffects may be accounted for by considering temperature-dependent sample dimensions inEquations 2.84a,b.Values for the thermal expansion coefficients of YBa2Cu3O6+x for all three axes (seeFigure 4.7) were taken from the high-resolution capacitance dilatometry measurements doneby the Meingast group at the Universita¨t Karlsruhe, published as Kraut et al. [115] (for964.5. Input parameters for ∆λ extraction0 10 20 30 40 50 60 70 80 90 1000510152025Thermal expansion  (10-5)Temperature (K) 6.59 a 6.93 a 7.00 a 6.59 b 6.93 b 7.00 b 6.61 c ( 2) 6.93 c ( 2) 7.00 c ( 2)Figure 4.7: The integrated thermal expansion ǫ(T ) for YBa2Cu3O6+x as a function oftemperature and doping for each axis. The x = 7.00 measurements are from Nagel et al.[114], while the rest are from Kraut et al. [115].6 + x = 6.59 [aˆ and bˆ], 6.61 [cˆ] and 6.93 [aˆ, bˆ, and cˆ]) and Nagel et al. [114] (6 + x = 7.00[aˆ, bˆ, and cˆ]). For the Kraut et al. data, the original raw ǫi(T ) data had graciouslybeen shared with our group by Christoph Meingast previously, and are used as-is; for theYBa2Cu3O7.00 data point, we use the reported thermal expansivities αi(T ) ≡ 1ℓi(T )dℓidT =ddT ln ℓi(T ) from Nagel et al. after conversion into integrated linear expansions ǫi(T ) (whereℓi(T ) = ℓi(0) [1 + ǫi(T )]) usingǫi(T ) = e´ T0 αi(T ′) dT ′ − 1, (4.4)via appropriate numerical integration of the αi(T ) data. The thermal expansion dataare used directly as temperature-dependent dimension modification factors in Equations4.3a,b; although the full dependence enters in a more complicated manner, one can see thatthe first-order artificial change in observed penetration depth due to thermal expansion issimply ∆λtherm.i (T ) ≃ ℓi(0)∆ǫi(T ). For temperatures below the lowest temperature datapoints available, where the contribution of thermal expansion is negligible, purely quadratic974.5. Input parameters for ∆λ extraction0 10 20 30 40 50020406080a (nm)Temperature (K) 6.49 6.80 6.998 6.49, no therm. exp. 6.80, no therm. exp. 6.998, no therm. exp.1 10 100110100100010000a (nm)Temperature (K) 6.49 6.80 6.998 6.49, no therm. exp. 6.80, no therm. exp. 6.998, no therm. exp.Figure 4.8: (Top) A demonstration of the sensitivity of extracted ∆λa to thermal expansion,with corrected (solid) and uncorrected (dashed) a-axis data shown for three representativedopings. (Bottom) The same data in a log-log scale, shown up through Tc; error bars havebeen omitted here to allow the lines to be distinguished.984.5. Input parameters for ∆λ extraction(ǫ(T ) = aT 2) extrapolations to zero at T = 0 are used.Since the doping dependence of the thermal expansion coefficients is relatively smooth– apart from a small feature near Tc, where thermal expansion corrections are perhaps theleast important correction – a simple interpolation scheme for ǫi(T ) is used: the value ofǫi(T ) is linearly interpolated as a function of oxygen content x between the T -interpolatedvalues for the three sets of measurements. This could alternatively have been done withhole doping p as the variable of interpolation, which might perhaps correlate better withthe lattice parameters than does the oxygen content [8]. However, the relative differences inǫi(T ) between the two treatments were found to be less than 11% for all dopings, axes, andtemperatures – not a large difference, considering our limited knowledge of the true dopingdependences of the ǫi(T ). A comparison of the data with and without thermal expansioncorrections is shown for ∆λa for three representative dopings in Figure 4.8; the correctionsare small, but non-negligible.4.5.4 Demagnetization effectsAs discussed in Sec. 2.3.4, the screening currents of the sample itself will modify the fieldsfound at the surface of the sample from the external applied field value. For an externalfield applied in the “thick” geometry, such effects can dominate the observed field values,whereas demagnetization effects vanish in the “thin” geometry in the limit of infinite aspectratio a/c. While all measurements presented in this thesis (including the low-energy µSRmeasurements) were taken in a thin (here: H⊥ cˆ) geometry, demagnetization effects areexpected to be observable for some of the thicker samples measured.As discussed in Sec. 2.3.4, a proper treatment of such effects in real, non-ellipsoidalsamples would require detailed numerical modelling for each unique sample shape and ori-entation. As an estimate of the magnitude of demagnetization effects, one most oftenapproximates the sample as an ellipsoid of the appropriate dimensions, the expressions forwhich were given in Eqs. 2.85a–c. Within this approximation scheme – expected to giveresults well within an order of magnitude of the true demagnetization factor – the largestof these estimated demagnetization factors for the measured samples was calculated to be∼ 3%.Although this ellipsoidal approximation provides an acceptable estimate for the size ofdemagnetization effects, it is not sufficiently accurate to allow for the true demagnetization994.5. Input parameters for ∆λ extraction0 10 20 30 40 5001020304050607080a (nm)Temperature (K) 6.49 6.80 6.998 6.49, demag correct. 6.80, demag correct. 6.998, demag correct.1 10 100110100100010000a (nm)Temperature (K) 6.49 6.80 6.998 6.49, demag. corr. 6.80, demag. corr. 6.998, demag. corr.Figure 4.9: (Top) A demonstration of the sensitivity of extracted ∆λ to demagnetization,with uncorrected (solid) and corrected (dashed) a-axis data shown for three representativedopings. (Bottom) The same data in log-log scale, shown up through Tc; error bars havebeen omitted here to allow the lines to be distinguished.1004.5. Input parameters for ∆λ extractioncontribution to be reliably divided off. Additionally, very near Tc, as λ approaches the sam-ple dimensions, the field will penetrate some parts of the sample before others, modifying thedemagnetization factor in a manner which highly depends on the particular sample geom-etry; this is a complication not dealt with here, and demagnetization factors are calculatedassuming zero field penetration at all temperatures. Nominally demagnetization-correctedλ(T ) data has been generated for the purposes of comparison (allowing estimation of theeffects of demagnetization propagating through the subsequent analysis), which is shownfor three representative dopings in Figure 4.9; just as with thermal expansion, the effects ofdemagnetization are small but non-negligible. However, given the substantial uncertainty inthese demagnetization factor corrections, they will not be used by default for what follows.4.5.5 TwinningWhile many of the YBa2Cu3O6+x samples measured for this thesis were successfully de-twinned completely, or very nearly so – that is, the entire sample volume possesses the samecrystallographic orientation – this was not always possible for certain samples. In such cases,a minority fraction of the sample being measured will be in domains of the opposite ori-entation (90◦-rotated) from the majority, introducing an admixture of the correspondinglydifferent surface impedance along the applied field direction. Assuming sufficiently largetwin domains45, measured values will be appropriately weighted averages of the propertiesfrom each orientation. This has already been accounted for in Equations 4.3a–b, where thetwin ratio R ≡ Amaj/Atot is the fraction of the sample in the “majority” crystallographicorientation.Measurements of sample twinning ratios are performed with an optical microscope withthe use of cross-polarized light, by which contrast between domains of different orientationsbecomes clearly visible. The relative areas of the majority and minority domains are thenmeasured from digital photographs of the samples. Because R can be measured quiteaccurately in this manner, and because the a–b anisotropy is relatively small, there isminimal uncertainty in this twinning correction analysis; it has therefore been applied toall processed data. For reference, the optimally doped YBCO 6.92 sample had a ratio R= 0.77, and the YBCO 6.998 sample had R = 0.85; all other samples measured had R ≥45Twinning at microscopic scales will render such a macroscopic average invalid, along with introducingdifferent problems related to scattering. Fortunately, when detwinning is not completely successful, theremaining domain structure is typically on a sufficiently large length scale.1014.6. Results0.95, for an average R of 0.945.4.6 Results4.6.1 Penetration depth shifts ∆λ(T )After accounting for all of the corrections and considerations detailed above, we arrive atthe results to be presented in what follows. We first present the corrected low-temperature∆λ data, shown in Figure 4.10 for both axes and at all dopings. The linear penetrationdepth at low temperature expected for a d-wave superconductor is shown quite clearly herefor most dopings. The 6+x = 6.49 data does show some downward curvature; this has beenfound to be very sensitive to the choice of c-axis penetration depth data used for correction.In Figure 4.11 we show the same data, plotted on a log-log scale to illustrate the similar-ity of the form of the data over a wide range, as well as allowing the display of the measureddata up to Tc. Below ∼10 K, The b-axis 6 + x = 6.85 data appears to diverge: a slightlow-temperature curvature here results in an extrapolation to zero which rapidly divergesfrom the other curves on this logarithmic scale.4.6.2 The inverse squared penetration depth λ−2(T ) and relatedquantitiesShown in Figure 4.12 is the inverse squared penetration depth λ−2(T ) as a function oftemperature, for both axes and at all dopings. We have already encountered many plotsof penetration depth in terms of λ−2(T ) rather than λ(T ); because of the simple relation-ship46 to the more physically meaningful nsm∗ , and the ability to represent the divergentpenetration depth near Tc (which instead shows as λ−2 → 0), we will continue to presentmost penetration depth data in this way.As a reminder, in the simple London electrodynamics picture [77], the London penetra-tion depth λL is expressed in terms of material properties byλL =√m∗µ0nse2⇒ 1λ2L= µ0e2nsm∗ . (4.5)The quantity nsm∗ – also known as the superfluid phase stiffness – is related to several other46Here we assume that λ ≈ λL – generally a valid approximation for clean cuprates in the local limit(ξ ≪ λ).1024.6. Results0 10 20 30 40 500102030405060708090100 6.49 6.67 6.75 6.80 6.85 6.89 6.92 6.97 6.998a-axisa (nm)Temperature (K)0 10 20 30 40 500102030405060708090100b-axisb (nm)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89 6.92 6.97 6.998Figure 4.10: The final corrected low temperature magnetic penetration depth shifts ∆λa(T )(Top) and ∆λb(T ) (Bottom) for the a and b axes, respectively.1034.6. Results1 10 100110100100010000 6.49 6.67 6.75 6.80 6.85 6.89 6.92 6.97 6.998a-axisa (nm)Temperature (K)1 10 100110100100010000b-axisb (nm)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89 6.92 6.97 6.998Figure 4.11: The magnetic penetration depth shifts ∆λa(T ) (Top) and ∆λb(T ) (Bottom)for the a and b axes, respectively, plotted in a log-log scale over the full temperature rangemeasured.1044.6. Resultsa-axis 6.998 6.97 6.92 6.89 6.85 6.80 6.75 6.67 6.491/a2 (m-2)Temperature (K)0.000.020.040.060.080.100.120.140.160.180.200.220.24(ns,a/np)/(m*/me)b-axis 6.998 6.97 6.92 6.89 6.85 6.80 6.75 6.67 6.491/b2 (m-2)Temperature (K)0.000.050.100.150.200.250.300.350.40(ns,b/np)/(m*/me)Figure 4.12: The inverse squared penetration depths 1/λ2a(T ) (Top) and 1/λ2b(T ) (Bot-tom) for the a and b axes, respectively. The ±1σ error bands (coloured bands) include– and are dominated by – uncertainty in λ(0), and are thus heavily correlated pointwiseacross a given curve. The right-hand axes show the equivalent dimensionless factor ns/npm∗/me ,the density of carriers per CuO2 plaquette divided by the mass enhancement factor.1054.6. Resultsphysical quantities of interest: further multiplication by a factor of e2 gives the superfluidspectral weight As presented in Section 2.1.4; dividing that quantity by ǫ0 gives the squaredplasma frequency ω2p (Equation 2.22). It is clear that, within factors of fundamental con-stants, λ−2(T ) can often be the most useful representation of the data.In lieu of error bars, uncertainties are shown here as coloured bands representing ±1σ foreach individual point, which includes the contribution due to λ(0). This latter contribution– which is pointwise correlated along the entire curve – dominates the uncertainty in thedata. Because of this heavy correlation, these uncertainties have been plotted as bands,within which one can visualize shifting the entire curve vertically, based on the value ofλ(0); this represents the underlying correlated uncertainty situation more clearly than wouldindividual error bars, and makes the sensitivity of these results to the value of the absolutepenetration depth more apparent.We see here in Figure 4.12 that the shape of the λ−2(T ) curves is quite similar acrossdopings, which will later be shown more clearly in Figure 4.13. We point out here a slightupturn at low temperatures for the highest dopings in the b-axis data (seen most obviouslyfor the YBCO 6.97 curve), which is not seen for the a-axis – the effects of which will becomeapparent soon.As mentioned above, λ−2 is proportional to the phase stiffness nsm∗ ; this quantity canitself be presented in a dimensionless form, by scaling to the natural parameters of thesystem. The effective mass m∗ is known to be of the same order as the electron mass me,within a small47 mass enhancement factor m∗me of order ∼2 [27, 58, 116–118], over much ofthe phase diagram. The superfluid density ns can be related to the density np of CuO2plaquettes in the material – of which there are two per unit cell for YBCO, giving np = 2Vc ,with Vc = abc being the volume of the conventional unit cell.We can therefore consider the quantitymenpµ0e2λ2L= ns/npm∗/me; (4.6)this is shown on the right hand scale of Figure 4.12.Another informative way in which to present the data is shown in Figure 4.13, in whichwe have plotted the normalized superfluid density λ2(0)λ2(T ) as a function of reduced temperature47Notwithstanding the large peaks in m∗ observed near the p1 and p2 quantum critical points [58, 67].1064.6. Results0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.70.80.91.0a-axis 6.49 6.67 6.75 6.80 6.85 6.89 6.92 6.97 6.998a2(0)/a2(T)T/Tc0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.70.80.91.0b-axis 6.49 6.67 6.75 6.80 6.85 6.89 6.92 6.97 6.998b2(0)/b2(T)T/TcFigure 4.13: The normalized inverse squared penetration depths ρa ≡ λ2a(0)/λ2a(T ) (Top)and ρb ≡ λ2b(0)/λ2b(T ) (Bottom) plotted as a function of reduced temperature T/Tc. Therelative uncertainties in ρ(T ) are far smaller fractionally than those in 1/λ2(T ) becauseλ(T ) is heavily correlated with λ(0).1074.6. ResultsTTc . With the exception of the 6 + x = 6.49 data, we see that the form of the temperaturedependence of the superfluid density remains relatively constant over this wide range ofdopings, for both axes. There is no clearly discernible doping trend with respect to the orderof the curves here, however. Note that the uncertainties here are significantly smaller thanthose in the λ−2(T ) data shown in Figure 4.12 because λ(0) appears in both the numeratorand denominator of the normalized quantity, heavily suppressing its error contribution.The 6 + x = 6.49 data are seen to differ significantly from the other dopings in Figure4.13. If absolute values of λa(0) and λb(0) are taken from Gd-ESR measurements ratherthan the LE-µSR measurements, the gap between this curve and the rest is roughly halved,but does not disappear; this demonstrates significant sensitivity to the input penetrationdepth data. Given the uncertainty in λ(0) values, it is not clear how different the penetrationdepth temperature dependence of YBCO 6.49 actually is from that of the higher dopings.In Figure 4.12 above, we included a normalized scale in terms of ns/npm∗/me ; we point outthat the low temperature intercepts of this quantity are comparable to the hole doping p ofthe samples – not surprising, since the hole doping is defined as the number of holes dopedin to the material per Cu atom in the CuO2 layer (or equivalently, per plaquette). If wedivide this quantity by p (i.e., ns/pnpm∗/me ) we can infer the effective superfluid density per dopedcarrier, renormalized by the mass enhancement factor; the results are shown in Figure 4.14.Interestingly, along the a axis the T → 0 intercept of this quantity is slightly less than1.0 for most dopings – with a drop-off at low dopings and an increase for high dopings –whereas for the b axis, this value sits closer to 1.4. This suggests an additional contributionfrom the CuOx chains, which run along the b direction [119]. It should be noted, however,that these intercepts are entirely determined by the input λ(T = 0; p) data, and the trendsseen here may depend strongly on the choice of doping interpolation employed therein.4.6.3 The low-temperature slopeA quantity which is often reported for penetration depth measurements in d-wave supercon-ductors such as the high-Tc cuprates is the slope of the linear low-temperature penetrationdepth dλdT , shown in the top of Figure 4.15; included for comparison are previous measure-ments by Hardy et al. [120], Carrington et al. [121], Kamal et al. [122], and Harris [123],with which the newest data presented here agrees reasonably well. Its variation with dopingis seen to be smooth, with some deviation at higher dopings. An interesting feature of this1084.6. Results0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.70.80.91.01.11.2a-axis 6.998 6.97 6.92 6.89 6.85 6.80 6.75 6.67 6.49(ns,a/pnp)/(m*/me)T/Tc0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.01.21.41.61.82.0b-axis 6.998 6.97 6.92 6.89 6.85 6.80 6.75 6.67 6.49(ns,b/pnp)/(m*/me)T/TcFigure 4.14: Estimated values of (ns/pnp)/(m∗/me), the effective number of superconduct-ing carriers of mass me per doped hole, for both axes, as a function of reduced temperature.(Top) a-axis results; (Bottom) b-axis results. Error bars have been suppressed here forclarity; relative uncertainties will be the same as for Figure 4.12.1094.7. Theory of low-energy electrodynamics and transport in the cupratesdata becomes apparent when plotting the ratio of the a-axis slope to the b-axis slope, shownin the bottom of the figure. For dopings less than p = 0.16 (optimal doping) the a axisslope is greater than that of the b axis, while above this doping the order is reversed. Asdiscussed in Chapter 1, the proposed pseudogap quantum critical point is believed to be ata higher doping p2 ≈ 0.18–0.19; additionally, c-axis conductivity has been shown to becomemetallic near optimal doping [110], the effects of which may be the cause of this step. Giventhe signal-to-noise ratio in this data, it is not certain that this step is significant, but it isquite suggestive nonetheless. We will be seeing more examples of this (possibly) interestingbehaviour above optimal doping shortly.A note on the extraction of slopesIt is important to point out that it is 1λ2(T ) which is truly linear over a significant temper-ature, and that the low temperature linearity of λ(T ) is just an approximation. We alsonote that the most accurately known quantity here is λ2(0)λ2(T ) , for which the contributions dueto the uncertainty in λ(0) are minimized. For all penetration depth derivative quantitiesextracted in this thesis, we have fit to the formλ2(0)λ2(T ) = 1− αT, (4.7)thus ddT(λ2(0)λ2(T ))= −α.For extracting dλdT instead, we use the limiting formlimT→0dλdT =α2 λ(0). (4.8)4.7 Theory of low-energy electrodynamics and transport inthe cupratesTo proceed further with our analysis, it is necessary to introduce some of the relevant theorybehind the low-energy electrodynamic properties underlying our measurements. We shallfirst discuss a model for the low-energy dispersion typical of high-Tc cuprate superconduc-tors, and then present results derived from this model in Fermi liquid theory, along withthe quantities which may be thereby extracted from measurement.1104.7. Theory of low-energy electrodynamics and transport in the cuprates0.08 0.10 0.12 0.14 0.16 0.18 0.200.00.20.40.60.81.01.2ab plane (Hardy 1993)a- b-axis (Carrington 1999)a- b-axis (Kamal 1998)a- b-axis (Harris 2003)d/dT [T0] (nm K-1)Hole doping pThis thesis: a axis b axis0.08 0.10 0.12 0.14 0.16 0.18 0.200.70.80.91.01.11.21.31.4(da/dT)/(db/dT) [T0]Hole doping pFigure 4.15: (Top) The low-temperature slope of λ(T ) as a function of doping for both axes,as a function of reduced temperature; included for comparison are previous measurements byHardy et al. [120], Carrington et al. [121], Kamal et al. [122], and Harris [123]. (Bottom)The penetration depth slope anisotropy dλadT /dλbdT ; note the possible appearance of an abruptjump and reversal of order at p ≈ 0.16.1114.7. Theory of low-energy electrodynamics and transport in the cuprates4.7.1 Quasiparticle dispersion near the nodal pointsThe low-energy electrodynamics and transport properties of the cuprates (at least for T ≪Tc) are primarily determined by the quasiparticle excitation spectrum in the vicinity of thefour gap nodes (points where the Fermi surface crosses the zone-diagonal nodal lines of thegap function ∆(k)); these nodal regions of the Brillouin zone contain the only accessiblequasiparticle excitation states at low temperature.Near these nodes, the quasiparticle spectrum Ek is found to be well described by ananisotropic Dirac-cone-like dispersion centred at each node. An often-used minimal phe-nomenological model which adequately captures this physics [124–127] employs a simpletight-binding model 2D electronic dispersion ǫk (neglecting dispersion along cˆ) in conjunc-tion with a “conventional” dx2−y2 superconducting gap function ∆k, given asǫk = −2tf [cos(kxa) + cos(kyb)] , (4.9)∆k =∆02 [cos(kxa)− cos(kyb)] , (4.10)where tf is a tight-binding energy scale and ∆0 is the maximum gap amplitude. Since thelattice parameters a and b are identical (or very nearly so) for the cuprates, we will setb = a for simplicity. The resulting quasiparticle dispersion Ek is then given byEk =√(ǫk − µ)2 +∆2k, (4.11)where µ is the chemical potential. Note that Ek is then zero only where both ǫk − µ and∆k are zero, which defines the nodal points kn which we may obtain from Eqns. 4.9–4.10,yieldingkn,x = ±[ π2a +1a sin−1( µ4tf)](4.12a)≃ ±[π2a +1aµ4tf+O(( µ4tf)3)], (4.12b)kn,y = ±kn,x. (4.12c)Here the ± signs of Eqns. 4.12a and 4.12c are independent, yielding four nodal points1124.7. Theory of low-energy electrodynamics and transport in the cuprateskn ≈ (± π2a ,± π2a).It is most convenient to work with a local momentum coordinate system k′ = k − knin the vicinity of the nodes; here we arbitrarily choose the node at kn ≈ (+ π2a ,+ π2a), butthe results will apply to all four nodes (with the appropriate changes). Furthermore, wewill find it useful to work in a rotated coordinate system {kˆ1, kˆ2}, where kˆ1 = 1√2(1, 1)and kˆ2 = 1√2(−1, 1), chosen such that kˆ1 and kˆ2 are the normal and tangent vectors to theFermi surface at kn, respectively. In this frame, kn =√2[π2a + 1a sin−1(µ4tf)]kˆ1. Rewriting(kx, ky) in terms of our rotated and offset coordinates (k′1, k′2) and substituting back intoEqns. 4.9 and 4.10 after some trigonometric substitutions yieldsǫk = −4tf cos(ak′1√2+ akn)cos(ak′2√2)(4.13a)∆k = ∆0 sin(ak′1√2+ akn)sin(ak′2√2); (4.13b)after some further trigonometric algebra, this becomesǫk =µ cos(ak′1√2)+ 4tf√1−( µ4tf)2sin(ak′1√2) cos(ak′2√2)(4.14a)∆k = ∆0√1−( µ4tf)2cos(ak′1√2)− µ4tfsin(ak′1√2) sin(ak′2√2). (4.14b)Expanding to first order in k′1 and k′2 near the node, and assuming the limit whereµ ≪ 4tf , we findǫk = µ+ 2√2tfak′1 (4.15a)∆k =∆0√2ak′2. (4.15b)Plugging these into Equation 4.11 yieldsEk =√(2√2tfa)2k′21 +(∆0a√2)2k′22 (4.16a)= ~√v21k′21 + v22k′22 , (4.16b)where the anisotropic Dirac cone dispersion is now made manifest, with velocities v1 =1134.7. Theory of low-energy electrodynamics and transport in the cuprates2√2tf a~ and v2 =∆0√2a~ along kˆ1 and kˆ2, respectively. As earlier discussed, this result appliesto the dispersion near all four nodal points in the Brillouin zone, given the appropriate trivialsign changes and permutations.We finally note that v1 is equal to the Fermi velocity vF ≡ ~−1|∇kǫk| near the node;similarly, v2 is equal to the “gap velocity” v∆ ≡ ~−1|∇k∆k| near the node. In keeping withthe most common notation in the published literature, we shall refer to these as vF and v2in what follows.4.7.2 Low-energy electrodynamics and transport via Fermi liquid theoryIf the model presented in the previous section for the quasiparticle dispersion of a cupratesuperconductor is treated in the framework of Fermi liquid theory [125, 126, 128, 129],48one can derive certain “universal” expressions that can be related to measured quantities.The normal fluid density nn(T ) ≡ ns(0)− ns(T ) is given bynn(T )m∗ =2 ln 2π1~2NpℓcvFv2αsFL2 kBT, (4.17)where Np is the number of CuO2 planes per unit cell, ℓc is the c-axis lattice constant, vFand v2 are the Fermi and gap velocities (respectively) as defined in Section 4.7.1, and αsFL isthe spin-symmetric Fermi liquid current renormalization factor [126].49 As its name wouldsuggest, the latter expresses the renormalization of the bare charge current Je0 to an effectivecharge current Je = αsFLJe0 in the presence of interactions. We note that – at least withinthe formalism used [126] – both the superconducting and normal currents are renormalizedin this manner.For a single-component, Galilean-invariant, two-dimensional system possessing a cylin-drical Fermi surface of perfectly circular cross-section, the current renormalization factormay be expressed as αsFL ≈ 1 +F s12 , where F s1 is the spin-symmetric ℓ = 1 Landau Fermiliquid parameter [127]. For the case of real Fermi surfaces which deviate from circularity,higher-order Landau parameters enter, and such an expansion of αsFL becomes less infor-mative.A very important consideration here is that, within Fermi liquid theory, for a single-48Ref. [125] lays the groundwork, but does not include the Fermi liquid corrections of Refs. [126, 129].49Note that most of the relevant theory literature expresses density in terms of the effective planar densityin the CuO2 planes; the factor of Npℓc here is used to convert to volumetric density (as in [129]).1144.7. Theory of low-energy electrodynamics and transport in the cupratescomponent, isotropic, Galilean-invariant system, the mass renormalization factor m∗m dueto interactions50 will perfectly cancel the associated current renormalization in the T → 0limit due to “backflow” corrections, leaving the zero-temperature currents (and thus pene-tration depths) unrenormalized by Fermi liquid interactions [130, 131]. Thus, while the fullλ(T ) for finite temperatures may contain information regarding Fermi liquid interactions,λ(0) cannot, given the aforementioned restrictions. Assuming that the quasiparticle massenhancement associated with quantum critical fluctuations – and observed in both cuprates[58, 67] and pnictides [132] – can be treated within such a Fermi liquid framework, onewould not expect a corresponding enhancement of either λ(0) or limT→0 dλdT as a functionof doping near the quantum critical point, whereas a na¨ıve application of London electro-dynamics (where λ ∼ √m) would predict a peak in both quantities as m∗ diverges at eachQCP.In reality, such a divergence has been observed in both λ(0) and dλdT for the pnictidesuperconductor BaFe2(As1−xPx)2 [64] near the putative antiferromagnetic quantum criticalpoint at x = 0.30, calling the validity of the cancellation of renormalization effects (or theunderlying assumptions) into question. On the other hand, similar evidence for such diver-gences in the cuprates is conspicuously lacking, raising the possibility of partial cancellation– a point we shall return to in Section 4.8. For our purposes here, we shall assume theFermi liquid treatment of Durst et al. [126] to be correct, although acknowledging both theuncertainty in its use and the need for further theoretical work to resolve these issues.Although we shall not need it until Section 5.3.5, application of the same considerationsused to derive Equation 4.17 lead to the existence of a low-temperature, low-frequencylimiting “universal conductivity” defined as σ00 ≡ limω,T→0 σ1(ω, T ), which has been shownto beσ00 =e2~π2NpℓcvFv2βVCαsFL2 , (4.18)where βVC is a vertex correction factor, shown to depend on the nature of the scatteringpotentials but not the density of impurities [126].Similarly, the thermal conductivity κ is given byκT =k2B3~Npℓc(vFv2+ v2vF). (4.19)50Note that for our purposes here, m is the band mass, which includes no corrections due to interactions– but is not (in general) the same as the free electron mass me.1154.7. Theory of low-energy electrodynamics and transport in the cupratesThe universal conductivity σ00 carries both the vertex correction and Fermi liquid renor-malization factors; in contrast, the normal fluid density depends only on Fermi liquid renor-malization, and the thermal conductivity depends on neither. While in principle, measure-ments of all three (or equivalent measurements of related parameters) should allow each ofvFv2 , βVC, and αsFL2 to be disentangled, the dominant large uncertainty in λ(0) limits theaccuracy to which this can be accomplished.To relate the above expressions to the penetration depth data, we first plot the quantityαsFL2 vFv2 , as extracted from Equation 4.17 (top of Figure 4.16). This parameter is relativelyconstant over the intermediate doping range, dropping away at lower dopings, and increasingagain near p = 0.18. Plotting the ratio of this quantity taken from b-axis measurements toa-axis measurements (bottom of Figure 4.16) we once again see an increase in anisotropyas one approaches p = 0.19. The possible “step” at p = 0.16 appears less significant here,but still remains.In Figure 4.17, we compare this same data with results analyzed using different pene-tration depth input data (∆λc from YSZ data, and λa(0) and λb(0) from Gd-ESR measure-ments) to demonstrate the sensitivity to these input parameters. We see that the upturnat higher doping is fairly robust to the input parameters, but the lower doping downturn ismuch smaller for the Gd-ESR λ(0) data – we therefore do not have a great deal of confidencein this feature.In Figure 4.18, we compare our a-axis αsFL2 vFv2 data (black squares; left scale) with thevFv2 values obtained from thermal conductivityκ0T data by Sutherland et al. [129] usingEquation 4.19 (blue diamonds; right scale). The left and right scales are set to be identicalto allow direct comparison of their magnitudes to estimate αsFL2.Previous estimates from [129] of αsFL2 obtained from comparison of thermal conductivitywith penetration depth data suggested values for αsFL2 of 0.6–0.7 at p = 0.09 and 0.4–0.5 atp = 0.16. Our estimate for αsFL2 here is larger, consistent with αsFL2 ≈ 1 or slightly greater;however, there is insufficient data here to make a strong comparison.1164.7. Theory of low-energy electrodynamics and transport in the cuprates0.08 0.10 0.12 0.14 0.16 0.18 0.2005101520253035404550FL2 vF/v2Hole doping p a axis b axis0.00.20.40.60.81.01.21.4-d(-2)/dT [T0] (m-2 K-1)0.08 0.10 0.12 0.14 0.16 0.18 0.201.01.21.41.61.82.02.22.42.62.83.0Ratio b/a: FL2 vF/v2Hole doping pFigure 4.16: (Top) The modified nodal dispersion anisotropy ratio αsFL2 vFv2 as a function ofdoping; this is proportional to the inverse penetration depth slope, shown on the right-handaxis. (Bottom) The ratio (b/a) of these values for the two planar directions.1174.7. Theory of low-energy electrodynamics and transport in the cuprates0.08 0.10 0.12 0.14 0.16 0.18 0.20051015202530354045505560FL2 vF/v2Hole doping p a axis b axis a axis, YSZ c b axis, YSZ c a axis, Gd-ESR ab b axis, Gd-ESR abFigure 4.17: A comparison of the modified anisotropy ratio αsFL2 vFv2 for both axes, fordifferent penetration depth input data.1184.7. Theory of low-energy electrodynamics and transport in the cuprates0.08 0.10 0.12 0.14 0.16 0.18 0.2005101520250510152025vF/v2FL2 vF/v2Hole doping p a axis 2vF/v2 a axis vF/v2 (0/T)Figure 4.18: A comparison of the a-axis modified anisotropy ratio αsFL2 vFv2 with a-axisthermal conductivity measurements of vFv2 from Sutherland et al. [129] as a function ofdoping. Except at the lowest doping, the data are consistent with an αsFL2 value slightlygreater than 1.1194.8. Summary and discussion4.7.3 The gap ratioIn addition to the relations presented above, one may also relate the low-temperature slopeof λ2(0)λ2(T ) to the “gap ratio”2∆(0)kBTc , via the expression [56]2∆(0)kBTc≃ 4 ln 2Tc[ ddT( λ2(0)λ2(T ))]−1. (4.20)Note that this expression makes several (possibly questionable) assumptions, includingisotropy of the Fermi surface, as well as a gap function of the form ∆(φ) = ∆0 cos(2φ). Theseassumptions impose a relationship (in fact, a proportionality) between the gap maximum∆(0) and the slope of the gap near the nodes which will not exist in general. However,in the absence of more concrete knowledge of the details of the Fermi surface and the gapfunction, we shall use Equation 4.20 in its present form solely to provide this oft-calculated“quantity of interest” for future reference and comparison, without attaching to it anyundue significance regarding its relationship to the true gap ratio.Before returning to this relation, we first plot the low-temperature slope of λ2(0)λ2(T ) , shownin Figure 4.19. We notice that the a and b axes track each other well until optimal doping,at p = 0.16, above which the anisotropy which we have previously encountered develops,clearly seen in the ratio plot at the bottom of the figure. The slopes are in fact consistentwith being equal until optimal doping.Returning to the issue of the gap ratio, an estimate of 2∆(0)kBTc extracted via Equation4.20 is shown in the top of Figure 4.20, with the weak coupling d-wave BCS value of 4.28highlighted. (Note that this calculated value rests on similar assumptions to those discussedabove [133], and is to be interpreted accordingly.) Shown at the bottom of the figure are thesame data processed using Gd-ESR values of λ(0); while the anisotropy at higher doping isunaffected, the large upturn for the lowest-doping sample disappears.4.8 Summary and discussionIn this chapter we have shown the results of the microwave penetration depth portionof the doping dependence survey, with some of the data presented in alternative formsfor convenient future reference. The most interesting results observed here concerned thechanges in behaviour at and above optimal doping, with the anisotropy observed to have a1204.8. Summary and discussion0.08 0.10 0.12 0.14 0.16 0.18 0.20456789101112d/dT [2(0)/2(T)]  (10-3 K-1)Hole doping p a axis b axis0.08 0.10 0.12 0.14 0.16 0.18 0.200.80.91.01.11.21.31.41.51.61.7Ratio b/a: d/dT [2(0)/2(T)]Hole doping pFigure 4.19: (Top) The low-temperature slope of λ2(0)λ2(T ) for the a and b axes, as a functionof doping. (Bottom) The ratios (b/a) of these slopes for the two planar directions.1214.8. Summary and discussion0.08 0.10 0.12 0.14 0.16 0.18 0.203.03.54.04.55.05.56.06.57.0Est. 20/kBTcHole doping p a axis b axis4.280.08 0.10 0.12 0.14 0.16 0.18 0.202.53.03.54.04.55.05.56.06.5Est. 20/kBTcHole doping p a axis b axis4.28Figure 4.20: (Top) Estimates of the gap ratio 2∆(0)kBTc using Equation 4.20. (Bottom) Thesame gap ratio analysis performed using λ(0) data from Gd-ESR. Although the upturnfor the lowest doping is no longer present, the anisotropy at and above optimal doping(p = 0.16) remains.1224.8. Summary and discussionpossible step near p = 0.16, and the values of slopes increasing further for the highest twodopings.As previously stated, while the appearance of a step-like feature near p = 0.16 is certainlysuggestive of an interesting change in electronic behaviour near optimal doping, it is byno means certain that this feature is truly present; given the signal-to-noise ratio, thepossibility remains that this is simply an improbable fluctuation in the data.51 However, itspresence would not be inconsistent with previous observations, including the change in c-axis conductivity seen in far infrared measurements [110], possibly associated with a changein in-plane anisotropy due to increased coupling to the one-dimensional chain bands.Perhaps most notably, a very recent STM doping dependence study of the hole-dopedcuprate (Bi,Pb)2(Sr,La)2CuO6+δ [134] has shown evidence for a change in Fermi surfacetopology slightly below optimal doping, which they claim does not affect the pseudogap,and is thus not associated with the pseudogap quantum critical point p2. This points toa scenario involving a third phase transition and associated quantum critical point underthe dome (“p1.5”?). Although we cannot firmly make any such bold claims solely on thebasis of the data presented here, this could perhaps hold up as early evidence from transportproperties of such a scenario. Further measurements in this doping range have been plannedto investigate this feature, and to confirm (or rule out) its existence.We turn instead to the question of the low-temperature penetration depth – both theT → 0 limit and the slope – at the lowest and highest dopings of the range of samples mea-sured, which approach the putative quantum critical points p1 ≈ 0.08–0.09 and p2 ≈ 0.18–0.19. First considering the low doping end, we see from Figure 4.2 that the low-energy µSRvalues of λ−2(0) appear to extrapolate to zero near this doping for both axes, consistentwith the divergent52 effective mass m∗ →∞ that has been previously observed [58, 137]. Onthe other hand, the presence of this feature in the low-energy µSR data is in disagreementwith the previous Gd-ESR measurements, which show no such signs of a vanishing λ−2(0)in this region. In terms of the low-temperature limiting slope limT→0 dλdT (see top of Figure4.15), the data for both axes also suggest a possible upturn as p1 is approached from above,consistent with quasiparticle mass enhancement at this point; however, atop a background51In other words, an example of pareidolia: the universal human tendency to seek order in stimuli evenin its absence.52For the purposes of this discussion, one should read “divergent” as “strongly enhanced,” since thedivergence of m∗ is expected to be truncated by a cutoff in real systems [135, 136].1234.8. Summary and discussionwhich is already sloping upward with underdoping, this upturn is not very dramatic, andthe evidence is not quite conclusive given the current data.Switching our focus at last to the highest dopings, expected to be in the vicinity of thepseudogap quantum critical point p2, we find that any possible signs in this region of eithera peak in dλdT or a suppression of λ−2(0) are weak at best, given the existing data. For λ−2(0)(Figure 4.2), both axes show a continued increase, if not an upturn, with increasing doping– the opposite of what would be expected from the diverging quasiparticle mass measuredvia quantum oscillations [67]. From the slopes (Figure 4.15), there is also little evidence fora strong enhancement due to such a divergent mass (except for a possible slight increasenear p = 0.18). Although further data points in this doping region would be desirable(and thus at present a peak in penetration depth that is very sharp in doping cannot beruled out) the expected effects on the penetration depth associated with the broader massdivergence seen by quantum oscillation measurements appear to be weak or nonexistent.To discuss this further, we return to the issue of the effects of Fermi liquid renormal-ization effects on the superconductivity, previously discussed in Section 4.7.2. As touchedon there, for a single-component, isotropic, Galilean-invariant system treated within Fermiliquid theory, the effects of mass renormalization are perfectly cancelled in the T → 0 limitdue to backflow corrections [130, 131], and thus the zero-temperature penetration depthλ(0) (and the limiting slope limT→0 dλdT ) would be unrenormalized by Fermi liquid inter-actions. To the extent to which the divergence of the effective mass m∗ due to quantumcritical fluctuations can be treated within a Fermi liquid framework, and given the validityof our assumptions, one should see little to no effect on the low temperature penetrationdepth associated with approaching a quantum critical point.However, these idealistic assumptions of isotropy, Galilean invariance, and a single com-ponent will not hold in real materials. The degree to which the zero-temperature cancel-lation of renormalization effects persists when these assumptions are violated remains anactive topic of discussion. Early work on the heavy fermion superconductor UBe13 sug-gested that the cancellation does not hold [138, 139], which was explained to be due tothe two-component nature of the conductivity in such heavy fermion systems [140]. Themore recent observation [64] of a dramatic peak in both λ(0) and limT→0 dλdT as a functionof doping for BaFe2(As1−xPx)2 near the putative antiferromagnetic quantum critical pointat x ≈ 0.30 [141] casts further doubt on the validity of this cancellation and its underlying1244.8. Summary and discussionassumptions. In fact, the effective mass enhancement inferred from specific heat, de Haasvan Alphen, and penetration depth measurements are all in agreement [132], pointing to theabsence of the cancellation of Fermi liquid renormalization. Just as for the case of UBe13,the presence of multiple bands in the pnictides has been proposed as the “loophole” whichspoils cancellation [136, 142].In this light, one would expect to see a similarly dramatic enhancement of λ and dλdTin YBa2Cu3O6+x (and other cuprates) in the vicinity of both the p1 and p2 quantumcritical points, particularly given the large mass enhancements observed near both QCPs[58, 67, 137]. While our data supports such an enhancement (if not conclusively) near p1 ≈0.08–0.09, any similar effect near p2 ≈ 0.18–0.19 is conspicuously weakened or absent –suggesting at least partial cancellation of Fermi liquid renormalization effects in the caseof the cuprates. There are several possible reasons for differences in this regard betweenBaFe2(As1−xPx)2 and the cuprates, mostly pertaining to the different electronic structureof the systems, along with the sensitivity of renormalization to such details.Levchenko et al. [136] and Nomoto and Ikeda [142] have calculated the effects of spindensity wave and antiferromagnetic order QCPs (respectively) on the magnetic penetrationdepth, with a focus on the BaFe2(As1−xPx)2 results, and come to similar conclusions. Bothgroups emphasize that the renormalization depends on the details of the Fermi surface, thesuperconducting gap, and the relevant ordered phase. Nomoto and Ikeda also consider thecase of a single-band model intended to model the cuprates, and find a finite enhancement ofλ(0) at the critical point, both for hole- and electron-doped cuprates – but the enhancementis far weaker and broader in doping than for the pnictides; they attribute this fact to alimited renormalization of current and Fermi velocity, which they find to be confined to the“hot spots” at 20◦ and 70◦ on the cuprate Fermi surface, rather than occurring along thefull surface as in the pnictides. However, no similar calculations are known for the case ofa charge order QCP – as is believed to be the relevant case here for p2 – and it is unclearwhat differences this would make for the penetration depth.Introducing further uncertainty, Chowdhury et al. [143] consider the penetration depthnear several different types of QCPs (including SDW and nematic order) and predict nopeak in penetration depth, only a change in its second derivative with respect to doping.Consequently, they claim that the peak observed in BaFe2(As1−xPx)2 [64] must occur insidethe ordered phase, rather than at the QCP, and must involve different physics from the1254.8. Summary and discussionfluctuations which they consider. It is apparent that the theory of penetration depth neara QCP remains somewhat unsettled, and theoretical clarification on this front is badlyneeded. In the meantime, it is difficult to evaluate the significance of our non-observationof penetration depth enhancement near the upper quantum critical point p2, althoughthe absence of any such strong enhancement in our observations should constrain relevanttheoretical work in the future.126Chapter 5YBa2Cu3O6+x conductivity dopingdependence survey II:Surface resistance5.1 Experimental procedureMuch the same as for the cavity perturbation measurements, a broadband bolometric spec-troscopy experiment begins with the mounting of the sample on the sapphire plate, using aspecial mounting jig for sample alignment as well as sapphire plate protection. A minimalquantity of Dow Corning High Vacuum Grease (a silicone grease with low vapour pressure)is used to secure the sample on the plate, as well as provide good thermal contact. Thelatter is far more important for bolometric measurements than for cavity perturbation, aspoor thermal contact will result in unacceptable temperature lags; while for cavity pertur-bation this would merely decrease the rate at which temperatures may be changed duringa series of measurements, for bolometry this will destroy the crucial equivalence betweenthe calibration heater and sample absorption heat signals. In the case of the reference-sidebolometry stage, the Ag:Au metallic reference sample is glued down with Stycast 1266 clearepoxy (measured to have low microwave loss) for permanent, high-quality thermal contact.(Because the Ag:Au reference sample is never changed, the reference-side bolometry stageis seldom removed, and usually only the sample-side bolometry stage requires remountingas described here.)Both bolometry stages are then mounted on the probe, with their sapphire plates andthe attached mounted samples inserted into the cavity body (see Figure 3.6 for a cutawayview of the assembly). Continuous copper cold fingers, running from outside the probedirectly immersed in the helium bath and then passed through the walls of the vacuum1275.1. Experimental procedureenclosure, are used to provide the thermal connection to the liquid helium bath for eachbolometry stage, with copper braid used for the reference side, and large-diameter gold-coated rigid copper wire used for the sample side. These are clamped tightly (but carefully)to the two copper anchors of each stage – one at the end where the weak link is anchored,and one mounted farther along the Vespel tube – using Cu-impregnated thermal grease forimproved thermal contact. This step is important because the quality of thermal contactto the bath impacts the lowest attainable base temperature as well as the thermal responsetime constants for bolometric detection.With all stages mounted and electrical connections tightened and checked, the probe’svacuum can is sealed with an indium O-ring and evacuated to below 10−5 torr with aturbomolecular vacuum pump. The probe is then cooled down in the same manner asdescribed in Section 4.2, with the same option available here for direct-immersion liquidnitrogen precooling of the probe for a faster turnaround.After the probe has cooled to ∼1.2 K in the pumped, temperature-regulated superfluidhelium bath, and after a settling time of one to two hours (required for all of the thermalmasses in the probe to reach their base temperatures), the experiment is then preparedfor measurement. All heater and thermometer resistances, including their individual leadresistances, are remeasured at this base temperature (which may vary between cooldowncycles); these are required both for direct use in signal calibration, as well as for diagnosticpurposes.As described in Section 3.5.4, the Haller-Beeman NTD thermometers are used not onlyfor temperature measurements and heat-signal detection, but also as the primary heatersused to set the temperatures of the bolometry stages. A low-noise bias voltage source – inthis case provided by either 9 V or 1.5 V “AA” alkaline batteries – connects across a biasresistor RB in series with the Haller-Beeman thermometer RHB; the large (relative to RHB)bias resistance acts as a ballast to provide a constant current through the thermometer.Given a fixed bias voltage (usually 9 V, except for measurements at dilution refrigeratortemperatures requiring lower heating power) the bias resistance is chosen to set the currentthrough the thermometer – and thus the Joule heating power I2BRHB – required to reachthe desired temperature THB.This is a very nonlinear process, since both the thermometer resistance RHB and thethermal conductance of the bolometry stage’s thermal weak link have strong, nonlinear tem-1285.1. Experimental procedureperature dependences. These are thankfully monotonic at low temperatures, precluding thepossibilities of history-dependence and metastability in setting temperatures. Nevertheless,temperature setting is typically an iterative procedure, starting from an initial bias resis-tance “guess” chosen based on previous runs,53 and adjusting until the desired temperatureis reached. The recent addition of an easily-adjustable, magnetically-shielded decade resis-tor box for use as the bias resistance on the sample stage has made such iterations simple; formost measurements done in this thesis (and for all reference-stage temperature settings), amore onerous process of selecting and measuring axial-leaded resistors was performed, withseries or parallel “trimming resistors” soldered in for fine adjustment as necessary.To measure the temperature of the thermometers – without disturbing the bias cur-rent responsible for setting this temperature – both the voltage across the bias resistor aswell as that across the thermometer (which includes the lead resistances) are measured.Given the knowledge of the measured lead resistances and the bias resistance, this givesthe thermometer resistance; known calibration values of RHB(T ) are then used to calcu-late the temperature. Once the desired temperatures have been set for both the sampleand reference bolometry stages, the calibration heater resistances (which have weak butnon-negligible temperature dependences) are remeasured.While the HP 83630A synthesized sweeper used for the broadband bolometry measure-ments is capable of operating from 10 MHz to 26.5 GHz, its power output capabilities overmuch of that range are lower than desired, resulting in a poor signal-to-noise ratio. Thesituation may be easily ameliorated through the use of broadband microwave amplifiers,producing sufficient power over their respective operating ranges. For the measurementsdone in this thesis, two different amplifiers were used to cover their respective ranges: theHP 8347A RF amplifier (100 kHz – 3 GHz) and the HP 8349B microwave amplifier (2–20GHz). For frequencies from 20–26.5 GHz, no amplification capability was available, andthus sparse frequency grids with long averaging times were necessary over this range.The consistency of data between the different amplifier configurations was often checkedover their ranges of overlap. It was found that a spurious, non-smooth frequency dependencecould often be observed at low frequencies below ∼3 GHz, with high variability betweenamplifier configurations. Testing showed that this was likely due to both the effects of53Variations in helium bath temperature and the nonrepeatable thermal braid connections are believedto be the primary causes of variability between cooldowns.1295.2. Data analysisreflections and standing waves in the microwave line, as well as higher-harmonic generationin the amplifiers (which is often known to be introduced or exacerbated by the presenceof reflected power). In an attempt to improve this, various configurations of isolators,attenuators, and low-pass filters were inserted in-line at the output of the amplifiers to checkthe results. The optimal combination of spurious signal suppression along with minimalpower attenuation was realized with the use of microwave low-pass filters; from 2–4 GHz,an inline 4 GHz low-pass filter was used, while below 2 GHz an additional 2 GHz low-passfilter was inserted. This suggests that higher harmonic content in the signal, in combinationwith an Rs which increases strongly with frequency, contributes to the problems at lowfrequency.5.2 Data analysis5.2.1 Basic processingFrom the measurement process, we are already provided with a final54 set of Rs values asa function of the frequencies, temperatures, and microwave power levels provided; at this“Rs stage” of analysis, only minor data processing is required. Unlike with cavity per-turbation measurements, which depend strongly on the helium bath temperature stability,the bolometry measurements are quite robust to slow bath temperature variations; exceptat the lowest temperature setpoints, the temperatures of the bolometry stages are largelydetermined by the Ohmic heating of the thermometers, which dwarfs any changes in heattransfer from the bath. The slow, diffusive thermal connections to the bath, combined withthe use of lock-in detection of the sample signal, also serve to decrease the measurementsensitivity to bath temperature fluctuations. In almost all cases, only the data collectedafter the liquid helium bath temperature has become unstable need to be discarded; evenafter this point the data often appear scarcely different (particularly for higher temperaturesettings), but are discarded as a precaution nonetheless.The data are then checked for power, amplifier, and filter dependence, by comparison ofmeasurements taken with different power settings, and with different amplifiers where theyoverlap (as described in Section 5.1). Preliminary checks are typically performed during54Corrections for demagnetization and for the frequency-dependent sample/reference ratio functionnotwithstanding; the latter is corrected for at the beginning of the Mathematica processing notebook,while the former is neglected for the reasons outlined in Section 4.5.4 (p. 99).1305.2. Data analysisthe course of the experiment, as the presence of any substantial power dependence can beindicative of a problem with the apparatus (or the sample). In the case where a powerdependence is observed, but only above a sufficiently high onset power level, care is takento use measurements only below that level. In most cases, measurements at different powerlevels are found to be in agreement, in which case the higher power (and thus higher signal-to-noise) measurements are retained. For overlapping measurements employing differentamplification and filter configurations, data are first checked for agreement over their rangesof validity before simple concatenation and unweighted averaging.5.2.2 Self-consistent extraction of σ1 from RsAt this stage the Rs data is essentially in its final form, but the fundamental quantity wewish to obtain is not Rs but rather σ1. As described in Section 2.2, and examining Equation2.42 in particular, full knowledge of σ2(ω) (or equivalently, Xs(ω)) is required to extractσ1(ω) from measurements of Rs(ω). While measurements of the penetration depth in thelow-frequency limit can determine the superfluid screening σ2s, they do not account for thequasiparticle conductivity screening contribution σ2n.Since the normal fluid screening σ2n(ω) is Kramers-Kronig related to σ1n(ω), we canextract its value from our measurements of Rs(ω) (which is dominated by σ1n). Unfortu-nately, the Kramers-Kronig relations are “nonlocal” in frequency: the value of σ2n at anygiven frequency depends on values of σ1n at every frequency, including those outside theRs measurement window. One way out is to make use of the knowledge that σ1(ω) forYBa2Cu3O6+x has been found to be very well described by a phenomenological modifiedDrude model, given by the expressionσ1n(ω, T ) =σ0(T )1 + (ω/Γ(T ))y(T )+ σ1D(T ); (5.1)this model – motivated by the observed Drude-like lineshape, but with a high-frequencyrolloff slower than ω−2 – fits the Rs data for YBCO quite well for all samples, in both axes,at all temperatures, over most of the measured frequency range.55In comparison with the conventional Drude lineshape of Equation 2.18a, a frequency55Some low-frequency deviation is often seen, and has previously been noted (in data taken using thissame probe) as a possible observation of a collective mode [144]; however, given that many factors (includingcavity-to-bolometer heat leakage and signal crosstalk) can produce large spurious signals below ∼1 GHz, thelow frequency results should be treated with caution.1315.2. Data analysisexponent parameter y(T ) is allowed to differ from its Drude value of 2, modifying theshape of the line both at its ω → 0 peak and in its power-law asymptotic fall-off at highfrequency. As a result, the width parameter Γ(T ) may no longer be rigorously identifiedwith a true scattering rate (or an inverse lifetime τ−1), but it still plays the role of acharacteristic frequency scale for conductivity. An additional constant-in-frequency termσ1D(T ) is included as well; its inclusion has been found necessary to model the b-axisconductivity, for which a very broad conductivity contribution from the 1D chain bands isbelieved to exist [119, 145].56Assuming that Equation 5.1 adequately models the observed data – and moreover,provides an accurate extrapolation of σ1n(ω, T ) outside the measurement frequency window– we may use Equation 2.9 to infer the quasiparticle imaginary conductivity σ2n(ω, T ),associated with inductive screening due to the normal fluid. Rather than immediatelysolving the equation, we instead fit the equation with the modelled σ1n to the measuredRs(ω) data to extract the model parameters. We then use the modelled σ2n along withthe measured Rs to extract σ1 using Equation 2.42.57 This deals with the correction self-consistently, using only the assumption that the modified Drude model describes σ1n wellenough to extrapolate – an assumption proven to be well-founded by post hoc consistencychecks. As an added benefit, this analysis method generates the best-fit parameters for ourmodified Drude model as a byproduct.Because the uncertainty in σ1 is dominated by the contribution from that of λ, whichprimarily enters as an overall scale factor with large fractional uncertainty (since σ1 ≃2Rs/(µ20ω2λ3)), a more sophisticated Monte Carlo treatment of error propagation – such asthat used in Chapter 4 – would be of little use here. In fact, since this large uncertainty(which can be ∼50% for some sources of λ(0) data) predominantly affects only the overallscale of the σ1(ω) curves, with minimal effect on their shapes, we suppress its uncertaintycontribution by using only the mean values of λ data in calculations. Explicit inclusion ofuncertainty from λ would obscure the shapes of σ1(ω) curves and their relative variations,56Contamination from c-axis conductivity can also result in such a frequency-independent contribution –however, the observation of its value being significantly greater along the b axis than along the a axis for arange of a :b aspect ratios suggests it is dominated by a true component of the b-axis conductivity, whateverits origin.57In principle, the process can be iterated ad infinitum, extracting an updated nth σ2n from which the(n + 1)th σ1n estimate can be obtained; in practice, this increases the already-lengthy computation timesubstantially, and the correction is sufficiently small that changes beyond the first order of iteration arenegligible (and well within statistical error bars).1325.3. Resultsas well as requiring careful treatment of the large correlation of uncertainties between dat-apoints at different frequencies, temperatures, and even different axes and dopings (sinceλ values from the same method have large correlated systematic uncertainties, and dopinginterpolations of λ introduce further correlation). Nevertheless, any proper comparisons ofabsolute conductivity data must account for this.5.3 Results5.3.1 Rs(ω, T ) and σ1(ω, T ) resultsThe measured Rs(ω, T ) curves are shown for each axis and doping in Figures 5.1 (a axis)and 5.2 (b axis). Note that the frequency-dependent ratio has not yet been corrected hereat this stage, since this correction will be accounted for during the extraction of σ1 from Rs.The YBCO 6.49 sample was measured on the dilution refrigerator, and thus measurementsof this were taken at much lower temperatures than for the others, but were not taken athigher temperatures. The YBCO 6.993 sample was not part of the doping scan, and waspreviously measured by Patrick Turner and Richard Harris, but with data only availablefor the a-axis – we show this data to illustrate the behaviour at the highest dopings.As previously discussed, we are less interested in Rs(ω, T ) itself as a quantity than asa means for obtaining σ1(ω, T ), the real part of the conductivity. Following the procedureof 5.2.2, we extract values of σ1(ω, T ); the resulting σ1 conductivity curves are shown inFigures 5.3 (a axis) and 5.4 (b axis). All curves shown were fit with all model parametersfree, with a fit range of 1–20 GHz. It is important to note that since Rs(ω) ≃ 12µ20ω2λ3σ1n(ω)(Equation 2.45, the extraction of σ1 from Rs carries a factor of λ3 and thus the verticalscale here is very sensitive to the chosen values of λ(0); however, apart from this largeuncertainty on the overall vertical scale factor, the shape of the curves is known to greaterprecision. We also note that since the normal fluid screening corrections are small, whiledifferent fit configurations can give quite different fit parameters they do not visibly alterthe extracted σ1(ω, T ) results.Over the range of dopings measured one observes a clear change in the temperaturedependence of the σ1(ω) lineshape, from the constant height and temperature-dependentwidth of x=6.49 to the constant width and temperature-dependent height of x=6.998.However, any possible smooth progression as a function of doping is spoiled by several1335.3. Results0 5 10 15 20 25051015202530354045506.49 a-axisRs (10-5 /sq.)Frequency (GHz) 4.27 K 2.95 K 1.90 K 1.41 K 1.17 K 0.94 K 0.71 K 0.62 K 0.33 K 0.23 K0 5 10 15 20 250102030405060Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.67 a-axis0 5 10 15 20 250102030405060Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.67 a-axis (disordered)0 5 10 15 20 2501020304050Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.75 a-axis0 5 10 15 200510152025Rs (10-5 /sq.)Frequency (GHz) 7 K 5 K 3 K 1.4 K6.80 a-axis0 5 10 15 20 25051015202530Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.85 a-axis0 5 10 15 20 25051015202530Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.89 a-axis0 5 10 15 20 2505101520Rs (10-5 /sq.)Frequency (GHz) 9 K 7 K 5 K 3 K 1.3 K6.998 a-axisFigure 5.1: The extracted a-axis Rs(ω, T ) for several dopings of YBCO.1345.3. Results0 5 10 15 20 250510152025306.49 b-axisRs (10-5 /sq.)Frequency (GHz) 4.27 K 2.94 K 1.90 K 1.42 K 1.16 K 0.95 K 0.73 K 0.62 K 0.35 K 0.23 K0 5 10 15 20 25051015202530Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.67 b-axis0 5 10 15 20 2505101520253035Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.67 b-axis (disordered)0 5 10 15 20 250510152025Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.75 b-axis0 5 10 15 2002468101214Rs (10-5 /sq.)Frequency (GHz) 7 K 5 K 3 K 1.4 K6.80 b-axis0 5 10 15 20 2502468101214161820Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.85 b-axis0 5 10 15 20 250246810121416Rs (10-5 /sq.)Frequency (GHz) 12 K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2.2 K 1.4 K6.89 b-axisFigure 5.2: The extracted b-axis Rs(ω, T ) for the measured dopings of YBCO.1355.3. Resultsintermediate-doping points, including the a-axis YBCO 6.67 and 6.75 measurements (to bediscussed extensively throughout the rest of this chapter), as well as the YBCO 6.85 data(which appears somewhat anomalous in this regard, bearing a greater resemblance to dataat lower dopings than to its neighbours).5.3.2 Conductivity width Γ(T )As useful byproducts of the procedure for extracting σ1(ω, T ) data from measured Rs(ω, T )data, one obtains the fit parameters Γ, σ0, y, and σ1D from our phenomenological modi-fied Drude conductivity lineshape model. One of the most important (and most physicallymeaningful) parameters is the generalized linewidth Γ (or equivalently, the scattering rateor inverse scattering lifetime τ−1). While its use with a frequency exponent different from2 precludes formally identifying it with a true scattering rate, it is nonetheless the charac-teristic frequency (or time) scale of the quasiparticle conductivity.In Figures 5.5 (a axis) and 5.6 (b axis) we see comparison between the results for theconductivity widths Γ(T ) from fits with all free parameters (top) with constrained fits withthe fixed parameters y = 1.5 and σ1D = 0. In both cases, and for both axes, the width Γ(T )is linear in temperature, extrapolating to a finite zero-temperature width, as previouslyobserved for 6 + x = 6.5 and 7.0 [145]. The scatter in the data is noticeably reduced forthe constrained fits compared to the free fits; the width Γ is heavily correlated with both yand σ1D in fits, and leaving them free may present too large a parameter space to reliablyextract fit parameters. In the b axis data, this does appear to make a quantitative differencein the intercepts.The upturn at low temperature for the 6.49 sample is correlated with a low temperaturedownturn observed in the exponent parameter y. This interesting behaviour observed atdilution refrigerator temperatures has been investigated, but still requires further experi-mental work. We point out here the large widths Γ(T ) seen for the 6.67 and 6.75 data, butonly in the a axis – a topic we shall return to in Section 5.4.5.3.3 Frequency exponent y(T )With a frequency exponent y = 2, our modified Drude model would simply reduce to theconventional version. However, the data is fit poorly to y = 2, and fits far better to y ≈ 1.5,as we will now see.1365.3. Results0 5 10 15 20 250123456789101 (106 -1 m-1)Frequency (GHz)6.49 a-axis 0.23 K 0.34 K 0.62 K 0.72 K 0.95 K 1.17 K 1.4 K 1.9 K 3 K 4.27 K0 5 10 15 20 250510152025301 (106 -1 m-1)Frequency (GHz)6.67 a-axis 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K 12 K0 5 10 15 20 250510152025301 (106 -1 m-1)Frequency (GHz)6.67 a-axis (disord.) 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K 12 K0 5 10 15 20 25051015202530351 (106 -1 m-1)Frequency (GHz)6.75 a-axis 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K 12 K0 5 10 15 20 25010203040501 (106 -1 m-1)Frequency (GHz)6.80 a-axis 1.4 K 3 K 5 K 7 K0 5 10 15 20 250204060801 (106 -1 m-1)Frequency (GHz)6.85 a-axis 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K 12 K0 5 10 15 20 2501020304050601 (106 -1 m-1)Frequency (GHz)6.89 a-axis 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K 12 K0 5 10 15 20 250204060801001201 (106 -1 m-1)Frequency (GHz)6.993 a-axis 1.3 K 3 K 5 K 7 K 9 KFigure 5.3: The extracted a-axis σ1(ω, T ) for several dopings of YBCO.1375.3. Results0 5 10 15 20 2505101520251 (106 -1 m-1)Frequency (GHz)6.49 b-axis 0.23 K 0.34 K 0.62 K 0.72 K 0.95 K 1.17 K 1.4 K 1.9 K 3 K 4 K 4.27 K0 5 10 15 20 25010203040506070801 (106 -1 m-1)Frequency (GHz)6.67 b-axis 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K 12 K0 5 10 15 20 25010203040506070801 (106 -1 m-1)Frequency (GHz)6.67 b-axis (disord.) 12 K 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K0 5 10 15 20 250204060801001 (106 -1 m-1)Frequency (GHz)6.75 b-axis 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 10 K 12 K0 5 10 15 20 2501020304050601 (106 -1 m-1)Frequency (GHz)6.80 b-axis 1.4 K 3 K 5 K 7 K0 5 10 15 20 25010203040506070801 (106 -1 m-1)Frequency (GHz)6.85 b-axis 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K 12 K0 5 10 15 20 25010203040501 (106 -1 m-1)Frequency (GHz)6.89 b-axis 1.4 K 2.2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K 10 K 12 KFigure 5.4: The extracted b-axis σ1(ω, T ) for several dopings of YBCO.1385.3. Results0 2 4 6 8 10 1202468101214a (1010 s-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89 6.993a-axis, free y and 1D0 2 4 6 8 10 1202468101214a (1010 s-1)T (K) 6.49 6.67 6.75 6.80 6.85 6.89 6.993a-axis, y = 1.5, 1D = 0Figure 5.5: Measurements of Γ(T ) in the a-axis, for (top) free and (bottom) constrainedfits.1395.3. Results0 2 4 6 8 10 1201234567b (1010 s-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89b-axis, free y and 1D0 2 4 6 8 10 1201234567b (1010 s-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89b-axis, y = 1.5, 1D = 0Figure 5.6: Measurements of Γ(T ) in the b-axis, for (top) free and (bottom) constrainedfits.1405.3. ResultsIn Figure 5.7 are plotted the values of the shape exponent y for both the a- and b-axisdata, extracted from fully free fits. The results are somewhat scattered, but with mostcurves clustered around y ≈ 1.5, with a general positive slope with temperature. For somecurves, exponents below 1 are observed, a result which physically cannot hold up to highfrequency, as the resulting spectral weight (proportional to the integral of conductivity)would be infinite, as can be seen in Equation 5.2 below.We note in passing that a conductivity frequency exponent different from 2 has beenproposed as a signature of quantum critical [146] or “nodal metal” [147] behaviour. Thesestudies have been done in the optical regime, and it is uncertain to what extent theirconclusions apply here.5.3.4 Offset conductivity σ1D(T )Previous work has found that, in addition to the frequency-dependent Drude-like conductiv-ity term, a frequency-independent offset conductivity σ1D was needed to describe the data– but only for the b direction. This has naturally been ascribed to broad one-dimensionalconductivity contributions from the CuO chains – contributing to b-axis conductivity only– which have heavy scattering due to chain defects [119, 145]. This broad scattering isindistinguishable within our frequency window from a small constant offset. This term willalso serve to absorb the effects of any contamination from c-axis conductivity on the sidefaces of the sample, which would give a similarly broad contribution also presenting as aconstant offset.The resulting σ1D data, extracted from free fits, is shown in Figure 5.8 for both axes.These free-fit results are again rather noisy, but we can still see that only in the b axis isthere consistent evidence for a positive σ1D contribution, which is seen to have a positiveslope with temperature.Although such free fits provide – as they must – slightly better fits to the conductivitylineshapes (and have thus been used for extracting the σ1 from Rs for the curves shownin Figures 5.3 and 5.4), this additional freedom is seen to produce too much variance inthe heavily correlated parameters Γ, y, and σ1D, obscuring the underlying temperature anddoping dependence. For all of the data which follows, we shall therefore fix the exponentparameter to y = 1.5 and the conductivity offset parameter to σ1D = 0 for both axes.1415.3. Results0 2 4 6 8 10 120.60.81.01.21.41.61.82.02.22.4yTemperature (K) 6.49 6.67 6.75 6.80 6.85 6.89 6.993a-axis, free y and 1D0 2 4 6 8 10 120.60.81.01.21.41.61.82.0yTemperature (K) 6.49 6.67 6.75 6.80 6.85 6.89b-axis, free y and 1DFigure 5.7: Extracted values of y(T ) in both axes and all dopings, from free fits.1425.3. Results0 2 4 6 8 10 12-10-8-6-4-20241D,a (106 -1 m-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89 6.993a-axis, free y and 1D0 2 4 6 8 10 12-3-2-1012341D,b (106 -1 m-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89b-axis, free y and 1DFigure 5.8: Results for σ1D(T ) in both axes and all dopings, from free fits.1435.3. Results5.3.5 Conductivity height σ0(T )The extracted conductivity height σ0 is shown in Figure 5.9. Here, the height behaviourdescribed in Section 5.3.1 can be seen more clearly; the relative change in height withtemperature increases with doping. Note that σ0 scales as λ3, and is therefore highlydependent on λ(T ) (whose uncertainty is not accounted for here); this makes comparisonbetween the scale of curves at different dopings less reliable, but affects their overall shapesless strongly, as λ3(12 K)λ3(0 K) is known with greater certainty.This conductivity height data can also be shown as a function of doping for severaltemperatures, as shown in Figure 5.10. One can also extract an estimated zero temperatureintercept, shown in orange here. This quantity is equivalent (assuming the model fully de-scribes the data) to the σ00 defined in Equation 4.18 in Section 4.7.2. Using that expressionalong with values of αsFL2 vFv2 inferred from penetration depth (see Figure 4.16), and with anarbitrary value of βVC for the vertex correction factor, we find the resulting estimate of σ00to be comparable to the value extrapolated from surface resistance measurements shownhere. (Once again here, we find the YBCO 6.85 sample at p = 0.15 to be an anomaly.)If we divide the surface resistance extrapolated σ00 value by αsFL2 vFv2 (and appropriateprefactors) we can obtain an estimate for the vertex correction factor βVC, shown in Figure5.11.5.3.6 Integrated spectral weightAssuming that the modified Drude model of Equation 5.1 accurately models our data,we may integrate this analytical model to obtain the normal fluid (quasiparticle) spectralweight, as described in Section 2.1.4. This can be compared to the loss of superfluid spectralweight with increasing temperature extracted from penetration depth measurements tocheck the relationship of Equation 2.34.Integrating the modified Drude model without the σ1D term,58 we find for the spectralweight:2πˆ ∞0σ1n(ω, T ) dω =2πˆ ∞0σ0(T )1 + (ω/Γ(T ))y(T )dω = 2Γ(T )σ0(T )y(T ) sin πy(T ). (5.2)This resulting spectral weight has been extracted from the approprate fit parameters (for58Since the σ1D term is a component of unknown width (and thus area), and since it is believed to corre-spond to a chain conductivity channel that is not thought to exhibit superconductivity in most circumstances,we omit this term from the spectral weight integration.1445.3. Results0 2 4 6 8 10 120204060801001201400,a (106 -1 m-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89 6.993a-axis, y = 1.5, 1D = 00 2 4 6 8 10 1201020304050607080900,b (106 -1 m-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89b-axis, y = 1.5, 1D = 0Figure 5.9: Results for σ0(T ) in both axes and all dopings, from fits with fixed y = 1.5 andσ1D = 0.1455.3. Results0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.190102030405060700,a (106 -1m-1)Hole doping p 7.2 K 5 K 3 K 1.4 K Extrap. 0 K 00 from a; VC = 16a axis0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.190102030405060708090100b axis0,b (106 -1m-1)Hole doping p 7.2 K 5 K 3 K 1.4 K Extrap. 0 K 00 from b; VC = 16Figure 5.10: Results for σ0(T ) in both axes, plotted as a function of doping, from fits withfixed y = 1.5 and σ1D = 0. We also show an extrapolation of σ0 to T = 0, and a comparisonof this using Equation 4.18 with estimates from penetration depth αsFL2 vFv2 values, alongwith βVC = 16 as an example.1465.3. Results0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.1901020304050607080VCHole doping p a-axis b-axisFigure 5.11: Plot of the vertex correction factor βVC as a function of doping, as extractedfrom combining βVCαsFL2 vFv2 data from σ00 with values of αsFL2 vFv2 from low-temperaturepenetration depth slopes.1475.4. The effects of oxygen chain disorder on conductivityfits with fixed y = 1.5 and σ1D = 0), and is shown as a function of temperature in Figure5.12.We see that, in each case, the spectral weight nne2m∗ is linear in temperature, as expectedfrom the linear T dependence of nsm∗ seen from penetration depth measurements. Thespectral weights can be seen here to extrapolate to finite value at low temperature – aresidual spectral weight that has been previously noted [145]. Notably, over much of thedoping range there is little doping dependence observed in Figure 5.12.In Figure 5.13 we show the normal fluid spectral weight as a function of doping, forboth axes and at four temperatures. The spectral weights of the a and b axes track eachother well for all dopings and temperatures. An overall trend of increased spectral weightwith doping is observed, though with a plateau for intermediate doping.Finally, we compare the increase in normal fluid spectral weight with temperature withthe lost superfluid spectral weight from penetration depth, to check the validity of Equation2.34; the results are shown in Figures 5.14 and 5.15 for the a and b-axis measurements,respectively. Examining the a-axis first, we see that over most of the doping range, the lostsuperfluid spectral weight tracks the increase in normal fluid spectral weight reasonablywell, with a small residual normal fluid spectral weight at T = 0. The exception is the 6.99data, for which the normal fluid spectral weight growth with temperature outpaces the lossof superfluid spectral weight.For the b-axis data, however, the slope of the measured loss in superfluid spectral weightis greater than the corresponding gain in normal fluid spectral weight for all dopings. Whilethe reason for this disagreement remains unclear, some theoretical work has predicted vi-olations of this Ferrell-Tinkham-Glover sum rule within the Yang-Rice-Zhang model, [57],although in-plane anisotropy was not considered.5.4 The effects of oxygen chain disorder on conductivity5.4.1 Production and characterization of the disordered sampleWe now return to the observation from Section 5.3.2 that the measured conductivity widthsΓ(T ) are significantly larger for YBCO 6.67 and 6.75 than for neighbouring dopings –but only along the a axis. Leaving aside the issue of anisotropy for the moment, it wasunexpected that the YBCO 6.67 and 6.75 samples – nominally well-ordered ortho-VIII and1485.4. The effects of oxygen chain disorder on conductivity0 2 4 6 8 10 12051015202530354045505560nn,ae2/m* (1017 -1 m-1 s-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89 6.993a-axis, y = 1.5, 1D = 00 2 4 6 8 10 120510152025303540455055nn,be2/m* (1017 -1 m-1 s-1)Temperature (K) 6.49 6.67 6.75 6.80 6.85 6.89b-axis, y = 1.5, 1D = 0Figure 5.12: Results for the integrated spectral weight in both axes, plotted as a functionof temperature.1495.4. The effects of oxygen chain disorder on conductivity0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.190510152025303540nne2/m* (1017 -1m-1s-1)Hole doping p 7.2 K 5 K 3 K 1.4 Ka-axis: solidb-axis: dashedFigure 5.13: Results for the integrated spectral weight in both axes, plotted as a functionof doping.ortho-III samples, respectively – would have greater conductivity than the less well-orderedsamples of YBCO 6.80, 6.85, and 6.89, in any direction. In order to investigate the effectsof chain order, the same ortho-VIII YBCO 6.67 sample that had been previously measuredwas remeasured with the broadband bolometric spectroscopy probe after its chain oxygenswere disordered.To disorder the CuO chain oxygens, the sample was sealed into an evacuated quartztube and heated to ∼ 100 ◦C for one hour; at this temperature, the oxygen is sufficientlymobile to become fully randomized within the CuO chain layer, but will not leave thesample (and thereby alter the total oxygen content). After this time, the sample tubeis quickly quenched from 100 ◦C to 0 ◦C by immersion in an icewater bath, in order tofreeze in the disordered oxygen configuration. Because the oxygen atoms are sufficientlymobile at room temperature to begin reassembling into the ortho-VIII phase (which is thethermodynamically favoured phase for YBCO 6.67 below 45 ◦C), the sample must be keptin a freezer between measurements. Unfortunately, sample mounting and probe preparationinevitably requires that the sample spend time at room temperature between measurements.1505.4. The effects of oxygen chain disorder on conductivity012345678010203040506001020304050051015202530354005101520253035400 2 4 6 8 10 1205101520253035400 2 4 6 8 10 120102030405060nn,ae2/m* (1017 -1m-1s-1) Integrated 1a( ) (1/2(T))a-axis 6.49 Integrated 1a( ) (1/2(T))a-axis 6.67nn,ae2/m* (1017 -1m-1s-1) Integrated 1a( ) (1/2(T))a-axis 6.75 Integrated 1a( ) (1/2(T))a-axis 6.80nn,ae2/m* (1017 -1m-1s-1) Integrated 1a( ) (1/2(T))a-axis 6.85Temperature (K) Integrated 1a( ) (1/2(T))a-axis 6.89nn,ae2/m* (1017 -1m-1s-1)Temperature (K) Integrated 1a( ) [6.993] (1/2(T)) [6.998]a-axis 6.99Figure 5.14: Comparisons of integrated spectral weight from σ1 and ∆λ measurementsfor the a axis. For the 6+x=6.99 data, the YBCO 6.998 data from Chapter 4 are usedfor penetration depth, but previous measurements of YBCO 6.993 (by Patrick Turner andRichard Harris) are used for σ1.1515.4. The effects of oxygen chain disorder on conductivity024681012140510152025303540455001020304050010203040500 2 4 6 8 10 120510152025303540450 2 4 6 8 10 12051015202530354045nn,be2/m* (1017 -1m-1s-1) Integrated 1b( ) (1/2(T))b-axis 6.49 Integrated 1b( ) (1/2(T))b-axis 6.67nn,be2/m* (1017 -1m-1s-1) Integrated 1b( ) (1/2(T))b-axis 6.75 Integrated 1b( ) (1/2(T))b-axis 6.80nn,be2/m* (1017 -1m-1s-1)Temperature (K) Integrated 1b( ) (1/2(T))b-axis 6.85Temperature (K) Integrated 1b( ) (1/2(T))b-axis 6.89Figure 5.15: Comparisons of integrated spectral weight from σ1 and ∆λ measurements forthe b axis.1525.4. The effects of oxygen chain disorder on conductivityThe temperature history of the sample – in particular, the cumulative time spent at roomtemperature – was thoroughly documented at all times.Shown in Figure 5.16 are measurements of the Tc of the disordered YBCO 6.67 sample(from which we can extract the hole doping p, as described in Section 4.1.2), taken witha SQUID MPMS dc magnetometer at regular intervals, as a function of the cumulativetime spent at room temperature after disordering. To understand the observed increase indoping with time, we note that a single isolated oxygen atom in the CuO chain layer withoutany neighbouring oxygens along the chain does not contribute any holes to the CuO2 planelayers; however, an isolated pair of chain oxygens contributes a single hole, three oxygens ina row contribute two holes, and so on – i.e., each nearest neighbour pairing of chain oxygenatoms contribute one hole. Therefore, as isolated oxygens begin to order and form chains,the hole doping will increase.In Figure 5.16 it is shown that the time dependence of the Tc resulting from the chainoxygens reordering with time follows a stretched exponential (also known as a Weibull) form,characteristic of systems with dynamics on a distribution of timescales. Unsurprisingly, wesee that the spontaneously reordering sample will not regain the original degree of disorderproduced by careful annealing, since it is asymptotically approaching a Tc which is ∼0.7 Klower than its original value measured before disordering. We use the observed stretchedexponential data to interpolate for estimates of the doping of the sample at the time of Rsmeasurement, which differed between the two orientations due to the unavoidable time atroom temperature to rotate the orientation of the sample. The appropriate Tc values of thesamples interpolated for the time of measurement are show in green in the bottom plot ofFigure 5.16.The results of the conductivity measurements of the disordered samples were alreadyshown in Figures 5.1 (a-axis Rs), 5.2 (b-axis Rs), 5.3 (a-axis σ1) and 5.4 (b-axis σ1); tohighlight the difference, in Figure 5.17 we show only the 6.67 data at 3 K, for the orderedand disordered states and for both axes. The results are striking – for the a-axis, thedisordered sample has a smaller conductivity width (equivalently, scattering rate) thanthe ordered sample! On the other hand, the b-axis conductivity is changed much lessdramatically. (Note that a narrow conductivity feature below ∼1 GHz is seen for both theordered and disordered samples, but this is not the width being considered.)This unexpected and counterintuitive behaviour is at odds both with common experience1535.4. The effects of oxygen chain disorder on conductivity0 600 1200 1800 2400 3000 3600 4200 4800 5400 600063.063.263.463.663.864.064.264.464.6  Tc (K)Time at ~295 K after quench (minutes)Tc [K] = 64.013 - (64.013 - 61.957) e-(t/332 min. )0.600Ordered Tc =  (64.664 ± 0.007) K100 100062.862.963.063.163.263.363.463.563.663.763.863.964.064.1Time (min) Tc (K)250 ± 10 63.1303 ± 0.0049367.5 ± 17.5 63.2971 ± 0.0041607.5 ± 17.5 63.5428 ± 0.0085967.5 ± 17.5 63.7018 ± 0.00781567.5 ± 17.5 63.8447 ± 0.00994935 ± 22 65.0001 ± 0.0045325-355 min.63.27 ± 0.02 K  Measured Tc Stretched exponential fit Exponential fitbTc (K)Time at ~295 K after quench (minutes)a63.03 ± 0.02 K195-210 min.0.1120.1130.1140.1150.116Hole doping pFigure 5.16: Measurements of the time dependence of Tc for disordered ortho-VIII samplesreordering at room temperature. SQUID magnetometry measurements were performedbefore disordering and at various times after, to track the superconducting transition tem-perature (and therefore hole doping) as the samples reorder, to allow estimation of thedopings at the time of the bolometry measurements. (Top) A fit to the stretched exponen-tial (Weibull) time dependence of Tc. Note that the sample is asymptotically approaching alower Tc than its previous well-ordered state. (Bottom) A comparison of fits for stretchedexponential and simple exponential time dependence, and the estimates for the Tc valuesat the time of the bolometry measurements of the disordered sample.1545.4. The effects of oxygen chain disorder on conductivity0 2 4 6 8 10 12 14 16 18 200510152025301 (106 -1 m-1)Frequency (GHz) a-axis O-VIII ordered a-axis disordered b-axis O-VIII ordered b-axis disorderedYBCO6.67 at T = 3 KFigure 5.17: Comparison of the conductivity at 3 K for YBCO 6.67 before and after chainoxygen disordering, for both axes.1555.4. The effects of oxygen chain disorder on conductivityand with previous experiments on ortho-II YBCO 6.5 [65, 66]. For these experiments, thechain oxygens were disordered, but the conductivity was observed to increase for both axes.The explanation proposed here lies in the nature of the chain order in these systems.The top drawing of Figure 5.18 is an illustration of perfect ortho-VIII order in a CuOchain layer. In real samples, however, chain oxygen order forms in domains which are ofdifferent offset phases, as shown in the middle of Figure 5.18.59 These domain boundariesare extended defects which produce extended scattering potentials in the neighbouringCuO2 plane layers; such extended potentials have been shown to be extremely effectivescatterers [148, 149]. In contrast, while the disordered chain layer configuration (as seen inthe bottom of Figure 5.18) exhibits a far greater number of scatterers, these are pointlikeand more homogeneously distributed, and thus may produce overall weaker scattering inthe CuO2 planes than the domain boundaries of the ordered phase.In response to the question of why these effects are seen only in the a-axis conductivity,and only for the YBCO 6.67 and 6.75 samples, we turn to Figure 5.19, a plot of thecorrelation lengths for the oxygen chain order superlattice peaks from x-ray scatteringmeasurements (roughly identifiable with chain oxygen order domain size) for YBCO asa function of oxygen content and axis. In different terms, the correlation lengths showncorrespond to the average distance between domain walls in that direction. Over the fulldoping range shown, the b-axis correlation lengths are a factor of two or more greater thanthat along the a axis. For the particular oxygen contents in question (6.67 and 6.75) thea-axis correlation lengths are very short (10–25 A˚), while the b-axis correlation lengths aresignificantly longer. The claim here is that the a-axis scattering lifetime is limited by theshort distance between domain boundaries, whereas these are not the dominant scatteringmechanism for the b-axis conductivity. Therefore, by disordering the chain-layer oxygens,the a-axis scattering is decreased without significantly altering the b-axis conductivity.On the other hand, for YBCO 6.50, the oxygen ordering correlation lengths (and thusdistances between domain walls) are sufficiently large so as to not be a significant contri-bution to scattering for either axis; instead, the additional scattering from the chain endsof the disordered sample will give an overall increase in scattering upon disordering.Additional support for this scenario – and possibly where the results of this thesis work59Even this is an idealization; the reality, as revealed by topographic images taken by scanning tunnellingmicroscopy – shows very ragged boundaries, which precipitate many defects.1565.4. The effects of oxygen chain disorder on conductivityabFigure 5.18: A schematic picture contrasting extended and point defects for an ortho-VIIIordered sample. (Top) Perfect ortho-VIII chain ordering. (Middle) An extended domainboundary defect, where ordered domains with a phase shift meet. Such extended defects canproduce extremely strong scattering potentials in the neighbouring CuO2 planes. (Bottom)A partially disordered sample, retaining little of the original ortho-VIII chain order. Thescattering sites (chain ends) are far more numerous – but substantially weaker and morehomogeneous, particularly as seen a distance away in the adjacent CuO2 planes.1575.4. The effects of oxygen chain disorder on conductivity		 	   Figure 5.19: Measurements of the x-ray correlation length of the oxygen chain order super-lattice peaks – indicative of domain size – in YBCO as a function of doping for both axes.Open data points are from Zimmermann et al. [11], and solid data points are measurementsof Ruixing Liang taken at UBC [150]. The distance between domain boundaries is severaltimes longer along the b axis than for the a axis for all dopings. Until recently, all successfulquantum oscillations measurements of YBCO have been performed in the region with largecorrelation lengths near 6+x=6.5; since the cyclotron orbits for such measurements sam-ple both in-plane directions, the mean free path along the a direction may be the limitingfactor. Figure provided courtesy of Ruixing Liang.1585.4. The effects of oxygen chain disorder on conductivitywill have the greatest impact – comes from quantum oscillations measurements. Informedby this discovery that disordered samples may display decreased scattering, a recent projectto measure disordered samples has been carried out [67]. The amplitude of quantum os-cillations is exponentially damped by the inverse of the quasiparticle lifetime [118] (orequivalently, the mean free path), and thus even a small decrease in scattering could makea dramatic improvement in such measurements. Historically, strong quantum oscillationsmeasurements (in which samples are measured in very high magnetic fields of order tens oftesla) have been limited to the region of doping around 6 + x = 6.5, where electron meanfree paths are sufficiently long to allow for complete cyclotron orbits between scatteringevents; it is thus no coincidence that this region coincides with the region of large correla-tion lengths in Figure 5.19. In fact, the mean free path of ∼350 A˚ for YBCO 6.59 takenfrom quantum oscillation measurements [117] is quite close to the correlation length at thatdoping inferred from Figure 5.19, strongly suggesting that CuOx oxygen chain order domainboundary scattering is the dominant (and thus limiting) scattering mechanism even in thisregion of the phase diagram.The recent quantum oscillation work on disordered samples has found that the chain-oxygen-disordered YBCO samples do indeed demonstrate decreased scattering [67], allowingquantum oscillations measurements to be made up to much higher dopings, a previouslyinaccessible region of the phase diagram. This provides even stronger evidence for thehypothesis that disordered samples can show decreased scattering, extending this conclusionto higher oxygen contents.Furthermore, the results of this study – which has been submitted but still awaitspublication – show a divergent electronic effective mass m∗ as the doping p = 0.18 isapproached. This is striking evidence for the very quantum critical point which this projectsought to uncover, and certainly represents the largest immediate impact of this thesiswork; as discussed in Section 4.8, the absence of any large associated peak in the magneticpenetration depth upon approaching this doping places an important constraint on futuretheory.5.4.2 The relationship to charge density wave orderThe most clear final representation of the main result of this conductivity doping dependencescan – the behaviour of the conductivity width Γ(T ) as a function of doping and disorder1595.4. The effects of oxygen chain disorder on conductivity0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.190102030405060708090100110120ord.disord. (109 s-1)Hole doping p 7.2 K 5 K 3 K 1.4 K6.896.856.806.756.676.51 6.993a (solid)b (dashed)Figure 5.20: Plot of the conductivity width Γ as a function of doping for both axes. Fitswere performed from 1 to 20 GHz with fixed y = 1.5 and σ1D = 0, except for the x =6.51and 6.993 points (as described in the text).1605.4. The effects of oxygen chain disorder on conductivity– is shown in Figure 5.20. (For the purposes of this plot, we have used measurementsfrom different samples for the highest and lowest dopings from before; measurements of adifferent ortho-II YBCO 6.51 sample previously taken by this author at higher temperatureswere used, along with fit results for a different YBCO 6.993 sample taken for both axesby Jake Bobowski, but for which the original conductivity data was unavailable. Thesemeasurements agreed with those of the samples previously shown for these dopings wherethey overlap, but provide measurements of Γ(T ) at the temperatures we wish to show in thisplot, for which data was unfortunately not collected for the previously-discussed samples.)Ignoring the a-axis YBCO 6.67 and 6.75 data for the moment, we see the general trendof increasing Γ(T → 0) and decreasing dΓdT with increasing doping, progressing relativelysmoothly from 6.5 to 7.0. As has been discussed, this trend is interrupted by a large peakin conductivity width in the a axis in the doping range p = 0.12–0.13, which appearsto be at least attenuated for samples whose chain oxygens are disordered. The proposalabove explains this phenomenology in terms of the strong scattering associated with domainboundaries.However, an alternate proposal naturally presents itself when examining the location ofthe conductivity width peak of p ≈ 0.125 – this doping range is where charge density wave(CDW) order is known to be strongest in other cuprates, and where it has been recentlyobserved in YBCO [151–154]. Recent measurements have also shown an interplay betweenoxygen order and CDW amplitude [155]. It is therefore prudent to consider whether theCDW order could be influencing the microwave conductivity in this region.In light of the evidence from quantum oscillations measurements, it appears that domainboundary scattering plays an important role, and the CDW cannot be solely responsible forthe observed behaviour. However, we can’t exclude the possibility of charge order playinga role, however subtle – particularly since the oxygen chain order is known to have aninfluence on the charge order.161Chapter 6Low-energy µSR measurements ofYBa2Cu3O6+x andBa(Co0.074Fe0.926)2As26.1 Introduction to µSRThe measurement technique known as µSR – an acronym for muon spin rotation, resonance,or relaxation, depending on context – is one of the more exotic experimental techniquesin the toolbox of condensed matter physics. Unlike similar accelerator- or reactor-basedtechniques such as synchrotron light spectroscopy and neutron scattering, µSR requiresintense proton beam sources for muon production, currently found at only four facilities inthe world. However, it provides unique measurement capabilities unavailable to other bulkmagnetism probes.The basic premise of µSR involves the implantation of spin-polarized muons into asample, at depths which depend on energy, ranging from just several nanometres belowthe surface (for the sub-keV muons possible with low-energy µSR) to over a millimetre formore common high-energy (∼4 MeV) muons [156], often probing an entire sample nearlyuniformly. Once stopped60 in the sample, the muon spins interact with their surroundingmagnetic environment for the duration of their 2.2 µs lifetimes. When a muon decays, itemits a positron61 in an angular probability distribution biased towards the direction of itsspin at the time of decay. Such positrons can easily escape from even the thickest samples60Once slowed, muons may continue to diffuse through the sample, which can have interesting measure-ment consequences and can be an interesting topic in its own right [157]. However, with a hopping rate of≪ 1 µs−1 below room temperature in YBCO [158], this diffusion length scale will be at most several latticeconstants during the muon lifetime, much smaller than the magnetic penetration depth length scale beingprobed here.61Or an electron, for µ− – but nearly all µSR is performed with positive muons; µ− form muonic atomsin normal matter, and undergo nuclear capture at a rate comparable to decay, complicating both their useand production.1626.1. Introduction to µSRand even most cryostat walls, and measurements of the positron distribution asymmetrythus provide a window into the distribution and fluctuations of magnetic fields deep in thebulk of samples.This advantage is shared with neutron scattering, which also probes the bulk of a sam-ple, and which carries the added advantage over µSR of being an energy- and momentum-resolved probe. On the other hand, µSR is sensitive to magnetic fluctuations on timescaleslonger than that of neutron scattering. Additionally, because the muon stops at specificsites within a crystalline unit cell, it can provide insights into intra-unit-cell magnetic struc-ture when additional knowledge of the stopping sites is available. Thus µSR and neutronscattering measurements are very complementary techniques for probing bulk magnetism.Where the µSR technique truly finds its niche is in the study of superconductors: themuons can probe magnetic fields and their spatial and temporal variations deep in the bulksamples in the superconducting state, where most other probes of magnetism would befoiled by Meissner screening. The Meissner state itself is not typically used for high energybulk µSR measurements of the penetration depth – almost all muons stop deep in thesample where 〈B〉 = 0. For the purpose of measuring superconducting penetration depthsand coherence lengths, bulk µSR measurements are done in the vortex lattice state of thesuperconductor, in an applied magnetic field H ‖ cˆ well above the lower critical field Hc1.With knowledge of the lattice configuration of the vortices, the average in-plane penetrationdepth λab =√λaλb and the average in-plane coherence length ξab =√ξaξb may be extractedfrom fits to the measured distribution of fields experienced by the muons. While certainlya powerful technique, vortex lattice bulk µSR does not allow a determination of λa and λbseparately; it also relies on the correct identification of the vortex lattice configuration, andis therefore sensitive to any vortex lattice disorder which may be present in real samples[159]. Perhaps more problematic is the requirement of relatively high magnetic fields toproduce the ordered flux lattice (> 1 kOe, compared to the < 50 Oe fields possible forlow-energy µSR), and inevitable issues related to the required extrapolations to zero field;possible variations in the vortex lattice structure as a function of magnetic field, as well asthe known nonlinearity in field of Meissner screening in the cuprates [84, 160–162] rendersuch extrapolations uncertain.The low-energy µSR measurements to be described in this chapter take a very differentapproach. For low-energy µSR (LE-µSR), muons are implanted with adjustable energies1636.1. Introduction to µSRof 0–30 keV, rather than at fixed MeV energies. As we shall see, the mean depth ofmuon implantation varies with energy, over a range of ∼1–300 nm62 for the low-energymuons – a range which happens to coincide with the typical range of magnetic penetrationdepths for most superconductors. By measuring the muon precession signal as a functionof implantation energy, the magnetic field depth profile at the surface of a superconductorcan be reconstructed: a difficult feat to accomplish with any other technique.63In this chapter, low-energy µSR measurements of YBa2Cu3O6+x as well as of the pnic-tide superconductor Ba(Co0.074Fe0.926)2As2, will be presented. These measurements wereperformed with the group of Dr. Robert Kiefl. For the YBa2Cu3O6+x measurements, thepenetration depth values acquired are a key component of the doping scan work that is thefocus of this thesis. While the Ba(Co0.074Fe0.926)2As2 measurements are auxiliary to thismain theme, the microwave penetration depth measurements carried out in support of thelow-energy µSR studies complement the work done elsewhere in this thesis, and were in-strumental in the comparison of different measurement techniques and conductivity modelswhich resulted from this work.6.1.1 Muon decay asymmetryThe use of µSR as a local probe of the magnetic field inside a sample is made possible by theparity violation of the weak interaction which governs muon decay, manifesting itself in anasymmetric spatial distribution of daughter leptons (electrons or positrons) that dependsstrongly on the spin direction of the parent muon. Measurement of this decay asymmetrycan therefore provide information about the spin evolution of a muon, wherever it can beplaced.In the Standard Model, the differential decay rate for a muon of either charge (µ±) canbe written (in c = 1 units, averaging over outgoing e± spin polarization, and neglectingneutrino masses and radiative corrections) as [165, 166]d2Γdx d cos θ =mµ2π3W4eµG2F√x2 − x20 (FIS(x)± FAS(x)Pµ cos θ) , (6.1)62Implantation depth profiles vary somewhat with sample composition, mostly dependent on the densityand nuclear charge Z of its constituent atoms; for YBa2Cu3O6.52, we find a maximum depth of ∼200 nm(see Figure 6.5).63A closely related technique known as β-detected nuclear magnetic resonance (β-NMR) uses a verysimilar measurement method in which radioactive ions (such as 8Li) are implanted into a sample, with thenuclear decay asymmetry reflecting the spin polarization [163, 164]; this technique is quite complementaryto low-energy µSR.1646.1. Introduction to µSRW+µ+ν¯µνee+Figure 6.1: Feynman diagram of the dominant decay process for positive muons µ+ →e++νe+ ν¯µ; time proceeds from left to right. For µ− decay, all fermion arrows are reversedand all charges are conjugated.01|Pµ|Pµ+e+Figure 6.2: The asymmetric angular distribution of muon decay averaged over energy,as a function of muon spin polarization Pµ. In contrast to the maximally asymmetricdistributions for decays with the highest positron energy, the energy-averaged distributionshave a maximum asymmetry of 1/3 for a fully-polarized (Pµ = 1) distribution of muons.1656.1. Introduction to µSRwhere θ is the angle between the initial µ± spin polarization Pµ and the momentum of theoutgoing e±, Weµ ≡ (m2µ+m2e)/2mµ is the maximum e± energy, x ≡ Ee/Weµ is the reducede± energy, x0 ≡ me/Weµ = 9.67 × 10−3 is the reduced e± rest mass, and GF is the Fermicoupling constant. The energy-dependent quantities FIS(x) and FAS(x) are the isotropicand anisotropic distribution functions, given byFIS(x) =x2 −x23 −x206 (6.2)andFAS(x) =13√x2 − x20[x− 1 + 12√1− x20]. (6.3)We see that for the lowest-energy e± (i.e., x → x0), FAS vanishes and the distributionbecomes isotropic; for the highest-energy e± (i.e., x → 1), FAS ≃ FIS = 16 , and the distri-bution becomes maximally asymmetric, with the decay rate nearly vanishing for outgoinge± along (−) or opposite (+) the polarization of the parent µ±, respectively.Integrating over x – as is approximately the case experimentally, where the e± aredetected with an efficiency that is only weakly dependent on energy – the rate becomesdΓd cos θ ≃12τ−1µ(1± 13Pµ cos θ). (6.4)This energy-integrated angular distribution is shown, as a function of polarization Pµ, inFigure 6.2. For the real experimental situation of µSR, the spin polarization Pµ will betime-dependent in both direction and magnitude, resulting in a time-dependent effectiveangle θ → θ(t)− φ for an observer at fixed angle φ. Additionally, since real detectors havefinite spatial extent, experimental positron detector count rates will represent appropriateintegration over this distribution.6.1.2 Surface muonsThe parity violation found in weak processes is not only the key to detecting the spin dy-namics of spin-polarized muons – it is also instrumental in the production of spin-polarizedmuons. Muons produced from pion decay are 100% spin-polarized at formation. In Figure6.3 we show the decay process π+ → µ+ + νµ for a positive pion at rest; for pions, theintrinsic spin S = 0. Conservation of linear and angular momentum thus dictates that the1666.1. Introduction to µSRFigure 6.3: An illustration of the spin alignment of the decay products of pion decay.Because all neutrinos (antineutrinos) are left (right) handed, the spin of the νµ to be an-tiparallel to its linear momentum; because the π+ has S = 0, and as a consequence of theconservation of linear and angular momentum, the same spin anti-alignment is guaranteedfor the µ+ as well.initial linear momenta and spin orientations of the µ+ and νµ are equal and opposite.As a consequence of parity violation in the weak interaction, only left-handed neutrinos(and right-handed antineutrinos) are created through pion decay [167] – that is, all suchdecay neutrinos have helicity h ≡ p·S|p| |S| = −1 (= +1 for antineutrinos). This means thatthe spin of a neutrino is always aligned antiparallel to its momentum – and thus the samewill be true for the initial spin and momentum of a µ+ produced from π+ decay. In otherwords, such muons are 100% spin-polarized from birth.In “meson factory”–type accelerator facilities, a high-energy, high-intensity proton beamis directed into a target, with pions produced in the resulting nuclear collisions. Pions whichstop near the surface of the target produce nearly monochromatic decay muons at an energyof 4.1 MeV, which are 100% spin polarized as discussed; these monochromatic surface muonsare then collected by appropriate beam optics for transport to the experiment.6.1.3 Spin precession and the muon polarization signalJust as with other spin measurement techniques such as NMR and ESR, the central principlebehind µSR measurements is the Larmor precession of the magnetic moment of the muon(associated and co-aligned with its spin) in a magnetic field. While for NMR and ESR, theemf induced by the precessing moments of the spins under observation is the sole meansof detection, for µSR the individual muon decays are detected one by one, with the time-dependent polarization reconstructed from the asymmetric angular distribution statisticsbuilt up over many such decays.For most purposes (including for the work done in this thesis), µSR can be described1676.1. Introduction to µSRperfectly well by the classical dynamics of a magnetic moment µ = γS, where γ is thegyromagnetic ratio (for the muon γµ = 2π×135.54 MHz/T) and S is its spin (the onlyimport from quantum mechanics that will be required here). When a muon is placed in amagnetic field B, its spin will experience a torque τ :τ = dSdt = µ×B = γS×B. (6.5)This torque will result in the Larmor precession of the spin S about the field B, at theLarmor frequency ωL = γµB; if we assign zˆ to the direction of B, the field dependence ofthe classical64 spin components will be〈Sx(t)〉 = Sxy,0 sin(ωLt+ φ) (6.6a)〈Sy(t)〉 = Sxy,0 cos(ωLt+ φ) (6.6b)〈Sz(t)〉 = Sz,0, (6.6c)where Sxy,0, Sz,0, and φ are determined by the initial projection of S. For an ensemble ofmuons whose spin is initially orthogonal to B, this corresponds to a net polarization Pµ(t)which precesses in the plane orthogonal to B – with a corresponding sinusoidal variation inthe asymmetric distribution of decay positrons as observed by detectors in this plane.For a given experimental run, the number N (t) of positrons seen by a detector at time tafter muon deposition will be described byN (t) = N0e−t/τµ(1 +A(t)) +Nbg, (6.7)where N0 is a normalization amplitude proportional to the number of muons deposited,τµ is the 2.2 µs muon lifetime, Nbg is a constant background count depending on runtime,and A(t) ∝ P(t) · nˆ (with nˆ being the direction of the detector65) is the remaining time-dependent asymmetry function accounting for the desired polarization signal. In general,both N0 and A(t) will depend on detector geometry and efficiency.Given a particular homogeneous macroscopic magnetic field B, the asymmetry signal64Since we will be only be measuring decay distributions from an ensemble of spins – and not correlatingorthogonal spin projections of individual spins – this classical picture of well-defined spin projections is valid.65For detectors of finite extent, the appropriate angular integration over the detector should be takeninstead to determine an average polarization.1686.1. Introduction to µSRwill be of the form [3]A(t) = A0e−12 (σt)2 cos (γµBt+ φ); (6.8)here the Gaussian broadening factor σ accounts for inhomogeneous broadening, such asthat due to nuclear dipole moments or any field inhomogeneities that are unaccounted for[2]. If we instead consider an ensemble of muons experiencing a distribution of fields p(B),this becomesA(t) = A0e−12 (σt)2ˆdB p(B) cos (γµBt+ φ). (6.9)For our low-energy µSR measurements, this distribution of fields comes about through theconvolution of a depth-dependent magnetic field profile B(z) with a distribution ζz of muonimplantation depths z; the resulting signal is thenA(t) = A0e−12 (σt)2ˆdz ζ(z) cos (γµB(z)t+ φ). (6.10)Motivated by observations, rather than a pure London exponential profile for B(z), we havefound it necessary to model our data with a modified profile including a dead layer – a layerof depth d ∼ 5–30 nm below the surface in which no diamagnetic currents flow and noMeissner screening takes place – or at least these are heavily suppressed. This has beenpreviously observed for YBCO [168], and has been observed in this work for both YBCO[3] and iron pnictides [2]. The motivation for such a dead layer can be seen when examiningthe form of the mean magnetic field data as a function of mean implantation depth (see anyof Refs. [1, 3, 169]), where a suppression of screening near the surface is required to matchthe external field (as determined from normal state measurements) to the exponential decayobserved deeper in the bulk.While the full reason for the dead layer effect is unknown, it is not necessary thatthe surface be chemically distinct or degraded – the dead layer effect has been attributedby some authors as possibly due to surface roughness [170, 171], rendering such a surfaceineffective at carrying the full screening currents, effectively submerging the screening cur-rents by an apparent depth d. However, X-ray photoelectron spectroscopy measurements ofYBa2Cu3O6+x thin films exposed to ambient air [172] have revealed the growth of a chemi-cally degraded surface layer (composed of hydroxides and carbonates) whose thickness growswith a square root time dependence d ≈ (0.1nm/√h)√t, corresponding to approximately1696.2. Apparatus9.4 nm for one year of degradation in air; as some samples have been stored for comparabletime intervals between growth and measurement, this could be an important contributionto the observed dead layer. On the other hand, coherence length effects can be ruled out asthe sole source of the dead layer; while the superconducting order parameter will be sup-pressed within the length scale ξ from the surface, ξ has been measured to be between ∼1.5nm and ∼5 nm in YBa2Cu3O6+x over this doping range [63, 173]: an order of magnitudesmaller than typical values of d. It must be pointed out here that for most surface probessuch as STM, AFM, and ARPES, samples are cleaved immediately before measurement,leaving pristine, near-atomically-flat surfaces for which one would not expect to see such adead layer; microwave conductivity would also be insensitive to an insulating dead layer, asthe effective surface of the sample seen by microwaves would simply be moved below thesurface by a depth d.With the inclusion of the dead layer d, the London profile B(z) we use is thenB(z) =B0 z ≤ dB0e−z−dλ z > d, (6.11)where λ is the London penetration depth and B0 = µ0Hext is the magnetic field at thesurface of the superconductor.6.2 Apparatus6.2.1 Low-energy muon production and transportThe low-energy µSR measurements described in this thesis were performed at the SwissMuon Source (SµS) located at the Paul Scherrer Institute (PSI) in Villigen, Switzerland.The Swiss Muon Source is one of only four such muon spectroscopy facilities in the worldcurrently operating – along with those at TRIUMF in Vancouver, KEK in Japan, and ISISin the United Kingdom – and is at present the only facility with the slow muon capabilityrequired for low-energy µSR (LEM).The Swiss Muon Source is powered by the three-stage 590 MeV High Intensity ProtonAccelerator, housed at the main experiment hall at PSI. Starting from a 60 keV ion source,an initial 870 keV Cockcroft-Walton generator feeds protons into the “Injector 2” 72 MeV1706.2. Apparatusring cyclotron, which itself injects into the main 590 MeV ring cyclotron [174]. The high-energy protons are produced in bunches at a 50.63 MHz repetition rate,66 with an averagebeam current of 2.2 mA for a total beam power of ∼1.3 MW – the highest power in theworld for such a facility.Protons are extracted from the cyclotron for transport along a proton beamline to tworotating graphite disk target stations – 5 mm thick Target M, followed by 40 mm thickTarget E – through which approximately 70% of protons pass, only moderately slowed,towards either the spallation target of the SINQ neutron facility or a beam dump. AtTarget E, surface muons (as described in Section 6.1.2) are extracted into the µE4 beamlinewhich serves the LEM experiment. This beamline has been optimized for a large acceptanceof positive muons with momentum p ≈ 28 MeV/c, close to the typical energy of the 4 MeVsurface muons from the target [175]. The extracted surface muons are guided towards theLEM moderator target through an “E ×B” mass separator (also known as a Wien filter)employing orthogonal static electric and magnetic fields as a mass selector to filter out theunwanted positrons which contaminate the beam.67The muons then hit the moderator, where slow (“epithermal”) muon production takesplace. The backbone of the moderator – which is cooled by suspension from a liquid heliumcryostat – consists of a ∼125 µm-thick Ag film substrate which has been microstructuredon its downstream side with V-shaped grooves 20 µm deep, spaced 30 µm apart, of peakangle 70.5◦ [176]. A ∼100 nm layer of van der Waals solid gas is cryodeposited onto thisdownstream surface, taking its corrugated form; this is typically solid Ar capped with a thinlayer of s-N2, which is more stable for high-voltage operation. Other configurations of s-Ar,s-Ne, and/or s-N2 are sometimes used, including s-N2 alone, when the need for high-voltagestability is more important than moderator efficiency.Spin-polarized surface muons at 4 MeV enter the moderator, with about 50% stoppingin the Ag, and the vast majority of the remainder exiting the other side at energies toohigh to be moderated (E > 30 keV). However, a fraction ∼ 10−3 of the muons are slowedto below 30 keV in the Ag before entering the solid gas layer. Muons below this thresh-66Though most commonly referred to as a continuous beam, the protons are actually grouped into bunches∼0.3 ns wide, spaced 19.75 ns apart. The 26.0 ns lifetime of the pi+ responsible for surface muon productionsmooths out the µ+ rate out to a < 0.1% ripple, and the low rates and use of a trigger detector alreadyrender the experiment insensitive to µ+ arrival time structure.67The decay mechanism pi+ → µ+ + νµ – with a branching ratio of (1.230± 0.004)× 10−4 [165] – is onecontributor.1716.2. Apparatusold dissipate their energy through repeated interactions with electrons in the solid, untilreaching ∼15 eV, below which inelastic scattering cross-sections quickly decrease in suchlarge-bandgap insulating van der Waals solids. This results in a large µ+ escape depth(de ≈ 70 nm in s-Ar) from which slow muons may be extracted from the surface and ac-celerated electrostatically up to a maximum 20 keV. The surface corrugation increases thestopping volume within range of escaping from the surface, with a corresponding increaseof about 50% in the slow muon production rate. Because the total deceleration time ∼10 psdoes not allow sufficient time for spin exchange interactions to occur, the slow muons exitthe moderator with nearly 100% of their initial polarization intact [177]. Though optimiza-tions have improved the situation substantially since early work, the moderation process isnot particularly efficient; starting from an incoming surface muon rate of ∼ 2 × 108 µ+/s,the resulting output of low-energy moderated muons which eventually reach the sampleplate is ∼4500 µ+/s.The Ag foil of the moderator is held at a positive voltage of Vtrans ≈ 15 kV68 withrespect to the potential downstream, which is held near ground; the muons (with chargeqµ+ = e+) are thus accelerated downstream from the foil up to a transport kinetic energyof Etrans = +eVtrans, After being accelerated out of the moderation region, the muons reachan electrostatic mirror, also charged to Vtrans; slow µ+ are deflected by 90◦ to continuetowards the sample, while unmoderated muons and other beam contaminants pass throughto a microchannel plate for detection and removal.In the final few metres, a series of electrostatic beam optics elements transports andfocuses the muon beam through a trigger detector and towards the sample plate at theend of the apparatus. Muons passing through the 10-nm-thick carbon foil trigger detectorare degraded, losing ∼1 keV of energy and gaining an energy spread of ∼400 eV [178];this will be the dominant contribution to the implantation energy resolution of the system,but (except at the lowest sub-keV energies) will not significantly worsen the intrinsicallybroad implantation depth profiles. Through adjustments of the voltages provided to thefour quadrants of the conically-shaped ring anode, minor steering of the beamspot maybe accomplished; this can be useful for optimizing measurements of small samples, as wellas for compensating for deflection due to magnetic fields in the sample region – especially68The moderator voltage (and thus transport energy) can be adjusted slightly, over a range of 12–18 kVfor this work, to extend the window of implantation energies; this can be particularly important for reachinglow energies when difficulties with maintaining high positive voltage at the sample plate are encountered.1726.2. ApparatusMCP1Surface µ beam(E ~ 4 MeV)SpSpModeratorMirrorEinzel lensTrigger detectorEinzel lens(LN2 cooled)Gate valveConical lensHelmholtz coilsSample Sample cryostatPositron detectorsScintillatorEinzel lens(LN2 cooled)Low-energy µ+ beamFigure 6.4: A schematic diagram of the low-energy µSR apparatus at the Paul ScherrerInstitute. Note that this schematic does not show the spin rotator installed before theAugust 2012 beamtime. Diagram kindly provided by Zaher Salman from the LEM groupat PSI.1736.2. Apparatusimportant at low energies.6.2.2 Muon implantation and the sample environmentThe samples to be measured sit on a high-purity aluminum sample plate coated with 1 µmof Ni. While not thick enough to generate any substantial external fields, muons whichmiss the samples and are implanted into this Ni coating precess at very high frequency dueto hyperfine fields [179], giving a background signal well separated in frequency from thedesired signal from muons hitting the samples. For small samples or mosaics of sampleswhich do not fully cover the ∼12 mm diameter (FWHM) muon beamspot, this eliminateswhat would otherwise be a large background component precessing at a Larmor frequencycorresponding to the applied field.Samples are affixed to the sample plate using conductive silver paint, providing goodthermal contact and electrical grounding of the samples to the plate. The aluminum sampleplate is bolted to a sapphire plate at the end of the cold head of the cryostat, with anindium foil compressed in between to improve thermal contact. Several different cryostatswere used in the course of measurements, each of them liquid 4He flow cryostats of differentconfigurations. Most measurements were performed in cryostats with base temperatures of∼5 K; a special low-temperature flow cryostat used for some measurements is equipped witha gas/liquid phase separator for a lower minimum base temperature of ∼2 K. This latterlow-temperature cryostat was significantly more difficult to operate and keep stable at itslowest temperatures, and was not used in most cases; the minimal advantages of measuringλ(2 K) instead of λ(5 K) – particularly given the availability of more accurate techniquesfor measuring λ(5 K) − λ(2 K), such as microwave spectroscopy – would not make up forthe expected losses in runtime due to dealing with cryostat problems.The sample plate, which is connected to an adjustable ±12.5 kV high voltage source,is electrically isolated from the cryostat, the radiation shield, and the surroundings, whichare held near earth ground. In between the opening of the radiation shield (at V = 0)and the sample plate (at Vsamp) are two guard rings, charged to 13Vsamp and 23Vsamp, whichserve to provide a uniform electric field gradient in front of the sample plate for acceleratingor decelerating the incoming muons. The final implantation energy will be the sum of thekinetic energy of the muons after traversing the trigger detector (with associated spreading)1746.2. Apparatusand the energy due to the final acceleration:Eimp = Etrans −∆ETD − eVsamp= e(Vtrans − V)− (1 keV± 0.4 keV). (6.12)(This is not valid for |Eimp| . 1 keV, where a significant fraction of muons will be backscat-tered and thus a more careful consideration of the momentum distribution is warranted.)Muons which enter the sample must deposit their energy through various scatteringprocesses before stopping. Such processes are stochastic, and for a given monochromaticimplantation energy, a spread of implantation depths will result. To a lowest order approxi-mation, the mean stopping depth scales linearly with implantation energy, as we shall see inSection 6.3.1 (see the bottom of Figure 6.5). In imperfect vacuum conditions and at highervoltages, arcing can occur, which limits the voltage which may be applied and thus theachievable implantation depth range. Just as was the case in the moderator, muons do notmeasurably depolarize during the ∼10 ps stopping time, and remain nearly 100% polarizedupon settling into their final stopping locations, ready for precession and measurement.Finally, the sample environment magnetic field is generated by Helmholtz coils placedoutside the cryostat, centred on the sample plate. For the work discussed here, a verticalmagnetic field is used, parallel to the sample surface (i.e., in the ab–plane of the samplesbeing measured) and perpendicular to both the momentum and initial polarization of theincoming muons; the spin precession thus occurs in the horizontal plane parallel to theground. Before each series of runs, a degaussing procedure is carried out to ensure themagnetic field offset is minimal.6.2.3 MeasurementFor the continuous-wave muon beam operation of the low-energy µSR apparatus at PSI,slow muons reach the sample at random times (i.e., a Poisson process) with typical ratesof ∼4500 µ+/s, corresponding to average times between muon arrivals of ∼200 µs. In orderto start the “stopwatch” of the measurement of the muon at the time it is deposited intothe sample and begins precessing, an aforementioned carbon foil trigger detector is usedto provide this signal; electrons ejected by muons traversing the carbon foil are directed toa microchannel plate perpendicular to the foil for detection. In the case where a second1756.3. Analysismuon arrives before the previous muon decay has been observed (or alternatively, manymuon lifetimes have passed, implying that the decay positron was simply missed), thetrigger detector also allows for a “pileup veto,” where such events are discarded due tothe ambiguity in identifying which decay belongs to which arrival time. While such pileupissues can be serious considerations for conventional µSR – where the times between muonarrivals approach muon lifetimes – the low rates of LE-µSR make pileup events much lessfrequent, and render the pileup rejection procedure much less likely to influence the data.While the trigger detector provides the start signal for measurement, the positron de-tectors provide the stop signal. The detectors consist of scintillators placed outside thecryostat, surrounding the sample space; for the measurements described here, scintillatorswere placed to the left and right side of the sample, in the plane of the spin precession.Either photomultiplier tubes (PMTs) or avalanche photodiodes (APDs) may be used todetect the light signal generated by a positron traversing the scintillator. A sophisticateddata collection system is used to read out these signals and process the timing data; theend result for the user is a series of positron collection events for each detector as a functionof time after muon arrival.6.3 Analysis6.3.1 TRIM.SPIn order to extract the depth dependence of the fields from our low-energy µSR measure-ments, we need accurate knowledge of the muon implantation depth distribution for thematerial and energies in question. This is accomplished with a specially designed implan-tation simulation code called TRIM.SP [180].TRIM.SP has its origins in sputtering simulations, and was modified for study of lightion implantation into matter – later extended to the case of muons. This code simulatesindividual muons scattered by nuclear collisions, with straight-line motion in between; afterdropping below a threshold kinetic energy, the muon stops. A resulting implantation depthprofile is generated as a function of implantation energy, implantation angle, and spreads ineach. The details of the scattering depend on the material properties, and thus implanta-tion profiles must be calculated for the particular material being measured. The accuracyof this code has been tested experimentally on thin metallic films, with good agreement1766.3. Analysisdemonstrated [181].Implantation profiles calculated for YBa2Cu3O6.52 are shown in Figure 6.5; similar pro-files (qualitatively similar, but with slight quantitative differences) have been generated forall samples and energies studied here. The mean implantation depth varies roughly (butnot quite) linearly with implantation energy. The profiles are rather asymmetric, and arenot particularly well approximated by Gaussians; the actual TRIM.SP-generated profilesthemselves are used for all data fitting procedures.6.3.2 musrfit data analysisData analysis is performed with the µSR data processing suite known as musrfit [182].This software provides for “global” fitting of measured muon data at a series of implantationenergies, using Equation 6.10 with shared field profiles from Equation 6.11 and energy-dependent depth profiles calculated as above. Example histograms for real data from theortho-II YBCO measurements presented here – along with the corresponding London modelfits – are shown in Figure 6.6. The top of this figure demonstrates the signals from the leftand right detectors for a single implantation energy, shown to be (nearly) 180◦ out of phaseas expected. The bottom of the figure shows the right detector results alone for a range ofimplantation energies; we see that as energy (and thus depth) increases, the muon precessionfrequency (proportional to B) decreases, consistent with the exponential decay of fields asa function of depth from the surface of the sample.Certain issues with fit parameters are often encountered at this stage of analysis. Oneunavoidable issue is that of naturally heavy correlation between the dead layer d and thepenetration depth λ; even given the knowledge of the field at the surface, a measurement ofthe magnetic field at a single depth cannot distinguish between the two contributions, andthus a complete energy scan is needed to disentangle the two. Additional measurementsin the normal state of the superconductor are required in order to measure the true fieldat the surface, which can often differ by several gauss from the nominally set value. Forsingle-energy measurements of the penetration depth as a function of temperature (ratherthan a single-temperature measurement as a function of energy), the dead layer parameteris fixed to the results from the energy scan, and assumed to be temperature independent– while our previous measurements have not shown substantial temperature dependence ofthe dead layer, this is not necessarily always the case.1776.3. Analysis0 20 40 60 80 100 120 140 160 180 2000%2%4%6%8%10%12%Fractionalimplantationdensity(nm-1)Implantation depth z (nm)500 eV1 keV3 keV6 keV10 keV15 keV20 keV25 keV30 keV0 5 10 15 20 25 30020406080100120140160180Implantation depth z (nm)Implantation energy (keV)0.0% nm-11.0% nm-12.0% nm-13.0% nm-14.0% nm-15.0% nm-16.0% nm-17.0% nm-18.0% nm-19.0% nm-110% nm-1 Mean depth <z>  1zFigure 6.5: Muon implantation depth profiles as a function of implantation energy forYBa2Cu3O6.52, calculated with TRIM.SP [180]. (Top) Profiles for several energy values;error bars (shown as here as ±1σ bands) represent Poisson√N counting statistics. (Bot-tom) a density plot of the same profiles; mean depths (solid) and their standard deviations(dashed) are overlaid in cyan.1786.3. Analysiss)µtime (0.5 1 1.5 2 2.5 3 3.5 4 4.5 5asymmetry-0.08-0.06-0.04-0.0200.020.040.060.08 DetectorLeftRights)µtime (0.5 1 1.5 2 2.5 3 3.5 4 4.5 5asymmetry-0.08-0.06-0.04-0.0200.020.040.060.08Energy+µ1.9...1 .1 .11 .11 .1..Figure 6.6: Example raw histogram data for low-energy µSR on ortho-II YBCO, at 5 Kand Hext = 48 G, for H ‖ bˆ (measuring λa). (Top) The left and right detector signalsfor a single energy (25.02 keV), with the constant background subtracted. Fits are to theLondon model convolved with the simulated muon implantation profile. (Bottom) Theright detector signal for a range of implantation energies. As expected, at higher energies(corresponding to greater implantation depths) the muon precession frequency decreases,corresponding to lower fields, consistent with Meissner screening.1796.4. YBCO measurementsAdditionally, there are issues regarding the choice of parameters to leave fixed or freebetween energies for the purposes of fits; while fixing parameters known to be shared be-tween runs will reduce the variance in other fit parameters, for some of the phase parametersinvolved in the fits it is not clear how valid an assumption this may be. This has been dealtwith in detail in the Ph.D. thesis of Masrur Hossain [169]. Here we use results for whichthe individual phases are left free in the fits.6.4 YBCO measurementsLow-energy µSR measurements of YBa2Cu3O6+x at five different oxygen contents (x = 6.52,6.60, 6.67, 6.92, 6.998) were taken over the course of five years of beamtime allotment atPSI – an ambitious undertaking, as the average annual beamtime allotment for this projectwas about 10 days, part of which was to be shared with pnictide measurements. To fulfillthe requirement that the vast majority of the muons in the beamspot (∼12 mm FHWMdiameter) be implanted into the sample material rather than the Ni-coated backing plate,a total sample surface area of 50–100 mm2 is necessary; since high-quality YBa2Cu3O6+xsingle crystals with areas of even 10 mm2 are exceedingly rare, tightly packed mosaics of10–20 single crystals were used for these measurements.The samples of both the 6.52 and the 6.60 mosaics were ortho-II ordered; the 6.67 mosaicsamples were ortho-VIII ordered; and the 6.92 and 6.998 samples were ortho-I ordered.The large samples required for the mosaics required longer annealing times for oxygenhomogenization (particularly for the x=6.998 samples, for which one seeks to minimizethe remaining oxygen vacancies) than is normally required for smaller microwave samples;additionally, each sample required manual detwinning, a laborious process. These heroicefforts of Dr. Ruixing Liang were crucial to the success of this project.Because the resolution of the muon penetration depth is far coarser than that attainablein ∆λ(T ) from other methods, the focus of the measurements was always on measuringthe penetration depth accurately for a single temperature, providing an accurate referencepoint for the absolute value of λ(T ) to which these other methods could be referenced.However, the temperature dependence was still measured as an important consistency checkwith these other techniques. Figure 6.7 shows the results of a comparison of the LE-µSRλ(T ) data on YBa2Cu3O6.998 with microwave penetration depth measurements presented1806.4. YBCO measurements0 10 20 30 40 50 60 70 80 90100150200250300 a-axis W ( (2 K) = 109 nm) a-axis LEM (free phase) b-axis W ( (2 K) = 72 nm) b-axis LEM (free phase) (nm)Temperature (K)YBCO6.998Figure 6.7: Penetration depth temperature dependence data for YBa2Cu3O6.998, comparingthe low-energy µSR data with microwave penetration depth ∆λ(T ) data fixed to the sameextrapolated λ(0); excellent agreement is demonstrated.1816.4. YBCO measurementsin Section 4.6 (and originally collected for this purpose); the microwave data λ(0) valueshave been fixed to the LE-µSR results here. The data are in very good agreement up tothe highest temperatures. Measurements for the other YBCO samples (and the pnictidemeasurements) have also shown good agreement between microwave and LE-µSR resultsat low temperatures; at higher temperatures, deviation is typically observed where thelower critical field Hc1(T ) approaches the 25 or 50 G LE-µSR measurement fields, withresulting flux penetration. However, the agreement provides us with added confidence inthe reliability of the absolute values of λ(T ) – at least at the lowest temperatures – extractedfrom low-energy µSR measurements.The absolute values of the penetration depths themselves are reported in Tables 6.1 and6.2 for the a- and b-axis measurements, respectively. Unlike the temperature dependencemeasurements, these values are extracted from global fits to scans at multiple energies, toreliably extract the depth dependence of field and thus the penetration depth. It should bepointed out here that, while the author of this thesis was heavily involved in the low-energyµSR measurement and data analysis, the final λ(Tmeas) fit results which I have chosen toreport here are those from the fits of Masrur Hossain [1, 169, 183], the member of thecollaboration who was responsible for the careful considerations of fit parameter treatmentas discussed above.69 His results agree with the results of independent fits done by theauthor, but with slightly different numerical values due to different fit starting conditions.It was decided for the sake of consistency to maintain the use of the same “canonical”penetration depth values at the measurement temperature.In Tables 6.1 and 6.2 we also report the values of the dead layer thickness d extractedfrom the London model fits. For three of the dopings, a “global” fit was performed to theLEM data from both the a and b axes simultaneously, with a shared dead layer for bothaxes enforced; this is indicated by italics in the table. Note that the uncertainties reportedfor d are the raw estimates of statistical uncertainties extracted from the fits, which likelyunderestimate the true uncertainty for some cases.For the extrapolation of λ(T ) results to λ(0), we differ from Ref. [169] in using microwavepenetration depth measurements (taken by the author, and presented in Section 4.6) to ex-trapolate to T = 0, rather than using the lower-resolution LE-µSR temperature dependence69The exception is for the YBCO 6.60 results, with which Dr. Hossain was not involved; here, the finalresults of Zaher Salman were used instead.1826.4.YBCOmeasurementsx Tc p Tmeas λLEMa (Tmeas) ∆λµWa (Tmeas) λa(0) d(K) (K) (nm) (nm) (nm) (nm)6.52 57.5 ± 0.5 0.0992 ± 0.0011 5.0 303.2 ± 14.7 4.3 ± 0.7 298.9 ± 14.7 22.9 ± 0.26.60 62.0 ± 0.2 0.1091 ± 0.0005 4.2 170.5 ± 6.0 3.3 ± 0.5 167.2 ± 6.0 14.5 ± 0.76.67 66.5 ± 0.5 0.1222 ± 0.0009 5.0 152.8 ± 5.4 4.0 ± 0.5 148.8 ± 5.4 27.7 ± 1.96.92 94.1 ± 0.1 0.1649 ± 0.0013 8.0 128.9 ± 3.2 2.9 ± 0.4 126 ± 3.2 10.3 ± 0.56.998 89.2 ± 0.4 0.1856 ± 0.0010 5.0 107.5 ± 4.9 1.8 ± 0.1 105.7 ± 4.9 17.2 ± 1.5Table 6.1: Low-energy µSR a-axis penetration depth measurements for YBa2Cu3O6+x, along with microwave ∆λa(T ) extrapolations toT = 0, and the extracted values of the dead layer thickness (dead layer values in italics are from global fits to both axes with a shareddead layer).x Tc p Tmeas λLEMb (Tmeas) ∆λµWb (Tmeas) λb(0) d(K) (K) (nm) (nm) (nm) (nm)6.52 57.5 ± 0.5 0.0992 ± 0.0011 5.0 231.3 ± 15.8 3.6 ± 0.5 227.7 ± 15.8 22.9 ± 0.26.60 62.0 ± 0.2 0.1091 ± 0.0005 4.2 155.5 ± 6.0 2.3 ± 0.5 153.2 ± 6.0 12.50 ± 0.016.67 66.5 ± 0.5 0.1222 ± 0.0009 5.0 123.5 ± 4.3 2.8 ± 0.5 120.7 ± 4.3 28.2 ± 1.76.92 94.1 ± 0.1 0.1649 ± 0.0013 8.0 108.4 ± 3.2 3.2 ± 0.5 105.2 ± 3.2 10.3 ± 0.56.998 89.2 ± 0.4 0.1856 ± 0.0010 5.0 84.0 ± 4.5 1.8 ± 0.2 82.2 ± 4.5 17.2 ± 1.5Table 6.2: Low-energy µSR b-axis penetration depth measurements for YBa2Cu3O6+x, along with microwave ∆λb(T ) extrapolations toT = 0, and the extracted values of the dead layer thickness (dead layer values in italics are from global fits to both axes with a shareddead layer).1836.5. Measurements of Ba(Co0.074Fe0.926)2As2measurements. The results for these extrapolated penetration depths are shown in Tables6.1 and 6.2 for the a- and b-axis measurements, respectively.These results are shown in the form of λ−2(0) – more readily comparable to the physicallysignificant quantity ns/m∗ than is λ(T ) itself – in Figure 6.8. Here we also compare to theresults from the Gd-doped ESR measurements of Pereg-Barnea et al. [109], one of the fewknown dependence studies which has resolved λa and λb separately for several dopings. Itcan be seen that the two techniques appear to agree at the higher dopings, the LE-µSRresults show a downturn at the lowest dopings which appears to deviate from the Gd-ESRdata.Figure 6.9 shows the penetration depth a–b anisotropy, in terms of the ratio λ2a(0)/λ2b(0),for both the LE-µSR and Gd-ESR data. Both techniques are consistent with a nearlydoping-independent anisotropy value of around 1.5, although it is possible that there issome structure visible here for p . 0.12.For comparison with a larger spread of techniques which cannot resolve λa and λbseparately, we instead plot in Figure 6.10 the in-plane geometric average λ−2ab = λ−1a λ−1b . Inaddition to the LE-µSR and Gd-ESR data, we plot data from bulk µSR [184–186], microwavecavity perturbation [187], magnetic force microscopy [188], far-infrared spectroscopy [189],and estimates from measurements of the upper critical field Hc1 [190]. It is clear that thereis some unresolved disagreement between techniques.Finally, in Figure 6.11 we construct a plot of Tc as a function of the inverse penetrationdepth λ−2 (= µ0nse2m∗ ), in order to test the validity of the Uemura relation nsm∗ ∝ Tc, proposedby Uemura as a universal scaling law for underdoped cuprates [191]. Here we confirmprevious observations that this relation does not appear to hold up for YBa2Cu3O6+x [109],at least over the doping range that has been measured.706.5 Measurements of Ba(Co0.074Fe0.926)2As26.5.1 Penetration depth measurement and a comparison of techniquesIn addition to the measurements of YBCO, some of the UBC low-energy µSR beamtime wasalso set aside for measurements of the iron pnictide superconductor Ba(Co0.074Fe0.926)2As2.70The Uemura relation also specifies that this proportionality hold across different families of cuprates;when considering the penetration depths at optimal doping, at least this aspect appears to hold.1846.5. Measurements of Ba(Co0.074Fe0.926)2As20.08 0.10 0.12 0.14 0.16 0.18 0.20020406080100120 Low-energy SR Gd-doped ESRa-2(0 K) (m-2)Hole doping pa axis0.08 0.10 0.12 0.14 0.16 0.18 0.20020406080100120140160180200b axis Low-energy SR Gd-doped ESRb-2(0 K) (m-2)Hole doping pFigure 6.8: The measured inverse squared penetration depths λ−2(0) for YBCO as a functionof doping, for both axes. Gd-ESR measurements are also shown as a comparison. (Top)a-axis measurements. (Bottom) b-axis measurements. Lines connecting points are onlyguides to the eye. (Repeat of Figure 4.2, reprinted here for convenience.)1856.5. Measurements of Ba(Co0.074Fe0.926)2As20.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.190.51.01.52.02.53.0 LE- SR Gd-ESRAnisotropy a2/b2(0 K)Hole doping pFigure 6.9: Comparison of the a–b anisotropy for λ−2(0) as a function of doping, for theLE-µSR and Gd-ESR measurements. Both datasets are consistent with each other and witha nearly temperature-independent penetration depth anisotropy, although some structuremay be visible here at low doping.1866.5. Measurements of Ba(Co0.074Fe0.926)2As20.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24020406080100120140ab-2(0 K) (m-2)Hole doping p Low-energy SR Gd-doped ESR  (Pereg-Barnea 2004) Bulk SR (Sonier 2007) Bulk SR, Ca-doped (Bernhard 2001) W (Broun 2007) W (ibid.,  extrapolated) MFM (Luan 2011) Far-IR (Homes 1999) Hc1 (Liang 2005)0.000.050.100.150.200.250.30(ns,ab/np)/(m*/me)Figure 6.10: The average in-plane inverse squared penetration depth λ−2ab (0) as a functionof hole doping p for YBa2Cu3O6+x, as determined by several techniques. The black dataare from low-energy µSR measurements at the Paul Scherrer Institute (see Chapter 6). Thered data were collected via microwave ESR measurements of Gd-doped YBCO by Pereg-Barnea et al. [109]. The magenta data are vortex-state µSR measurements by Sonier et al.[184]. The light green data are vortex-state µSR measurements on Y0.8Ca0.2Ba2Cu3O6+xby Bernhard et al. [185, 186]. The blue data are 2.64 GHz microwave cavity perturbationmeasurements from Broun et al. [187], extrapolated merely as a guide to the eye; whilesuggestive, a drop below linearity between the Broun et al. data and p ≈ 0.09 – as suggestedby the µSR data – is not precluded. The orange data are magnetic force microscopy(MFM) measurements from Luan et al. [188]. The red data are far-infrared spectroscopymeasurements by Homes et al. [189]. The dark green data are λ−2ab estimates derived fromthe Hc1 data of Liang et al. [190].1876.5. Measurements of Ba(Co0.074Fe0.926)2As20 20 40 60 80 100 120 1400102030405060708090100 Low-energy SR Gd-doped ESR Bulk SR (Sonier 2007) W (Broun 2007) MFM (Luan 2011) Far-IR (Homes 1999) Hc1 (Liang 2005) Tc ~ -2 ~ ns (Uemura) Tc ~ -1 ~ ns1/2Tc (K)1/ab2 ( m-2)Figure 6.11: A plot of Tc as a function of λ−2ab (0) for several techniques, as a test of theproposed Uemura scaling relationship Tc ∝ 1/λab(0) [191]; both dashed lines drawn arefixed to the LEM optimal doping data point.1886.5. Measurements of Ba(Co0.074Fe0.926)2As2As previously mentioned, while this work was not a part of the YBCO doping scan, itis closely related in terms of tools and techniques, and will be presented here as anotherexample of the power of combining low-energy µSR and microwave measurements to yielda complete picture of the penetration depth λ(T ) of a superconductor. These results havebeen previously published [2], and will only be summarized here.The sample of Ba(Co0.074Fe0.926)2As2 was provided by the Canfield group at the AmesLab at Iowa State University, and was grown by a self-flux method [192]. It was measuredvia SQUID magnetometry to have Tc = 21.7 K, with a transition width of 0.8 K. Aftermounting on the sample plate, the top face of the sample was cleaved in situ under a flowof dry N2 gas, before quickly loading into the LE-µSR apparatus and pumping down; thiswas designed to minimize – though not eliminate – the exposure to air for these air-sensitivesamples. Measurements were performed as described above.In addition to these low-energy µSR measurements, a small piece of the sample wasbroken off for microwave penetration depth measurements, using the same apparatus andtechniques described in Chapter 4. The sample was cut and cleaved on all faces immediatelybefore mounting on the sapphire sample plate of the 1 GHz loop-gap resonator probe,encapsulated with a thin protective layer of vacuum grease, which has been found to decrease(but not eliminate) surface degradation.The results (from 0 to 18 K) of these measurements are shown in Figure 6.12. Themicrowave ∆λ(T ) data has been fixed to the low-temperature LE-µSR λ(T ) data, and usedto extrapolate to λ(0) = (250.5 ± 2.6) nm. We again see good agreement in the temperaturedependence of the microwave and LE-µSR data.In addition to our own measurements, Figure 6.12 also shows data from tunnel dioderesonator (TDR) measurements of ∆λ(T ) from the group of Ruslan Prozorov at the AmesLab at Iowa State University [193], taken on a very similar sample grown in the samelaboratory, as well as magnetic force microscopy (MFM) measurements from Lan Luanof the group of Kam Moler at Stanford [194], taken on a nominally equivalent samplefrom a different crystal grower. The TDR ∆λ(T ) data were referenced to the same zero-temperature penetration depth of λ(0) = (250.5 ± 2.6) nm as the microwave data; whilethe MFM data were absolute measurements of penetration λ(T ), which extrapolated to aslightly lower λ(0) ≈ 246 nm, they have been shifted up by a constant ∼4.4 nm to thesame λ(0) as the other curves for comparison. We see that the TDR data agrees reasonably1896.5. Measurements of Ba(Co0.074Fe0.926)2As20 2 4 6 8 10 12 14 16 18240260280300320340360380400 (nm)Temperature (K) Low-energy SR W ( (0) = 250.5  2.6 nm) TDR ( (0) = 250.5  2.6 nm) MFM ( (0) = 250.5  2.6 nm)Figure 6.12: A comparison of Ba(Co0.074Fe0.926)2As2 in-plane penetration depth measure-ments from different techniques. Microwave (µW; this author) and tunnel diode resonator(TDR; Ames Lab [193]) measurements are of penetration depth shifts ∆λ(T ), here refer-enced to the absolute value λ(0) = (250.5 ± 2.6) nm obtained from combining low-energyµSR λ(T ) data with extrapolations of the microwave data. The magnetic force microscopy(MFM; from Luan et al. [194]) data are of the absolute penetration depth λ(T ), but havebeen shifted up by a constant ∼4.4 nm to extrapolate to the same λ(0) for the purposes ofcomparison.1906.5. Measurements of Ba(Co0.074Fe0.926)2As2well with the microwave and LE-µSR datasets, with the slight discrepancy well withinexpectations of variations between crystal growth batches. On the other hand, the MFMdata shows a substantially weaker temperature dependence.Publication of this comparison [2] – with the suggestion that there may be differencesbetween local measurements of λ(T ) (MFM) and surface-averaged global ones (LEM, mi-crowaves, and TDR) – prompted further investigation by the Moler and Prozorov groups,where measurements were performed using both techniques (MFM and TDR) on the samesample. The published results [195] showed good agreement between MFM and TDR datameasured on the exact same samples. While the consistency of results between all fourtechniques bolsters confidence in their validity, it has been made clear that discrepanciesbetween nominally identical samples can be a major issue for the reproducibility of mea-surements on iron pnictide samples.6.5.2 Fits and analysisOne main goal of the Ba(Co0.074Fe0.926)2As2 measurements were to clarify the temperaturedependence of the penetration depth, a quantity which can distinguish between differentmodels of the superconducting gap structure and its symmetry, as well as provide informa-tion regarding the nature of scattering.In order to clarify these questions, the penetration depth data has been fit to variousmodels of its temperature dependence below – a power-law model, a phenomenological two-gap s-wave “α model,” as well as fits done to a self-consistent two-gap “γ model.” Theresults of the fits (and their residuals) are shown in Figure 6.13.Power-law behaviour of λ(T ) at low temperatures had previously been observed in sev-eral different families of pnictide superconductors, and has been ascribed to the behaviourexpected for a sign-changing s± gap function in the presence of nonmagnetic impurities[196]. For such a model, the normalized superfluid density ρ(T ) is given byρ(T ) ≡ λ2(0)λ2(T ) = 1− α( TTc)n. (6.13)As seen in Figure 6.13, the power law model (shown in green) provides an excellent fit to thedata at low temperatures, but fails to describe the high-temperature behaviour – which isunsurprising, given its initial motivation as an empirical description of the low-temperature1916.5. Measurements of Ba(Co0.074Fe0.926)2As20.00.10.20.30.40.50.60.70.80.91.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-1012(T) = 2(0)/2(T) W + LE- SR  model  model power law0.00 0.05 0.10 0.15 0.20 0.25 0.300.920.940.960.981.00Residuals (10-2)T/TcFigure 6.13: Fits to the measured Ba(Co0.074Fe0.926)2As2 data, with several models shown.(Top) The data and fits, with an inset zoomed in at low temperature. (Bottom) Theresulting fit residuals.penetration depth. While the exact fit parameters are fairly sensitive to the temperaturerange of data included in the fit, a range of upper fit cutoff temperatures from 5 K to 12K gives an exponent n = 2.51 ± 0.02 and a coefficient α = 1.39 ± 0.03, where the quotederrors correspond to the parameter variation over this spread of ranges. The model curveshown in Figure 6.13 is for a fit cutoff of 8 K.For the two-gap s-wave “α model” fits, we assume a picture of two independent super-fluid components i ∈ S,L with different energy gap values ∆i(T ), where ∆S and ∆L arethe Small and Large gaps, respectively. This model is not rigorously valid, as interactionsbetween bands are not considered, but it has been shown to provide a good phenomeno-1926.5. Measurements of Ba(Co0.074Fe0.926)2As2logical description of the temperature dependence of superconductivity in two-gap s-wavesuperconductors such as MgB2 [197]. In this model, the normalized superfluid density takesthe formρ(T ) = 1− xδn(S)s (∆S(T ), T )n(S)s (0)− (1− x)δn(L)s (∆L(T ), T )n(L)s (0), (6.14)whereδn(i)s (∆i(T ), T )n(i)s (0)= 2kBTˆ ∞0dǫ f(ǫ,∆i(T ), T ) [1− f(ǫ,∆i(T ), T )] , (6.15)with f being the appropriate Fermi functionf (ǫ,∆i(T ), T ) =11 + e√ǫ2+∆2i (T )kBT. (6.16)For the temperature dependence of the gaps, we use an empirical relation which has beenshown to be an adequate model over the full temperature range [56, 139]:∆i(T ) = ∆0,i tanh[πkBTc∆0,i√ai(TcT − 1)]. (6.17)The resulting model, shown fit to the data (in red) in Figure 6.13, describes the temper-ature dependence very well over the entire temperature range from 0 < T < Tc. Here wefound y= 0.097±0.001, indicating that the bulk of the superfluid in this model is describedby the large gap. The resulting gap values from the fits (expressed in terms of the gapratios 2∆0kBTc ) were2∆0,LkBTc = 3.46 ± 0.10 (consistent with the weak-coupling limit BCS valueof 3.528) and 2∆0,SkBTc = 1.20 ± 0.07: both in good agreement with previous bulk µSR work[198]. While the “shape parameter” ai was fixed to aS = 1 for the small gap (for reasonsof fit stability; also done in [194]), the value for the large gap was found to be aL = 0.83 ±0.03.Finally, the results of a fit to a different two-gap s-wave model [199] (referred to as the“γ model”) – are also shown in Figure 6.13 (in blue), provided by Ruslan Prozorov [193].Unlike the α model, the γ model fully accounts for the effects of interband coupling in aself-consistent fashion.Comparing the three models, we see that, at low temperature, the power law providesthe best fit, but it does not do significantly better than the α model, particularly in light ofthe fact that the power law fit is restricted to the low temperature data. 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