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Precision electrodynamics of unconventional superconductors : microwave spectroscopy and penetration… Bobowski, Jake Stanley 2010

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Precision Electrodynamics of Unconventional Superconductors Microwave Spectroscopy and Penetration Depth by Jake Stanley Bobowski B.Sc., The University of Manitoba, 2002 M.Sc., The University of British Columbia, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2010 c
 Jake Stanley Bobowski 2010 Abstract Precision measurements of the electrodynamics of unconventional superconductors are reported. Measurements of the magnetic penetration depth temperature depen- dence are made via cavity perturbation and a novel microwave spectrometer gives the surface resistance continuously from 0.05 to 26.5 GHz. The microwave conductivity of single crystals of YBa2Cu3O6:50 has been measured using the broadband spectrometer. Conductivity spectra were measured after prepar- ing the crystal in the ortho-II phase in which the Cu-O chain oxygens are ordered into alternating full and empty chains. These spectra exhibit features expected for quasiparticle scattering from dilute weak impurities in a d-wave superconductor. The measurements were repeated after reducing the degree of oxygen order in the Cu-O chains. The disordered spectra retain the weak-limit scattering features, however, increased quasiparticle scattering broadens the widths. The conductivity of a new generation of samples, with an order of magnitude lower impurity concentrations, are unchanged from those of the older generation samples. It is shown that in both gen- erations of crystals, even with ordered Cu-O chains, the spectral widths are largely determined by residual disorder in the chains. The electrodynamics of single crystal Ba0:72K0:28Fe2As2 and Ba(Fe0:95Co0:05)2As2 have been investigated. Measurements of (T ) are used to extract the super
uid density 2(0)=2(T ) which is observed to approach Tc linearly indicative of mean-eld behaviour. At low-temperatures, the super
uid density obeys the power law 2(0)=2(T ) = 1 (T=T ?)n with n varying from 2.1 to 2.7 for the three crystals studied. In all three samples, at T=Tc  0:04, there is an anomalous step-like feature of order 1 A in (T ) that is of unknown origin. The surface resistance of two Ba0:72K0:28Fe2As2 crystals reveal a sample-dependent extrinsic loss. The spectra, however, share a common temperature dependence and follow !2 when the extrinsic loss is removed. This frequency dependence translates to a 
at quasiparticle conductivity and implies a high scattering rate ( 200 GHz). The microwave spectroscopy of a Ba(Fe0:95Co0:05)2As2 sample reveals more anomalous behaviour. Even after subtracting o the extrinsic loss, the measuredRS(!; T ) rapidly ii Abstract increases above ~!=kBTc  0:04. The resistivity of the Ba(Fe0:95Co0:05)2As2 sample is an order of magnitude larger than that of the K-doped samples suggesting the anomalous behaviour may be activated by enhanced impurity concentrations. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1 Conventional Superconductivity . . . . . . . . . . . . . . . . . . . . . 1 1.1 Nearly a Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 More than Perfect Conductivity . . . . . . . . . . . . . . . . . . . . 1 1.3 The London Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 The Isotope Eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 The Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . 7 1.5.1 Length Scales & the GL Parameter . . . . . . . . . . . . . . . 8 1.6 Type-II Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7.1 Cooper Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7.2 BCS Ground State . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 The Josephson Eect . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.8.1 The dc Josephson Eect . . . . . . . . . . . . . . . . . . . . . 18 1.8.2 The ac Josephson Eect . . . . . . . . . . . . . . . . . . . . . 19 1.8.3 The dc SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.9 Beyond Conventional . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Unconventional Superconductivity . . . . . . . . . . . . . . . . . . . 26 2.1 Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.2 Doping Phase Diagram . . . . . . . . . . . . . . . . . . . . . 28 iv Table of Contents 2.2 Mott Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Mott Metal-Insulator Transition . . . . . . . . . . . . . . . . 29 2.2.2 The Large U=t Limit . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.3 Oxygen p-Orbitals . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.4 Doping a Charge Transfer Insulator . . . . . . . . . . . . . . 35 2.3 dx2y2 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 Delicate Superconducting State . . . . . . . . . . . . . . . . . 38 2.4 The \Normal State" . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.1 Overdoped . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.2 Underdoped . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5 Pnictide Superconductivity . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.1 Qualitative Assessment . . . . . . . . . . . . . . . . . . . . . 43 3 Superconductor Electrodynamics . . . . . . . . . . . . . . . . . . . . 49 3.1 Surface Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.1 The Thin Limit . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.2 RS(!) and XS(!) . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Two-
uid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Power Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Experimental Techniques: Penetration Depth . . . . . . . . . . . . 59 4.1 Magnetic Penetration Depth . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Ac Susceptometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Cavity Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Sub-kelvin Cavity Perturbation . . . . . . . . . . . . . . . . . . . . . 71 4.5 Loop-Gap Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Robinson Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5 Experimental Techniques: Microwave Spectroscopy . . . . . . . . 80 5.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 Rectangular Coaxial Line Assembly . . . . . . . . . . . . . . . . . . 83 5.3 Sample Stage Design/Sample Loading . . . . . . . . . . . . . . . . . 85 5.4 Measurement Technique . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5 Thermal Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.6 Signal Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.7 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 v Table of Contents 5.8 Signal Acquisition and Processing . . . . . . . . . . . . . . . . . . . 107 5.8.1 Lock-in Detection . . . . . . . . . . . . . . . . . . . . . . . . 111 5.9 Power Ratio and RS(!) . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.10 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.10.1 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . 131 6 Normal State Microwave Spectroscopy . . . . . . . . . . . . . . . . 137 6.1 Frequency-Independent Conductivity . . . . . . . . . . . . . . . . . . 137 6.2 RS(!) in the Thin-Limit . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.3 Drude Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.4 The Anomalous Skin Eect . . . . . . . . . . . . . . . . . . . . . . . 142 6.5 RS(!; T ) of a Superconductor . . . . . . . . . . . . . . . . . . . . . . 147 7 Cu-O Chain Defects in YBa2Cu3O6+y . . . . . . . . . . . . . . . . . . 149 7.1 Charge Carrier Doping in YBa2Cu3O6+y . . . . . . . . . . . . . . . . 150 7.2 Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 . . . . . . . . . . 156 7.3 Microwave Spectroscopy of Disordered YBa2Cu3O6:5 . . . . . . . . . 163 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.5 Chain Defects in YBa2Cu3O6:50 . . . . . . . . . . . . . . . . . . . . . 172 8 Microwave Electrodynamics of K- and Co-doped BaFe2As2 . . . . 181 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.3 Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 8.4 Microwave Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.4.1 Sources of the Extrinsic Microwave Loss . . . . . . . . . . . . 203 8.5 Microwave Spectroscopy of Ba(Fe0:95Co0:05)2As2 . . . . . . . . . . . . 205 8.6 Iron-Arsenide Summary . . . . . . . . . . . . . . . . . . . . . . . . . 210 9 Ultra-Low-Temperature Microwave Spectroscopy . . . . . . . . . . 212 9.1 Dilution Fridge Mounting . . . . . . . . . . . . . . . . . . . . . . . . 213 9.2 Preliminary Measurements: Sr2RuO4 . . . . . . . . . . . . . . . . . . 219 9.3 Quartz Tube (T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 9.4 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 9.4.1 Ortho-I YBa2Cu3O6:99 . . . . . . . . . . . . . . . . . . . . . . 224 9.4.2 Universal Conduction . . . . . . . . . . . . . . . . . . . . . . 225 9.4.3 Heavy Fermions . . . . . . . . . . . . . . . . . . . . . . . . . 228 vi Table of Contents 9.4.4 Cooling by Adiabatic Demagnetization . . . . . . . . . . . . . 229 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Appendices A Oscillator Strength Sum Rule . . . . . . . . . . . . . . . . . . . . . . 242 B Properties of CuGa Alloy at 1 K . . . . . . . . . . . . . . . . . . . . 244 B.1 Resistivity and Thermal Conductivity . . . . . . . . . . . . . . . . . 244 B.2 Specic Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 C Cascaded Low-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . 247 D Ortho-I YBa2Cu3O6:50 Doping . . . . . . . . . . . . . . . . . . . . . . 250 D.1 Numerical Conrmation . . . . . . . . . . . . . . . . . . . . . . . . . 250 D.2 Combinatorics Analysis of Ortho-I YBa2Cu3O6+y Doping . . . . . . 254 D.3 Ortho-I YBa2Cu3O6+y Doping from Expectation Values . . . . . . . 260 E Extracting 1(!) from Measurements of RS(!) . . . . . . . . . . . . 262 E.1 Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 E.2 Extracting 1n(!; T ) from RS(!; T ) . . . . . . . . . . . . . . . . . . . 264 F Sr2RuO4 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 F.1 Floating Zone Crystal Growth . . . . . . . . . . . . . . . . . . . . . 268 F.2 O2 Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 vii List of Tables 2.1 Ionization and electron congurations of the HiTc parent compounds. 31 2.2 Allowed singlet pairing states for a square CuO2 plane. . . . . . . . . 37 2.3 The iron-arsenide 1111 and 122 superconductors. . . . . . . . . . . . 45 2.4 Pnictide 1111, 122, 111, and 011 superconductors. . . . . . . . . . . . 46 2.5 Ionization and electron congurations of the parent compound BaFe2As2. 46 5.1 Experimental determination of the magnitude of H at the sample sites. 93 5.2 Physical dimensions of the spectrometer low-pass thermal lters. . . . 97 5.3 Numerical values of the spectrometer thermal capacitors and resistors. 98 5.4 Noise analysis of several op amps. . . . . . . . . . . . . . . . . . . . . 106 5.5 Surface resistance thin-limit t results. . . . . . . . . . . . . . . . . . 116 5.6 Cuto frequencies of the TE01 and TE10 modes. . . . . . . . . . . . . 118 5.7 Summary of sample thermal stage frequency response t parameters. 127 6.1 The important anomalous skin eect parameters for pure tin. . . . . . 145 7.1 Chainlet lengths and CuO2 holes contributed for ortho-I YBa2Cu3O6:5. 156 7.2 Details of the four measured YBa2Cu3O6:5 single crystals. . . . . . . . 162 7.3 Spectral widths of high-purity and ultra-high-purity YBa2Cu3O6:5. . . 163 8.1 Iron arsenide sample transition temperatures and dimensions. . . . . 184 8.2 Power law and two-gap super
uid density t parameters. . . . . . . . 194 8.3 Spectral width estimates for Ba0:72K0:28Fe2As2 sample B. . . . . . . . 200 8.4 Iron-arsenide sample t and  from ts to normal state RS(!; T ). . . . 209 9.1 Dilution refrigerator low-pass lter temperatures and time constants. 217 9.2 Extracted t parameters from normal state Sr2RuO4 RS(!; T ) spectra. 220 9.3 Estimates of the ab-plane residual resistivity in Sr2RuO4. . . . . . . . 222 B.1 Resistivity and thermal conductivity of a CuGa alloy. . . . . . . . . . 244 D.1 Fractional holes of half-lled Cu-O chains with 8 oxygen ion sites. . . 252 viii List of Tables D.2 Probabilities of the four pairs of states of neighbouring oxygen sites. . 261 ix List of Figures 1.1 Timeline of a century of superconductivity. . . . . . . . . . . . . . . . 2 1.2 Distinguishing a superconductor from a perfect conductor. . . . . . . 4 1.3 Type-II superconductor phase diagram. . . . . . . . . . . . . . . . . . 11 1.4 Distinguishing type-I and type-II superconductivity. . . . . . . . . . . 12 1.5 Phonon mediated electron-electron interaction. . . . . . . . . . . . . . 15 1.6 Josephson junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7 Flux quantization in a superconducting ring. . . . . . . . . . . . . . . 19 1.8 The dc SQUID. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.9 The dc SQUID screening and critical currents. . . . . . . . . . . . . . 22 1.10 The dc SQUID I-V characteristic and voltage response. . . . . . . . . 23 1.11 Classication of the known superconductor families. . . . . . . . . . . 25 2.1 The generic ABX3 perovskite unit cell. . . . . . . . . . . . . . . . . . 27 2.2 Unit cell of La2xSrxCuO4. . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Unit cells of YBa2Cu3O6 and YBa2Cu3O7. . . . . . . . . . . . . . . . 29 2.4 Generic HiTc (cuprate) T -p phase diagram. . . . . . . . . . . . . . . . 30 2.5 The lower and upper Hubbard bands. . . . . . . . . . . . . . . . . . . 33 2.6 The CuO2 plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7 The superexchange interaction. . . . . . . . . . . . . . . . . . . . . . 34 2.8 Charge transfer insulator. . . . . . . . . . . . . . . . . . . . . . . . . 35 2.9 Superconducting gaps: s-wave and d-wave. . . . . . . . . . . . . . . . 37 2.10 Distorted d-wave superconducting gap. . . . . . . . . . . . . . . . . . 38 2.11 Fermi arcs in the pseudogap phase of underdoped cuprates. . . . . . . 42 2.12 Fermi pockets in underdoped YBa2Cu3O6:5. . . . . . . . . . . . . . . 43 2.13 Ba0:72K0:28Fe2As2 platelet single crystals. . . . . . . . . . . . . . . . . 44 2.14 The CuO2 and FeAs planes. . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 The thin limit eective surface impedance. . . . . . . . . . . . . . . . 50 3.2 Normal metal thin limit eective surface resistance. . . . . . . . . . . 53 3.3 LR circuit analogy for losses in a superconductor. . . . . . . . . . . . 54 x List of Figures 3.4 The conductivity of a superconductor. . . . . . . . . . . . . . . . . . 57 4.1 Density of states of s-wave, d-wave, and dirty d-wave superconductors. 60 4.2 Exponential temperature dependence of (T ) of Pb0:95Sn0:05. . . . . 61 4.3 Linear temperature dependence of (T ) of YBa2Cu3O6:95. . . . . . . 61 4.4 Quadratic temperature dependence of (T ) of YBa2 (Cu1xZnx)3O6:95. 62 4.5 A platelet sample in a homogeneous external magnetic eld. . . . . . 63 4.6 Primary and secondary coil arrangement of an ac susceptometer. . . . 64 4.7 Expulsion of magnetic eld from a platelet sample. . . . . . . . . . . 65 4.8 Typical sample thermal stage. . . . . . . . . . . . . . . . . . . . . . . 67 4.9 Cartoon of the cavity perturbation measurement. . . . . . . . . . . . 69 4.10 Sub-kelvin cavity perturbation. . . . . . . . . . . . . . . . . . . . . . 72 4.11 Microwave loop-gap resonator. . . . . . . . . . . . . . . . . . . . . . . 74 4.12 Assembled loop-gap resonator. . . . . . . . . . . . . . . . . . . . . . . 75 4.13 Robinson oscillator circuit schematic and digital photograph. . . . . . 76 4.14 Oscillator frequency stability. . . . . . . . . . . . . . . . . . . . . . . 78 4.15 Extracting f from the raw oscillator data. . . . . . . . . . . . . . . 79 4.16 f and  versus T extracted from oscillator data. . . . . . . . . . . 79 5.1 Microwave spectrometer cross-section schematic. . . . . . . . . . . . . 81 5.2 Dimensions of the rectangular waveguide cross-section. . . . . . . . . 82 5.3 Design of the rectangular coaxial line. . . . . . . . . . . . . . . . . . . 84 5.4 Digital photograph of septum before and after PbSn coating. . . . . . 84 5.5 Mating the semirigid coaxial cable with the rectangular coaxial cable. 85 5.6 Cross-section of coaxial line mating. . . . . . . . . . . . . . . . . . . . 86 5.7 Digital photograph of the sample thermal stage. . . . . . . . . . . . . 87 5.8 Wiring of the sample stage. . . . . . . . . . . . . . . . . . . . . . . . 87 5.9 Digital photograph of the sample stage wiring. . . . . . . . . . . . . . 88 5.10 Side plate design for loading the sample stage. . . . . . . . . . . . . . 89 5.11 Sample loading and the microwave spectrometer. . . . . . . . . . . . 90 5.12 Design of the rectangular waveguide. . . . . . . . . . . . . . . . . . . 91 5.13 Equivalent circuit: Low-pass thermal lters. . . . . . . . . . . . . . . 96 5.14 Haller-Beeman bolometer calibration. . . . . . . . . . . . . . . . . . . 99 5.15 Low-temperature Haller-Beeman bolometer calibration. . . . . . . . . 100 5.16 Setting and determining the sample temperature. . . . . . . . . . . . 100 5.17 Bolometer signal conditioning electronics. . . . . . . . . . . . . . . . . 102 5.18 Non-inverting amplier noise analysis. . . . . . . . . . . . . . . . . . 103 xi List of Figures 5.19 Eective noise temperature. . . . . . . . . . . . . . . . . . . . . . . . 103 5.20 Calibration circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.21 Typical calibration curve. . . . . . . . . . . . . . . . . . . . . . . . . 108 5.22 Lock-in detection: software signal processing. . . . . . . . . . . . . . . 109 5.23 Low- and high-pass square-wave ltering. . . . . . . . . . . . . . . . . 110 5.24 Schematic diagram of the spectrometer detection system. . . . . . . . 111 5.25 The measured power spectrum and RS(!) ratio. . . . . . . . . . . . . 113 5.26 RS(!) ratio of two platelet AgAu alloy samples. . . . . . . . . . . . . 114 5.27 Thin limit ReS ratio of two platelet AgAu alloy samples. . . . . . . . 115 5.28 The AgAu ratio samples mounted on the sapphire plates. . . . . . . . 116 5.29 Rectangular coax cross-section and E- and B-eld congurations. . . 117 5.30 Mapping TE modes of a rectangular coax to ridged waveguides. . . . 118 5.31 Background power absorption of the sapphire plate. . . . . . . . . . . 120 5.32 Eective surface resistance background of the sapphire plate. . . . . . 121 5.33 Cartoon drawing of the sample hot nger. . . . . . . . . . . . . . . . 122 5.34 Calculated normalized frequency response of the sample stage. . . . . 125 5.35 Measured frequency response of the sample stage at 1.2 K. . . . . . . 126 5.36 Fit to the measured frequency response of the sample thermal stage. . 127 5.37 Sample stage frequency response as a function of temperature. . . . . 128 5.38 Quartz tube thermal diusivity B as a function of temperature. . . . 129 5.39 The temperature-independent reference stage frequency response. . . 130 5.40 Numerically calculated sample stage frequency response. . . . . . . . 133 5.41 Numerically calculated sample stage response to a dc power. . . . . . 135 5.42 Numerically calculated sample stage frequency response at 7 K. . . . 136 6.1 The normal state RS(!) of Sr2RuO4. . . . . . . . . . . . . . . . . . . 138 6.2 The thin-limit normal state RS(!) spectrum of Ba0:72K0:28Fe2As2. . . 139 6.3 Normal state RS(!) of SrTi1xNbxO3 and the Drude model. . . . . . 141 6.4 Cartoon picture of the anomalous skin eect. . . . . . . . . . . . . . . 142 6.5 Measured (T ) and (T ) of pure Sn from 1.2 to 10 K. . . . . . . . . 144 6.6 Normal state RS(!) and the anomalous skin eect in pure tin. . . . . 146 6.7 Superconducting surface resistance spectra of pure Sn. . . . . . . . . 148 7.1 Hole doping of YBa2Cu3O6+y by manipulation of the Cu-Oy chains. . 151 7.2 A half-lled Cu-O chain layer in the ortho-II phase. . . . . . . . . . . 152 7.3 Crystal structure of ortho-II YBa2Cu3O6:5. . . . . . . . . . . . . . . . 153 7.4 YBa2Cu3O6+y Tc as a function of doping p. . . . . . . . . . . . . . . . 154 xii List of Figures 7.5 A half-lled Cu-O chain layer in the ortho-I phase. . . . . . . . . . . 155 7.6 Surface resistance spectra of ortho-II ordered YBa2Cu3O6:52. . . . . . 158 7.7 Conductivity spectra of ortho-II ordered YBa2Cu3O6:52. . . . . . . . . 159 7.8 Conductivity spectra of YBa2Cu3O6:52 and the oscillator strength. . . 160 7.9 Comparison of RS(!; T ) of four YBa2Cu3O6:5 single crystals. . . . . . 161 7.10 Low-temperature 1(!) of four dierent YBa2Cu3O6:5 single crystals. 163 7.11 Frequency shift and low-temperature a(T ) of YBa2Cu3O6:5. . . . . 165 7.12 YBa2Cu3O6:50 a-axis RS and 1 for ordered/disordered Cu-O chains. . 166 7.13 YBa2Cu3O6:50 b-axis RS and 1 for ordered/disordered Cu-O chains. . 168 7.14 Quasiparticle conductivity spectral widths of YBa2Cu3O6:50. . . . . . 169 7.15 The oxygen ordered phases of YBa2Cu3O6+y. . . . . . . . . . . . . . . 174 7.16 Cu-O chain ordering temperature program. . . . . . . . . . . . . . . . 175 7.17 YBa2Cu3O6:33 resistance as a function of annealing time. . . . . . . . 175 7.18 Schematic cross-section of the annealing apparatus. . . . . . . . . . . 176 7.19 Detailed views of the sample resistance during annealing. . . . . . . . 177 7.20 Possible ortho-II domain boundary defects. . . . . . . . . . . . . . . . 179 7.21 Temperature gradient annealing scheme. . . . . . . . . . . . . . . . . 180 8.1 Ba0:72K0:28Fe2As2 and Ba(Fe0:95Co0:05)2As2 single crystal photographs. 185 8.2 Field dependence of the Ba0:72K0:28Fe2As2 superconducting trasition. 186 8.3 Measured (T ) of all three iron-arsenide samples studied. . . . . . . 187 8.4 Extracted super
uid density of the three iron-arsenide samples. . . . 188 8.5 Super
uid density of the three iron-arsenide samples near Tc. . . . . . 190 8.6 Measured low-temperature (T ) of the iron-arsenide samples. . . . 191 8.7 Power law and two-gap ts to the low-temperature super
uid density. 193 8.8 Raw low-temperature frequency shift of sample B. . . . . . . . . . . . 195 8.9 Low-temperature (T ) feature. . . . . . . . . . . . . . . . . . . . . 196 8.10 Microwave surface resistance of Ba0:72K0:28Fe2As2 samples A and B. . 198 8.11 RS(!; T ) of the Ba0:72K0:28Fe2As2 samples A and B. . . . . . . . . . 199 8.12 Spectral width estimates for Ba0:72K0:28Fe2As2 sample B. . . . . . . . 201 8.13 RS(!; T ) at 13 GHz as a function of T=Tc for several superconductors.202 8.14 SEM images of delaminated Ba0:72K0:28Fe2As2 crystal edges. . . . . . 204 8.15 Measured RS(!; T ) spectra of the Ba(Fe0:95Co0:05)2As2 crystal. . . . . 206 8.16 RS(!; T ) of the Ba(Fe0:95Co0:05)2As2 crystal. . . . . . . . . . . . . . 207 8.17 Normal state RS(!; T ) spectra of samples B and C. . . . . . . . . . . 209 9.1 Mounting the microwave spectrometer onto the dilution refrigerator. . 213 xiii List of Figures 9.2 Thermal lters used in the dilution refrigerator mounting scheme. . . 215 9.3 Estimating the relevant dilution refrigerator operating temperatures. 216 9.4 Stability of the temperaure-controlled mixing chamber temperature. . 218 9.5 Digital photographs of the measured Sr2RuO4 single crystal. . . . . . 219 9.6 Normal and superconducting state RS(!; T ) spectra of Sr2RuO4. . . . 221 9.7 Signal response at sub-kelvin temperatures. . . . . . . . . . . . . . . . 223 9.8 The measured a-axis conductivity spectra of ortho-I YBa2Cu3O6:99. . 225 B.1 Determining the specic heat of the CuGa alloy. . . . . . . . . . . . . 245 C.1  eT2.eTB of the spectrometer low-pass thermal lter. . . . . . . . . . . 248 C.2  eT1.eTB of the spectrometer low-pass thermal lter. . . . . . . . . . . 249 D.1 The 20 ways to ll six Cu-O chain sites with three oxygen ions. . . . 251 D.2 The 70 ways to ll eight Cu-O chain sites with four oxygen ions. . . . 253 D.3 The ratio ny=n1 approaches y 2 as the Cu-O chain length is increased. 254 D.4 Combinatorics for one chainlet and for two chainlets. . . . . . . . . . 255 D.5 Splitting the m lled Cu-O chain sites into q chainlets. . . . . . . . . 256 E.1 Conductivity spectra of YBa2Cu3O6:5 at 8.6 K. . . . . . . . . . . . . 266 F.1 Digital images from a Sr2RuO4 crystal growth in an image furnace. . 268 F.2 Digital photograph of two as-grown Sr2RuO4 single crystals. . . . . . 269 F.3 T -dependence of 0 and 00 of as-grown and O2-annealed Sr2RuO4. . . 270 xiv Acknowledgements A marathon followed by a solo bike trip across Canada and the purchase of my rst home with my wife are the events that bookend my PhD degree. In between are my wedding, my rst trip across the Atlantic to Europe, a three month stay in Kyoto, and a move to Kelowna. Throughout it all there was, of course, experimental low- temperature physics. Sleepless nights spent in the lab tracking down a stubborn leak, or in front of a computer staring at a blank screen, or reading and rereading seemingly incomprehensible theory papers were exchanged for a few 
eeting moments of success that make it worthwhile (at least in hindsight). Needless to say, I have not achieved, nor would I have achieved, anything on my own. I have benetted from the continuous support and companionship of a large cast of people. A small fraction of these people are acknowledged below. First and foremost, I owe a great deal of gratitude to my supervisor Walter Hardy. Walter has the distinct ability to optimize all aspects of an experiment to generate the highest possible resolution data in the most ecient way. He is always willing to help and, even better, always follows through on his commitments. Doug Bonn is able to process the vast and wide-ranging superconductivity literature and succinctly communicate (verbally and in writing) the overall \big" picture and pinpoint key unresolved issues in a most understandable way. Discussions with Doug have always left me with a lasting impression and wanting more. It is no exaggeration to suggest that the past and present success of the UBC superconductivity group rests rmly upon the foundation of Ruixing Liang's meticulous crystal growth program. This will almost certainly continue to be the case in the foreseeable future. It has been an absolute pleasure to work and chat with Pinder Donsanjh. Pinder's surgical-like steadiness and precision have saved me on more than a few occasions. Special thanks to the UBC Physics and Astronomy machine shop. The majority of the work presented in thesis would have been impossible without their expertise. Thanks to my PhD committee members who have been charged with the undesirable task of reading this entire document. I have benetted from working alongside many talented graduate students and xv Acknowledgements postdocs, but none more so than Patrick Turner. Pat showed me the ropes when I rst arrived, but more importantly we became and remain good friends. I'll never forget the sound of Pat's leg breaking. However, all the more clearly, I'll remember Pat suggesting that he ride a bicycle to the nearest hospital. Any worthwhile work that I completed as a graduate student only builds upon achievements of past and current group members. A sincere thanks to: David Broun, Saeid Kamal, Chris Bidinosti, James Day, Rinat Ofer, Jordan Baglo, B. J. Ramshaw, Shun Chi, Lynne Semple, Darren Peets, Kevin Musselman, Geo Mullins, Marty Kurylowicz, Richard Harris, Ahmad Hosseini, Jennifer DeBenedictis, Andrea Morello, and honourary member Dan Beaton. Numerous summer students have passed through the superconductivity lab while I was a member. Those who have con- tributed directly to the work presented in this thesis include: Alexandre Rousseau, Antonio Elias, Marc L'Heureux, and Simon Hastings. A warm thanks to Maeno-san, Ishida-san, Murata-san, and the members of the Quantum Materials Lab at Kyoto University. You all went out of your way to make my stay in your laboratory comfortable and productive. I will never be totally com- fortable with the fact that I visited your lab as a foreigner (gaigin) only to force the entire group to communicate with me in my native language. Although, I suppose I was never uncomfortable enough to become competent in Japanese! Finally, I must extend a heartfelt thanks to my family. Your unwavering support has been invaluable. Aunty Joyce, your company and friendship while I've been in British Columbia are deeply appreciated. Sadly, Gannie Bobowski and Uncle Frank passed away during the course of my PhD. All you have done for me will always be remembered. Thank you. Mom and Dad, I could not have asked for any more from you. You have always been model parents. Hiro your relentless eort to continuously upgrade everything in your life is truly inspiring. You must often contemplate upgrading your husband, who could certainly use some improvements. But, please remember that I'm a work in progress and, although I don't show or say it nearly often enough, you are what I value most in life. xvi Chapter 1 Conventional Superconductivity 1.1 Nearly a Century Throughout its history superconductivity has proven to be an extremely challenging phenomenon to probe experimentally and understand theoretically. Indeed, super- conductivity would only be discovered after Kamerlingh Onnes successfully liquied helium in 1908, a considerable engineering accomplishment at the time. Three years later Onnes found that the resistivity of solid mercury abruptly vanished when cooled below 4.2 K [1]. Nearly a century later, superconductivity continues to be one of the most active elds of research in condensed matter physics. Along the way super- conductivity has consistently deed conventional wisdom and has proven to be ripe with exotic physics. The complexity and richness of the eld has fueled the develop- ment of a variety of new theoretical and experimental tools that will, no doubt, nd applications in other areas of condensed matter physics. A timeline of some of the major milestones in superconductivity research is given in Fig. 1.1. Breakthroughs that rejuvenate the eld tend to cluster and occur with a periodicity of approximately 20 years. The most recent breakthrough was the discovery of superconductivity in the pnictide materials, 22 years after the discovery of the HiTc cuprate materials. Nodal superconductivity and the pseudogap phase are included in the timeline be- cause they are of current interest and continue to occupy the bulk of the attention of both experimentalists and theorists in the eld. In the following sections each of the items highlighted in the gure, up to and including HiTc, will be discussed in varying amounts of detail. 1.2 More than Perfect Conductivity Below a critical transition temperature Tc, superconductors exhibit two hallmark features. One being zero electrical resistance and the other being the expulsion of magnetic 
ux from the bulk of the superconductor. For example, the lifetime of a persistent current 
owing in a superconducting ring has been estimated to be greater 1 1.2. More than Perfect Conductivity Figure 1.1: Timeline of signicant dates in the history of superconductivity research. More emphasis has been placed on developments of the past 20 years. Dates in blue occurred during the course of this thesis project. than the age of the universe1. The exclusion of magnetic elds from the interior of a superconductor is called the Meissner eect and it is this property that distinguishes superconductivity from perfect conductivity. The current density J 
owing in a conductor is related to the electric eld E 1This result holds for a superconducting ring well below Tc and whose diameter is large compared to the Cooper pair coherence length. 2 1.3. The London Equations inside the conductor by Ohm's law: J = E; (1.1) where the constant of proportionality is the electrical conductivity . For a perfect conductor,  ! 1, which guarantees that E = 0 within the bulk of the material. When combined with Faraday's law: rE = @B @t ; (1.2) one nds that @B=@t = 0, or in other words, that the magnetic eld B inside a perfect conductor is constant. In contrast, the magnetic eld inside a superconduc- tor is zero regardless of the initial magnetic elds present before entering into the superconducting state2. Figure 1.2 illustrates this distinction. 1.3 The London Equations The London equations phenomenologically describe lossless conduction and the Meiss- ner eect [2]. In the Drude model, the equation of motion of conduction electrons in a metal is assumed to be: m dv dt = eE mv  ; (1.3) where m is the electron mass, v is the average electron drift velocity, e is the electron charge, and  is a relaxation time. The probability that an electron will experience a scattering event in time interval dt is given by dt= [3]. In a normal metal, equilibrium is reached when v = eE=m and the current density J = nev = (ne2=m)E, where n is the electron number density. This expression is just Ohm's law (Eq. 1.1) with  = ne2=m. A simple way to model the perfect conductivity of a superconductor is to force the scattering rate 1= ! 0. In this case the current density obeys: dJ s dt = nse 2 m? E; (1.4) which is the rst London equation. The subscript \s" indicates that this equation is describing the charge carriers that contribute to the supercurrent and these charge carriers have mass m?. Rather than competing against a resistance as in a nor- 2In the mixed state, magnetic elds can thread through a superconducting material as vortices. 3 1.3. The London Equations Figure 1.2: The pink conductor in the top row enters into a state of perfect conductiv- ity ( !1) when cooled below Tc. Whereas, the blue conductor in the bottom row enters into a superconducting state ( !1 and the Meissner eect) when cooled be- low Tc. (a) While in their normal states, T > Tc, the two conductors are placed in an external dc magnetic eld Bext. In both cases the magnetic eld penetrates through the conductors undisturbed. (b) The externally applied eld remains in place and the conductors are cooled below Tc. There is no change in the magnetic eld through the perfect conductor, however the magnetic eld is expelled from the superconductor to within a penetration depth  from its surface. (c) With T < Tc the external magnetic eld is switched o. The magnetic eld in the bulk of the superconductor remains zero. For the perfect conductor, because @B=@t = 0, the internal magnetic eld is sustained at B = Bext. As a practical matter, the magnitude of the expelled 
ux from the interior of a superconductor can be reduced by imperfections in the sample. mal metal, in a superconductor any electric eld will accelerate the superconducting charge carriers. Equilibrium dJ s=dt = 0 is reached only when the internal electric eld is zero. The second London equation cannot be obtained from a purely classical argument. Nevertheless, the original insights of the London brothers will be followed to obtain 4 1.3. The London Equations the desired result. Taking the curl of Eq. 1.4 and applying Faraday's law leads to: @ @t  r J s + nse 2 m? B  = 0: (1.5) This equation allows for nonzero constant magnetic elds as in a perfect conductor, see Fig. 1.2(c). The London brothers were aware of the experimental results demonstrat- ing the Meissner eect and postulated that the expression inside the parentheses was not only constant, but identically zero. Using Ampere's law to rewrite 0J s = rB and the Maxwell equation r B = 0, one arrives at the second London equation: r2B = B 2L ; (1.6) where the London penetration depth 2L = 0nse 2=m? has been dened. An alternative quantum \derivation" of the London equation proceeds as follows: in the absence of an applied eld, the quantum ground state of the system has zero net canonical momentum: < p >=< m?v + eA >= 0; (1.7) whereA is the vector potential. Provided the system remains in its ground state in the presence of a eld, the electrons will be accelerated to a velocity < vs >= eA=m? and current density is given by: J s = nsevs = nse 2A m? : (1.8) Taking the time derivative of this expression reproduces the rst London equation (Eq. 1.4) and taking the curl gives the second (Eq. 1.6) [4]. For certain combinations of eld geometries and sample shapes, one can assume B only varies in one direction. In such a case, the 1D solution of Eq. 1.6 is: B = B0e x=L : (1.9) Magnetic elds are exponentially attenuated as they penetrate into a superconductor. For T  Tc, depending on the material, L is typically in the range 500-2000 A. The penetration depth is distinct from the skin depth  in normal metals in one very important way: it is independent of frequency, !. In contrast,  / !1=2 and is innite for dc magnetic elds. A dc magnetic eld will therefore completely penetrate a 5 1.4. The Isotope Eect normal metal as in Fig. 1.2(a), but is excluded from the interior of a superconductor3. In the simplest superconductors, the Meissner state persists until a critical eld Hc is reached4. Above Hc the superconductor is driven into its normal state. Hc is a maximum for T ! 0 and must approach zero as T ! Tc. Empirically it is found that a good approximation to Hc(T ) is [4]: Hc(T )  Hc(0) " 1  T Tc 2# : (1.10) The limiting T ! Tc behavior can be found by writing (T=Tc)2 as [(1 t) + 1]2  1 + 2t, where t  T=Tc. The result is: Hc(T )  2Hc(0)  1 T Tc  ; T / Tc: (1.11) 1.4 The Isotope Eect Occasionally advancements in the understanding of one phenomenon will rather quickly lead to the understanding of something much greater. The isotope eect was a catalyst for the development of a full microscopic theory for what we now know as conventional superconductivity. For many of the elemental superconductors one nds experimentally that: Tc; Hc /M1=2; (1.12) where M is atomic mass of the element and can be varied by studying isotopically enriched samples [4]. Motivated by the theoretical work of Frohlich [6], the isotope eect was independently discovered by Maxwell [7] and Reynolds et al. [8] in mercury5. When combined, these authors studied ve dierent samples with M ranging from 198 to 203.4 amu and demonstrated a systematic evolution of both Tc and Hc. These results were a clear indication that lattice vibrations were playing a vital role in the superconductivity of these materials. Attempts at similar studies have been made 3It will be shown later that L is not necessarily the quantity measured in experiments because it includes screening only from the super
uid fraction of the total electron density. The true pen- etration includes screening by the normal 
uid which can be non-negligible at high frequencies, or when T is close to Tc [5]. 4By convention, critical elds are written in terms of H = B=0 for nonmagnetic materials. 5I was pleasantly surprised to discover that each of references [7] and [8] were Letters to the Editor that appeared in Physical Review and were less than one page in length. They report brie
y on the experimental technique and then show the data. The presentation is not clouded by a forced interpretation of the experimental results. The modern day Physical Review Letters has evolved to become something quite distinct from its origins. 6 1.5. The Ginzburg-Landau Theory on the HiTc materials. However the results are much more dicult to interpret for a number of reasons, the most obvious being that these materials typically contain four or ve dierent elements arranged in a complicated crystal structure. A complete microscopic theory of HiTc will likely nd its stimulus elsewhere. 1.5 The Ginzburg-Landau Theory The Ginzburg-Landau (GL) theory [9] has rmly established its niche in the eld of superconductivity. It quantitatively treats the superconducting state in the presence of large spatial variations of ns, a situation in which other theories struggle. For example, in strong magnetic elds near Hc, superconductors enter into an intermedi- ate state where one encounters boundaries between the superconducting and normal states. In unconventional superconductors a mixed state is present even in relatively weak magnetic elds. Moreover, defects in these materials, that have complex chem- ical structures, are known to cause large spatial variations in the superconducting properties [10]. The limitation of GL theory is that it is only strictly applicable for T / Tc. The GL theory of superconductivity is an extension of Landau's general theory of second-order (continuous) phase transitions which is widely celebrated as a triumph of physical intuition [4, 11]. In general, the strategy is to expand the free energy density of some system that enters into an ordered state below a critical temperature Tc in terms of an order parameter : fs = fn +  2 +  2 4: (1.13) Here fs is the free energy density of the ordered state and fn > fs is the free energy density of the normal state. The order parameter has the property that it is zero in the normal state and nonzero in the ordered state. Odd powers of  are excluded to preserve the property that fs() = fs(). For example, for an Ising spin system that orders spontaneously below Tc, the order parameter is identied as the magnetization M . The free energy cannot depend on whether the spins align up or down. In Eq. 1.13, the constant  must be positive. If it were not, then fs would always be minimized when jj ! 1. Physically,  is required to be small just below Tc for the original expansion to remain valid. On the other hand,  must be negative in the ordered state and change sign at Tc. To this end,  is generally expanded about Tc such that, to rst order, (t) = 1(t 1) where 1 > 0. With these restrictions on 7 1.5. The Ginzburg-Landau Theory and , fs is minimized for small, but nonzero, values of  2 when T < Tc, and  = 0 for T > Tc. In the absence of an applied magnetic eld and for a uniform order parameter, Eq. 1.13 is the phenomenology proposed by Ginzburg and Landau for supercon- ductors. However, the order parameter is usually written as  to emphasize the macroscopic quantum properties of the superconducting state: fs = fn + 1(t 1) j j2 + 2 j j4 ; 1 &  > 0: (1.14) This free energy is minimized when j j2 = 1(t 1)=, such that: fs  fn =  2 1(t 1)2 2 = 0H 2 c 2 ; (1.15) where the dierence in the free energy in zero eld is known as the condensation energy and denes the critical eld. This simple analysis has recovered the empirical observation of Eq. 1.11 that close to Tc, Hc(T ) / (1 t). The real strength of the GL theory is its ability to account for the eects of an applied eld and spatial gradients of the order parameter. In this case, the full free energy density is written as: fs = fn + 1(t 1) j j2 + 2 j j4 + 1 2m? j(i~r e?A) (r)j2 + B 2 20 ; (1.16) where ~ is Plancks' constant h divided by 2. A variational minimization of this free energy leads to the two GL equations [4]:  +  j j2  + 1 2m? (i~r e?A)2  = 0; (1.17a) J = ie?~ 2m? ( r   r ) j j 2 e?2A m? : (1.17b) In the absence of gradients in the order parameter, the GL current density is reduced to J =  j j2 e?2A=m? which is equivalent to Eq. 1.8 provided we make the important identication j j2 = ns / (1 t). 1.5.1 Length Scales & the GL Parameter The rst GL equation is analogous to Schrodinger's equation for particles with wave- function  , mass m?, and charge e?. The solution of this equation reveals that 8 1.5. The Ginzburg-Landau Theory deviations of  from its equilibrium value exponentially decay over a characteristic length scale: 2(T ) = ~2 2m? j(T )j ; (1.18) where  is the GL coherence length. This quantity is related to, but distinct from, a temperature independent coherence length 0  ~vF=kBTc introduced by Pippard which represents the spatial extent of the wavefunctions of the superconducting charge carriers [12]. The quantities vF and kB are the Fermi velocity and Boltzmann's con- stant respectively. Writing  (r) = j (r)j ei'(r) and working within the London gauge where r' = 0, the second to last term in Eq. 1.16 can be reworked to read: 1 2m?  (~rj j)2 + (e?A j j)2 : (1.19) The second term is the kinetic energy associated with supercurrents and can be equated to ns (m ?v2s =2) using Eq. 1.8 to yield: 1 02L = e?2 j j2 m? : (1.20) Again, it is natural to make the replacement j j2 ! ns. Furthermore, this identica- tion predicts that L(T ) / (1 t)1=2 near Tc. The combination of j j2 = = and equations 1.15 and 1.20 allows the parameters  and  to be written purely in terms of experimentally accessible quantities: (T ) =  2 0e ?2 m? H2c (T ) 2 L(T ); (1.21a) (T ) = 30e ?4 m?2H2c (T ) 4 L(T ) : (1.21b) We are now in a position to introduce the GL parameter:   L(T ) (T ) = 2 p 20Hc(T ) 2 L(T ) 0 ; (1.22) where 0  h=e? is the 
ux quantum6. To rst order, this dimensionless parameter is temperature independent and it is used to distinguish between type-I and type-II superconductors as will be seen in the following section. 6This denition anticipates the BCS result of x1.7 that e? = 2e. 9 1.6. Type-II Superconductivity The GL theory is a mean eld theory that ignores 
uctuations of the order param- eter. For temperatures suciently close to Tc, these critical 
uctuations will become important and GL theory will break down. In particular, the exponents that gov- ern the temperature dependencies of various thermodynamic quantities near Tc are expected to deviate from their mean eld values and adopt the so-called critical expo- nent values. The temperature TG at which GL theory is expected to break down can be calculated and is known as the Ginzburg criterion. The value of TG is a sensitive function of the zero temperature coherence length (0). In three dimensions [11]:TG  TcTc  = 132 (cv=kB)2 6(0) ; (1.23) where cv is the jump in the specic heat associated with the phase transition from the normal state into the superconducting state. For aluminum, Tc = 1:2 K, cv  200 Jm3K1 [13], and (0)  1:6 m [14] which gives jTG  Tcj  1018 K! Therefore, for conventional superconductors like aluminum, deviations from the GL mean eld theory are expected only within a temperature range that is hopelessly close to Tc. However, the unconventional HiTc superconductors discussed in chapter 2 have much shorter coherence lengths (typically (0)  2 nm [4]) and critical 
uctuations are expected to be important within an experimentally accessible temperature range. Lobb calculated the critical exponents expected for a variety of physical quantities near Tc. In particular, within the regime of critical 
uctuations the penetration depth is expected to obey (T ) / (1  t)1=3, whereas in the GL mean eld theory the exponent is 1=2 as seen in Eq. 1.20 [15]. Critical 
uctuations were experimentally conrmed in the penetration depth measurements of Kamal et al. on YBa2Cu3O6:95 with Tc = 93 K for 1 t spanning 103 to 101 [16]. 1.6 Type-II Superconductivity The penetration depth of aluminum is L(0)  16 nm [14], making the GL parameter   102  1 which is typical of the superconducting elements. In 1957, Abrikosov used GL theory to investigate the properties of hypothetical superconductors with  1 [17]. He found that these materials, which he termed type-II superconductors, allowed for the presence of a novel phase now called the mixed state. Type-II super- conductors have a lower and an upper critical eld, Hc1 and Hc2 respectively. Below Hc1, the type-II superconductors behave no dierently than their usual type-I coun- terparts. Similarly, aboveHc2 the superconducting state is completely suppressed just 10 1.6. Type-II Superconductivity as in the type-I materials. Type-II superconductors with Hc1 < H < Hc2, remain perfect electrical conductors7, however, the Meissner eect is strongly modied. In the mixed state, the superconducting state coexists with patches of the normal state and there are strong restrictions on the properties and arrangement of the normal regions. Figure 1.3 shows the eld versus temperature phase diagram of a type-II superconductor. 0.0 0.2 0.4 0.6 0.8 1.0 Superconducting State Mixed State H c2   T/T c H c1 Normal State Figure 1.3: Phase diagram of a type-II superconductor. The mixed state occurs between Hc1 and Hc2. Early on, the energy cost of introducing a superconductor:normal conductor do- main wall into a pure superconductor was studied. The following qualitative argu- ments highlight the essential points:  Type-I: For   L, there exists a region near the domain boundary of order  =   L where both the internal magnetic eld H and the superconducting order parameter  are suppressed as in Fig. 1.4(a). The suppression of H is due to the diamagnetic response which makes a positive contribution to the domain 7A type-II superconductor in the mixed state is a perfect conductor only if the vortices are pinned. See the discussion in the last paragraph of this section. 11 1.6. Type-II Superconductivity wall energy. The order parameter  , on the other hand, is associated with the energy savings gained by entering into the superconducting state (condensation energy). In a type-I superconductor, the condensation energy is partially lost in the region dened by  and there is a positive net energy cost for introducing domain walls.  Type-II: For the case   L (Fig. 1.4(b)), the argument is reversed. The region  enjoys the full savings of the condensation energy, but the diamagnetic response is suppressed resulting in a negative domain wall energy. As a result, patches of the normal state proliferate until the energy gained is balanced by the cost of having large gradients of  (Eq. 1.16). (a)  HType-I  Normal State (b) Normal State Type-II  H  Figure 1.4: (a) In a type-I superconductor  =   L > 0 and the net energy associated with a domain wall is positive. (b) A type-II superconductor on the other hand, has  =   L < 0 and the domain wall energy is negative. Based on gures from references [4] and [14]. From this rough argument, the crossover from type-I to type-II superconductivity would be expected to occur when  =   L  0, or   1. A full quantitative analysis shows that the true crossover occurs when  = 1= p 2 [4]. The contribution of Abrikosov was to show that in the mixed state, the normal regions organize into a triangular lattice of 
ux tubes, or vortices, each of which per- mit a single quantum of 
ux 0 to penetrate through the sample [17]. The triangular lattice is the close-packed conguration and maximizes nearest neighbour distance between the repulsive vortices. A very large number of type-II superconductors have now been discovered and the triangular vortex lattice has been experimentally con- rmed [18]. All of the unconventional superconductors are type-II, and in many cases are referred to as extreme type-II when  & 50. 12 1.7. BCS Theory At Hc1, the vortices are spaced far apart and the eld at the centre of each vortex core is Hc1 and decays radially outward. The area occupied by a vortex is roughly 2 and hence Hc1 is governed by : 0Hc1  0 2 : (1.24) As the external eld is increased, the density of the vortices increases. At Hc2, the vortices are packed as closely as  permits. In this case, the 
ux carried by each vortex core is roughly 2Hc2 such that [14]: 0Hc2  0 2 : (1.25) When a current is passed through a clean type-II superconductor in the mixed state, it will interact with the vortices causing them to move, a phenomenon known as 
ow. The Lorentz force density f between the current J and the 
ux is given by f = J B. Motion of the vortices leads to power dissipation which is a serious problem for applications. For example, in magnets wound with superconducting wire, 
ow can lead to local heating causing a section of the wire to become normal. This section of wire will experience further heating due to usual ohmic losses causing the length of the wire in the normal state to rapidly grow. This runaway process is known as quenching and in some cases can cause irreparable damage to the magnet. To combat 
ow, defects such as dislocations and/or impurities are intentionally introduced into commercially produced superconducting wire. These imperfections cause local variations of , , or Hc2 resulting in preferred sites for vortices. These imperfections pin vortices and prevent or limit 
ow [4]. Large superconducting magnets (in excess of 20 tesla in some cases) are now commonplace in Magnetic Resonance Imaging (MRI) machines, particle accelerators, and nuclear magnetic resonance (NMR) spectrometers. 1.7 BCS Theory By 1957, the existence of an energy gap   kBTc between the low-energy excitations and the ground state of superconductors was experimentally established. For exam- ple, the electronic specic heat of vanadium was found to be exponentially suppressed when T < Tc [19]. Additionally, spectroscopic studies clearly showed a gap in the excitation spectrum of superconductors for frequencies ~! < 3kBTc [20]. Another 13 1.7. BCS Theory key feature of the superconducting state was revealed when Cooper showed that any attractive interaction between electrons, no matter how weak, results in an instability of the electron Fermi sea against the formation of bound pairs of electrons [21]. Armed with a knowledge of an energy gap, the isotope eect (pointing to the importance of phonons), and the possibility of electron pairs, Bardeen, Cooper, and Schrieer were able to formulate a full microscopic theory of superconductivity [22, 23]. The BCS theory of superconductivity is arguably the most successful theory in condensed matter physics. It beautifully explained a vast amount of experimental data and was able to make successful quantitative predictions. Moreover, BCS theory conrmed the phenomenological London equations (Eqns 1.4 and 1.6) and the mean eld GL theory was shown to be equivalent to BCS theory in the limit T ! Tc [24]. A simple understanding of the microscopic mechanism driving conventional su- perconductivity is given by imagining a conduction electron moving through a lattice of positive ions. The ions are attracted to the negative charge of the passing electron causing a slight distortion of the lattice. The disturbance of the lattice cations occurs on a time scale set by the Debye frequency8 2=!D  1013 s. During this time, the initial electron, traveling at the Fermi velocity vF  106 m/s, has moved a dis- tance   1000 A. This temporary region of excess positive charge attracts a second electron and in this sense the two electrons form a bound pair whose size is typically orders of magnitudes larger than the lattice constant of the host metal. The Coulomb repulsion of the two electrons is completely screened over a length of a fewangstroms and does not impede the attractive pairing interaction. The energy required to break the pair apart is 2(T ). In a normal metal, the resistance is due predominantly to conduction electrons scattering from lattice vibrations, or phonons. In the superconducting state, the Cooper pairs of electrons can be considered as composite particles that obey Bose- Einstein, rather than Fermi, statistics. Unlike fermions, multiple bosons can occupy the same quantum state. In fact, if n  1 bosons are already in a single quantum state, the probability that the nth boson will occupy the same state is enhanced by a factor n. It is the tendency for Cooper pairs to occupy the same state that accounts for lossless conduction. Once the Cooper pairs are collectively in a state of current 
ow, it is very dicult to remove even a single electron pair from this state [25]. The remainder of this section will be devoted to developing a greater appreciation for some of the essential features of the BCS state. However, care will be taken to avoid getting bogged down in lengthy calculations [4, 26]. 8~!D = kBD, where D is the Debye temperature. 14 1.7. BCS Theory 1.7.1 Cooper Pairing Consider adding two electrons to a lled Fermi sea at T = 0. Assume that these two electrons interact with each other but, outside of the Pauli exclusion principle, not with the electrons in the Fermi sea. The electrons have wavevectors k1 and k2, where k1; k2 ' kF and kF = mvF=~ is the Fermi wavevector. The electrons will exchange phonons subject to momentum conservation: k1 + k2 = k 0 1 + k 0 2 K; (1.26) where k01 = k1  q and k02 = k2 + q if electron 1 emits a phonon of momentum q which is then absorbed by electron 2. The maximum momentum qmax of the phonon is restricted to be consistent with the maximum phonon energy ~!D. The phase space available for the phonon mediated electron-electron interaction is shown in Fig. 1.5. The allowed interactions occur in the regions where the two rings overlap. Notice that Figure 1.5: Each ring represents the possible momentum values available to the electrons. The region where the rings overlap also satises conservation of momen- tum. The phonon mediated electron-electron interaction is greatly enhanced when k1 = k2. Based on a gure from Ref. [26]. the phase space available for phonon exchange is greatly enhanced when K = 0, or equivalently, when the electrons have equal, but opposite, momentum. It is common, and appropriate, to consider only Copper pairs made up of electrons with k1 = k2, an essential element of BCS theory. The total Cooper pair wavefunction is the product of the orbital and spin parts and must be antisymmetric under the exchange of the two electrons. The symmetric orbital wavefunction  cosk  (r1  r2) is preferred because there is a large probability for the two electrons to be near each other and 15 1.7. BCS Theory thus favours pairing. The spin state of the electron pairs is therefore expected to be the antisymmetric spin singlet. By adopting a plane wave form for the electron pair wavefunction and applying the Schrodinger equation, the energy cost of adding two electrons on top of the lled Fermi sea is found to be: E  2EF  2~!De2=N(0)V0 ; (1.27) where N(0) is the density of states at the Fermi energy. In the presence of an attractive interaction, the two electrons, rather than joining the Fermi sea, lower their energy by forming a bound pair. To arrive at this result, one assumes a simple form for the electron-electron interaction potential Vkk0 , which characterizes the strength of the potential for scattering a pair of electrons with momenta (k0;k0) to momenta (k;k): Vkk0 = ( 0; E > EF + ~!D V0; otherwise : (1.28) Furthermore, the so-called weak-coupling approximation, N(0)V0  1, has been made. Experimentally, this approximation has been found to be valid for most of the classic superconductors [4, 26]. 1.7.2 BCS Ground State BCS proposed the following ground state wavefunction that automatically enforces pairs with opposite momenta and spin: j  'i = Y k  jukj+ jvkj ei'cyk"cyk#  j 0i (1.29) where j 0i is the vacuum state. The operator cyk creates an electron of momentum k and spin  = " or #. The corresponding annihilation operator is ck. These operators obey the usual anticommutation relations for fermions. The probability that the state (k ";k #) is occupied is given by jvkj2 and the probability that it is unoccupied is given by jukj2 = 1 jvkj2. The coecients uk and vk dier by a phase factor ei'. The appropriate BCS Hamiltonian is given by: HBCS = X k "knk + X kl Vklc y k"c y k#cl#cl"; (1.30) 16 1.8. The Josephson Eect where "k = ~2k2=2m is the single particle kinetic energy and nk = cykck is the number operator. In the second term, Vkl is the potential to scatter an electron pair initially in a state with (l ";l #) to a state with (k ";k #). A minimization of h ' j HBCS j  'i reveals a gap in the quasiparticle excitation spectrum: Ek = q 2k +  2 k; (1.31) where k  "k EF measures the single-particle energy relative to the Fermi energy. The energy gap k is the minimum energy required for quasiparticle excitations. Assuming Eq. 1.28 fairly represents the potential and making using of the weak- coupling approximation the gap is given by: (0)  2~!De1=N(0)V0 : (1.32) The subscript k has been dropped because under these approximations the gap is independent of k and the zero in the parenthesis indicates that this is the gap value at T = 0. Note that (0) is nearly identical to the Cooper pair binding energy of Eq. 1.27. For a BCS superconductor, Tc is the temperature at which the gap magni- tude (T ) goes to zero. This condition forces the quasiparticle excitation spectrum Ek to match the normal state excitation spectrum k.The BCS weak-coupling result for the critical temperature is 2(0) = 3:5kBTc [4]. The BCS theory of superconductivity is a remarkably successful microscopic the- ory for conventional superconductors and goes far beyond the greatly simplied and abbreviated treatment given here. For example, one can calculate the temperature dependence of the superconducting energy gap and other thermodynamic proper- ties such as the specic heat and critical eld Hc. The electrodynamic absorption spectrum and the temperature dependence of the penetration depth (T ) can also be calculated. In each case BCS theory has either explained existing experimental results or been conrmed by experiments that postdate its predictions. 1.8 The Josephson Eect As a 22 year old graduate student B. D. Josephson9 showed that, in the absence of an applied voltage, a supercurrent will 
ow between two superconductors coupled through a thin insulating material. Even more astonishing, he predicted that when 9Dr. Josephson is currently the director of the Mind-Matter Unication Project of the theory condensed matter group at the Cavendish Laboratory, Cambridge. 17 1.8. The Josephson Eect a dc voltage is applied across the insulating junction, the resulting supercurrent os- cillates with a frequency proportional to the applied voltage. The Ginzburg-Landau formalism developed in x1.5 will be used to arrive at the essential results of Joseph- son's eect. This approach is insightful because the macroscopic quantum nature of the superconducting state is emphasized. In particular, the phase of the GL order parameter  = j j ei' will play a central role. The Josephson eect has also led to the development of the Superconducting QUantum Interference Device (SQUID), only the second widespread application of superconductivity, the rst being large magnets wound from superconducting wire. 1.8.1 The dc Josephson Eect Consider the geometry of Fig. 1.6. Two bulk superconductors are separated by an Figure 1.6: Superconductor A is connected to superconductor B through a thin in- sulating barrier. In superconductor A, the phase has been arbitrarily set to zero. The dierence in phase between superconductors A and B is '. A wildly schematic representation of a single Cooper pair is shown in superconductor A and only serves to emphasize that the coherence length  is much longer than the barrier width `. insulating barrier whose width ` is much less than the superconducting coherence length . This type of junction is known as a Josephson junction. The phase of the order parameter in either of the superconductors is arbitrary, however the dierence in the phase is not, and it is denoted '. The starting point for this analysis is the rst GL equation 1.17a. In the absence of an applied eld and in one dimension, this equation reduces to: 2 d2f dx2 + f  f3 = 0: (1.33) 18 1.8. The Josephson Eect Equation 1.18 has been used and the quantity f   = 1 has been dened, where  21 = = is the equilibrium value of  10. Aslamazov and Larkin showed that, provided =` 1, the rst term dominates [27], in which case the solution is: f =  1 x `  + x ` ei'; (1.34) where the boundary conditions f = 1 at x = 0 and f = ei' at x = ` have been imposed. Substituting this solution into the second GL equation 1.17b yields the dc Josephson eect: Js = Jc sin'; (1.35) where Js is the supercurrent induced and Jc  2e~ 21=m?` is called the critical current11. 1.8.2 The ac Josephson Eect In this simple derivation of the ac Josephson eect, the 
ux that threads through a superconducting ring (see Fig. 1.7) is rst considered. Quantum mechanically, the Figure 1.7: A superconducting ring joined by a insulating junction. Deep inside the ring (along the dashed line) the magnetic eld and current density are zero. 10The notation  1 is conventionally used because it denotes the value of the order parameter \innitely deep" inside the superconductor. 11The quantity e? in Eq. 1.17b has been replaced by 2e. 19 1.8. The Josephson Eect velocity of a Cooper pair is: vs = 1 m? (i~r 2eA) ; (1.36) and the current density is J s = 2e (r)vs (r). Inside the ring j (r)j2 = ns is constant and the order parameter can be written as  = n 1=2 s ei'(r), such that: J s = 2ens m? (~r' 2eA) : (1.37) Because of the Meissner eect the magnetic eld, and hence J s, are zero deep inside the ring such that: r' = 2e ~ A: (1.38) Integrating this result around the circumference of the ring (assuming that the width of the insulating barrier is negligible), making use of Stoke's theorem, and forcing to be single valued gives: ' = 2s = 2  0 ; (1.39) where s = 0; 1; 2; : : : is an integer,  is the 
ux through the centre of the ring, and 0 = h=2e is the 
ux quantum [28]. Taking the time derivative of ' and applying Faraday's law gives the second Josephson equation: d dt ' = 2 V 0 ; (1.40) where V is the voltage drop across the junction. When inserted into Eq. 1.35, the current density Js = Jc sin [(2V=0) t] oscillates at a frequency given by V=0. 1.8.3 The dc SQUID Superconducting magnets in persistent current mode and SQUIDs are arguably the only truly remarkable applications of superconductivity. Making use of the macro- scopic quantum nature of the superconducting state, SQUIDs have revolutionized how tiny magnetic elds are detected. The topic of this section is the dc SQUID, and its close cousin, the rf SQUID, will not be discussed. This treatment will be largely qualitative and attempts to give a simple description of the operation of a practical device. Figure 1.8 is a schematic of a dc SQUID, which is essentially a superconduct- ing ring with two Josephson junctions. Assume that the two branches of the ring are identical and each Josephson junction has a critical current Ic = JcA, where A 20 1.8. The Josephson Eect Figure 1.8: In a dc SQUID two Josephson junctions, marked by crosses, are fabricated into an otherwise superconducting ring. A bias current I is passed through the device and the voltage across it is measured as a function of an externally applied 
ux a that passes through the interior of the ring. Based on gures from Refs. [29] and [30]. is the cross-sectional area of the junction (see Eq. 1.35). The SQUID is biased with a current I, such that I=2 passes through the left and right Josephson junctions, as shown in the gure. A voltage V will appear when the current through a least one of the Josephson junctions exceeds Ic. The essential feature of the dc SQUID, as described below, is that the bias current I required to enter into the nonzero voltage state is a periodic function of the applied 
ux a. Case I: 0 < a < 0=2 When there is no external 
ux through the loop, the net current in the left and right branches of the SQUID are equal. Both Josephson junctions reach their critical currents simultaneously when I = 2Ic. Now, if a 
ux 0 < a < 0=2 passes through the SQUID, a screening current Is = a=L is induced, where L is the self inductance of the ring. This screening current exactly cancels the applied 
ux in order to satisfy the requirement that  = s0 (s being 0 in this case), where  is the total 
ux through the ring. The screening current will add to the bias current in one branch of 21 1.8. The Josephson Eect the SQUID and subtract from the other, as in Fig. 1.8. Consider the junction in the left branch, the critical current is reached when I=2 + Is = Ic. When this condition is met, the net current in the right branch is I=2 Is = Ic 2Is. Therefore, the total bias current required to generate a nonzero voltage across the SQUID is: I = 2Ic  2Is = 2Ic  2a L for 0  a  0 2 : (1.41) Case II: 0=2 < a < 0 For 0=2 < a < 0 it is favourable for the screening current to 
ip directions and increase the total 
ux through the loop until  = 0. In this case, the opposite Josephson junction is the rst to exceed its critical current and this occurs when: I = 2Ic  2(0  a) L for 0 2  a  0: (1.42) Following these arguments through as a is increased further, will show that both Is and the critical bias current oscillate as a function of a with a period of 0 (see Fig. 1.9). To operate a practical dc SQUID, a large enough bias current is used (a) 0 1 2 3 0 0  / 2L  I s  a 0 (b) 0 1 2 3 0  / L   a 0 2I c Figure 1.9: (a) The screening current Is oscillates as a function of the external 
ux a. As a passes through (s+ 1=2)0, the screening current reverses direction. (b) As a result of the changing screening current, the critical current of the SQUID oscillates at the same frequency. Based on gures from Refs. [29] and [30]. such that the critical current is always exceeded. Typical I-V characteristics of a dc SQUID are shown in Fig. 1.10(a) for a = s0 and a = (s+1=2)0. As a is ramped up (or down) the change in the voltage across the SQUID is V = IR=2, where R=2 is the parallel resistance of the two Josephson junctions. From Fig. 1.9(a), the 22 1.9. Beyond Conventional (a) I 2I c a =(s+1/2) 0   V a =s 0 2I c - 0  / L (b) 0 1 2 3   a 0 V Figure 1.10: (a) I-V characteristic of the dc SQUID. The critical current of the SQUID is maximum at 2Ic when a = s0 and is a minimum when a = (s+ 1=2)0. The operating bias current is shown as a dashed line. (b) At this bias current, the voltage will oscillate as a function of a. The peak-to-peak voltage swing is V = (0=L)(R=2). Based on gures from Refs. [29] and [30]. peak-to-peak change of Is is Ip2p = 0=L, so: Vp2p =  0 L  R 2  : (1.43) Knowing R=2 and L of a particular SQUID, it is possible to detect changes in a that are tiny fractions of 0 = 2  1015 Wb [29, 30]. Commercial SQUIDs, which nd use in medical applications, are routinely used as sensitive magnetometers with noise 
oors as low as 3 fT/ p Hz. Moreover, very small custom devices are being fabricated to scan the surfaces of type-II superconductors to study vortices in the mixed state [31]. 1.9 Beyond Conventional In 1986, the eld of superconductivity was rejuvenated. Bednorz and Muller discov- ered superconductivity in the ceramic oxide La2xBaxCuO4+ [32]. This unexpected superconductor boasted a record Tc = 29 K. The record would not last long. In the following year, superconductivity was reported in a close cousin YBa2Cu3O6+y with Tc = 93 K, above the boiling point of liquid nitrogen [33]. These and related materials form the family of high-temperature superconductors (abbreviated HiTc, or sometimes HTSC). A common feature shared by all the materials in this family are weakly-coupled stacks of 2-dimensional copper oxide planes. These CuO2 planes are believed to be the essential ingredient for the superconductivity and for this reason 23 1.9. Beyond Conventional these materials are also referred to as cuprate superconductors. There are numerous other novel superconducting families and others will, no doubt, continue to be discovered. Figure 1.11 classies some of the most common fam- ilies12. There is clearly a broad spectrum of superconducting materials all with unique properties. The term conventional superconductivity has become synonymous with standard BCS superconductivity: spin singlet electron pairs with zero net momentum and a uniform k-independent energy gap (T ). However, this denition is incom- plete. BCS theory does not specify the origin of the weak attractive electron-electron interaction, only that it exists. On the other hand, when one speaks of conventional superconductivity one is almost certainly referring to a phonon mediated attractive interaction between electrons. A popular denition of unconventional superconductivity is the following [34]:X k k = 0: (1.44) This is a useful denition for the cuprates, in which the electrons pairs are spin singlet, but the gap varies as k = 0  k2x  k2y  , so-called d-wave pairing13. However, this denition of unconventional superconductivity is not useful for a material like MgB2. Experiments have revealed that MgB2 is a two-gap superconductor with both gaps showing BCS-like behaviour [35]. The essential point is that the past several decades of research has spawned many new superconductors exhibiting exotic behaviours that include: multigap supercon- ductors, superconductivity coexisting with magnetism, spin triplet pairing, strange normal state properties (pseudogap physics), superconducting 
uctuations, quasi-1D transport, . . . This thesis focuses primarily on the cuprate cuprate superconductor YBa2Cu3O6+y and the newly discovered pnictide superconductors, the majority of which possess FeAs planes in place of CuO2 planes. 12Apologies if your favourite superconductor does not appear in Fig. 1.11. 13x2.3 will point out that, because of its orthorhombic structure, the d-wave pairing state of the cuprate superconductor YBa2Cu3O6+y is mixed with the s-wave state. The d-wave state is dominant and the presence of the s-wave state is a technical detail. 24 1.9. Beyond Conventional Figure 1.11: Classications and characteristics of the most common superconductor families. 25 Chapter 2 Unconventional Superconductivity 2.1 Cuprates The HiTc cuprates dier from conventional superconductors in many ways. A number of these dierences are highlighted below. 2.1.1 Crystal Structure The cuprate superconductors have a complicated unit cell made up of four (La2xSrxCuO4+, YBa2Cu3O6+y), ve (Bi2Sr2Can1CunO2n+6), or even six (Hg1xTlxBa2Ca2Cu3O8+) elements. The chemical structure of the HiTc materials is a tetragonal or orthorhombic defected perovskite. The generic perovskite chemical formula is ABX3. The A cations are generally metallic and larger than the nonmetallic B cations and both bind to the X anions (typically oxygen)14. The generic perovskite structure is shown in Fig. 2.1. It is made up of alternating planes of BX2 and AX. These planes are square and stack vertically along the c-axis. The X anions of the AX plane are directly above/below the B cations of the BX2 planes. Although the cuprate materials are not ideal perovskites, they do retain these important features. In the cuprates, the square BX2 planes are CuO2 planes which tend to be adjacent to AO planes. The A cation varies among the dierent cuprate materials and can be La, Sr, Ba, . . . Of the various cuprate materials, La2xSrxCuO4 has a particularly simple unit cell15 which is shown in Fig. 2.2. The shorthand nomenclature for La2xSrxCuO4 is LSCO and the chemical structure is known as 214. Given the unit cell of LSCO, the structure might more accurately be labeled 428. The 214 structure is doubled because adjacent CuO2 planes are oset by half a lattice constant. The oxygen atoms directly below or above a Cu atom are known as apical oxygen atoms because they form the tips of a double pyramid structure that surrounds each Cu atom (see the 14The ideal perovskite unit cell is cubic. The original perovskite is CaTiO3 and was named after the Russian mineralogist L. A. Perovski. 15The unit cell of LSCO contains a perovskite plus a rock-salt layer [37]. 26 2.1. Cuprates Figure 2.1: One unit cell of the generic perovskite ABX3 crystal structure. The c-axis is along the vertical direction. The cuprates and perovskites have some important features in common (see text). Based on a gure from Ref. [36]. Figure 2.2: Crystal structure of La2xSrxCuO4. Each unit cell contains two CuO2 planes in identical chemical environments. The CuO2 planes are emphasized by grey shading. The c-axis is along the vertical direction. Figure provided courtesy of Darren Peets [38]. single Cu atom in the central CuO2 plane of Fig. 2.2). These apical oxygen atoms are 2.4 A from the Cu atoms, whereas the inplane Cu-O distance is 1.9 A. Because of its long bond length, the apical oxygen does not play a signicant role in the essential physics of cuprates which is believed to be contained within the strongly 2- dimensional CuO2 planes [39]. The CuO2 planes are essential ingredients to all of the 27 2.2. Mott Physics cuprate superconductors and, given the wide variety of materials, what goes between the CuO2 planes does not seem to be of much consequence. The main function of the intervening atoms is to provide a charge reservoir for the CuO2 planes. In all of the cuprate materials, the charge carrier density within the CuO2 planes can be tuned. In La2xSrxCuO4 changing the charge carrier density (doping) is achieved by substituting the native La3+ cations, which are adjacent to the CuO2 planes with Sr2+. The substitution of one Sr2+ removes one electron from the CuO2 plane, or equivalently, adds one hole to the copper oxide plane. In this way, the doping p of La2xSrxCuO4 per unit cell per CuO2 plane is given by x. There is also a family of electron-doped cuprates Ln2xCexCuO4 where Ln=Nd, Pr, Sm, or Eu (LnCCO). The crystal structure of the electron-doped materials is similar to, but distinct from, that of LSCO. In particular, there are no apical oxygen atoms. Doping is achieved by substituting the trivalent lanthanide Ln3+ with Ce4+. In this case, an extra electron is donated to the CuO2 plane. Likewise, hole-doping of YBa2Cu3O6+y (YBCO) can be achieved by substitution of Ca2+ for the native Y3+. However, the more common way to change the carrier density in this material is by varying both the oxygen content and the oxygen ordering of the so-called Cu-Oy chains where 0 < y < 1. This method of doping oers important advantages over cation substitution and will be explored in detail in chapter 7. The unit cells of YBa2Cu3O6 and YBa2Cu3O7 are shown in Fig. 2.3. YBCO is called a bilayer material because there are two neighbouring CuO2 planes in each unit cell separated by a single yttrium atom. 2.1.2 Doping Phase Diagram The richness and complexity of the physics of the cuprates is evident from the generic temperature-doping phase diagram. The version adopted here is shown in Fig. 2.4. Almost every region of the phase diagram gives rise to unconventional (and therefore interesting) physics. In the sections that follow, some of the essential physics of each of the regions is explored. We start at zero doping. 2.2 Mott Physics At rst glance, the physics of the undoped cuprates might appear to be mundane. However, it oers the rst taste of unconventional strongly-correlated electron physics out of which the entire phase diagram arises. Consider the cuprate parent compounds La2CuO4 and YBa2Cu3O6. The valencies and electron congurations of the unit cell 28 2.2. Mott Physics (a) (b) Figure 2.3: Unit cells of (a) undoped YBa2Cu3O6 and (b) fully oxygenated YBa2Cu3O7. The 2D CuO2 planes are highlighted by horizontal grey shading and the Cu-Oy chains by vertical grey shading. The c-axis is vertical and the chains are aligned with the b-axis. In YBa2Cu3O6, all of the oxygen chain sites are empty and the compound is an antiferromagnetic insulator. With all the chain sites full, YBa2Cu3O7 is a slightly overdoped superconductor (see x2.4.1). The unit cell di- mensions of YBa2Cu3O7 are a = 3:920 A, b = 3:885 A, and c = 11:676 A. Figures provided courtesy of Darren Peets [38]. ions are given in Table 2.1. The planar Cu2+ cations have one unpaired electron in the dx2y2 orbital resulting in an incomplete atomic 3d-shell. All other atomic shells in the unit cell are completely lled. Then, at least for the single layer materials, there are an odd number of valence electrons per unit cell and band theory predicts that the parent compounds should be metallic. However, the undoped HiTc cuprates are antiferromagnetic insulators (AFI). 2.2.1 Mott Metal-Insulator Transition Mott was the rst to point out that the failure of band theory in the undoped cuprate superconductors was due to the neglect of electron-electron correlations. In materials where these correlations are particularly strong, band theory fails. A common path to the essential physics of the Mott metal-insulator transition is to consider Na ([Ne]3s1), a solid with one neutral atom per unit cell. Because of the half-lled 3s band, Na is expected to be metallic, and indeed it is. However, imagine a Na crystal with an extremely large lattice constant a. In this case, the system is just an array of isolated Na atoms and there can be no conduction. 29 2.2. Mott Physics Figure 2.4: Generic temperature versus doping phase diagram of the HiTc cuprates. Based on a gure from Ref. [40]. Each 3s1 electron is trapped to an individual Na ion and its energy determined by the atomic energy levels. Now slowly decrease the lattice constant. At some point the wavefunctions of neighbouring electrons will overlap and there is a nite probability for an electron to tunnel to a neighbouring site. There is a reduction in the energy of a single electron at site i given by: tij =  Z dr (r Ri) X k 6=i V (r Rk) (r Rj) ; (2.1) which is due to the overlap of the atomic wavefunctions  (r Ri) and  (r Rj) of electrons at sites i and j respectively and to the lattice of atomic potentials V (r Rk). This is the tight binding model and Eq. 2.1 is called the overlap inte- gral. Neglecting all but nearest neighbour tunneling (or hopping16), the tight binding 16The terms tunneling and hopping are used interchangeably despite their seemingly distinct connotations. 30 2.2. Mott Physics ion conguration La2CuO4 La 3+ [Kr]4d105s25p6 ([Xe]) Cu2+ [Ar]3d9 O2 1s22s22p6 ([Ne]) YBa2Cu3O6 Y 3+ [Ar]3d104s24p6 ([Kr]) Ba2+ [Kr]4d105s25p6 ([Xe]) Cu1+ [Ar]3d10 Cu2+ [Ar]3d9 O2 1s22s22p6 ([Ne]) Table 2.1: Valencies and electron congurations of the ions in the parent compounds of La2xSrxCuO4 and YBa2Cu3O6+y. In the case of YBa2Cu3O6, the ionization of the copper atoms in the CuO2 planes is Cu 2+ and is Cu1+ in the chains. In both compounds, all of the electron shells are completely lled with the exception of the 3d9 shell of the planar Cu atoms. Hamiltonian is: H = "0 X i X  cyi;ci;  t X <i;j> X   cyi;cj; + c y j;ci;  ; (2.2) where "0 is the on-site energy, c y i; creates electron at site i with spin  =" or #, and ci; is the corresponding annihilation operator. The notation < i; j > indicates that the sum involving the overlap term applies to only nearest neighbour sites [41]. Mott realized that electron tunneling must compete with the Coulomb repulsion U encountered when two electrons occupy the same site which is given by: U = 1 4"0 Z dr1 Z dr2 j(r1 Ri)j2 e 2 jr1  r2j j(r2 Ri)j 2 : (2.3) Here two electrons occupy the same ionic site and share the same ground state orbital, but have opposite spin. The model system of Na atoms can then be described by the single-band Hubbard Hamiltonian: H = t X <i;j> X   cyi;cj; + c y j;ci;  + U X i cyi;"ci;"c y i;#ci;#: (2.4) The constant on-site contribution has been dropped and the rst term is often referred 31 2.2. Mott Physics to as the kinetic term because of its association with electron hopping. For nite t and large on-site Coulomb repulsion U , the spins order antiferromagnetically which allows electrons to temporarily hop to a neighbouring site without violating Pauli's exclusion principle. When U=t 1, the cost of having doubly occupied sites is too great there is no conduction resulting in an insulator with a half-lled band. In the opposite limit U=t  1, electron correlations can be neglected and band theory is reliable. Near U=t  1, one expects to encounter the so-called Mott metal-insulator transition. In some materials it is possible to tune through the Mott transition. The parameter t is very sensitive to the lattice parameter size a, whereas U is less so. Through the application of pressure, a can be decreased causing a substantial decrease in U=t [42]. The material MnO, is a Mott insulator at ambient pressure but is suitably close to U=t & 1, that it goes through an insulator-to-metal transition when subjected to 90 GPa of pressure inside a diamond anvil cell [43]. 2.2.2 The Large U=t Limit In the large U=t limit at half lling, a unique band structure emerges from the Hub- bard model. The ground state has no doubly occupied sites and naively each electron has an energy "0, the on-site atomic energy. Consider the addition of a single electron to the system. The total energy is increased by "0+U because the newly introduced electron causes there to be a single doubly occupied site. This electron is said to be in the Upper Hubbard Band (UHB) which is separated by U from the lled Lower Hubbard Band (LHB). Conversely, removing an electron from the system adds a freely propagating hole to the otherwise lled LHB. Each of the bands can accommodate only one electron per site. A generic picture of the lower and upper Hubbard bands is shown in Fig. 2.5. Despite its sound physical motivation and apparent elegance, the Hubbard model has proven itself to be exceedingly dicult to handle theoretically. Further simpli- cation is often sought by considering the limit where the on-site Coulomb repulsion dominates inter-site hopping. After signicant manoeuvering (see Ref. [42] for exam- ple) one arrives at the so-called t J model: HtJ = t X <i;j> X  h (1 n̂i) cyicj (1 n̂j) + H:C: i + J X <i;j>  Si  Sj  n̂in̂j 4  : (2.5) Here n̂i = c y ici is the number operator for a particle at site i with spin  and 32 2.2. Mott Physics Figure 2.5: At half-lling, the ground state of the Mott insulator is made up of a lled LHB and an empty UHB. This cartoon shows the bands as a function of the density of states ("). For U=t 1, the bands are well separated and the activation energy needed to promote an electron to the UHB is ac  U W . The width of the bands W is given by t to within a numerical constant that depends on the number of nearest neighbours. n̂i = n̂i"+ n̂i# is the spin independent number operator. Si = c y ici are the usual spin operators,  being the Pauli spin matrices. The exchange-coupling constant is given by J = 4t2=U . The t J Hamiltonian does not allow for any doubly occupied sites. Hence, at exactly half lling, the t term is zero andHtJ becomes the well-known Heisenberg Hamiltonian: HHeis = J X <i;j> Si  Sj; (2.6) where ni = nj = 1 and the 1=4 constant term has been dropped. For positive J , HHeis is minimized for antiparallel neighbouring spins which leads to antiferromagnetic ordering. 2.2.3 Oxygen p-Orbitals The Hubbard and t J Hamiltonians (Eqns 2.4 and 2.5), were developed by consid- ering variations of the lattice constant separating neutral Na atoms. The essential physics of the CuO2 in the cuprates is also believed to be captured by these mod- els. However, a complication arises because neighbouring planar Cu2+ cations are 33 2.2. Mott Physics Figure 2.6: An ab-plane view of the CuO2 plane. Neighbouring Cu 2+ sites are sepa- rated by intervening O2 anions. separated by O2 anions with lled 2p-orbitals as shown in Fig. 2.6. In this case hopping between neighbouring Cu2+ sites is mediated via the lled 2p orbitals of the O2 anions as described in Fig. 2.7. This type of interaction is called superex- change and modies the parameters t ! te and J ! Je , but not the form of the Hamiltonians [42]. The presence of the lled 2p orbitals allows for the possibility that the gap between the UHB and 2p orbital is less than that between the UHB and the LHB. Figure 2.8 Figure 2.7: The 3dx2y2 orbitals of two neighbouring Cu2+ cations separated by the 2px orbitals of an intervening O 2 anion. The red lobes are 180 out of phase with the blue lobes. Interactions between the two 3dx2y2 orbitals are mediated by electrons hopping from the 2px orbtial. Interactions in the y-direction are likewise mediated by the 2py O 2 orbital. This type of exchange interaction is called superexchange. 34 2.2. Mott Physics Figure 2.8: (a) A true Mott insulator as originally shown in Fig. 2.5. The lled O2 2p orbital lies below the LHB. (b) The cuprates are more properly identied as charge transfer insulators. The top of the lled 2p band is above the top of the LHB. Excitations into the UHB come more easily from the 2p band. depicts this dierence. In the cuprates, at half lling, the rst lled band below the empty UHB is the O2 2p orbital. For this reason, the cuprates are not classic Mott insulators, but rather, are classied as charge transfer insulators [42]. 2.2.4 Doping a Charge Transfer Insulator Electron doping away from half lling does not distinguish between true Mott insu- lators and charge transfer insulators; in both cases electrons are added to the UHB. Conversely, hole doping does make a distinction. In a charge transfer insulator (i.e. the cuprates), the hole is introduced into the 2p band and in Mott insulators it goes into the LHB. For this reason, the cuprate doping phase diagram might not be expected to be electron-hole symmetric. Doping a Mott or charge transfer insulator away from half lling facilitates elec- tron hopping and the antiferromagnetic insulator state quickly gives way to new ground states. There is an incredible variety of the exotic ground states which vary from compound to compound and vary with doping within a single material. These ground states, many of which elude a theoretical understanding, include colossal mag- netoresistance, spin density waves (SDW), charge density waves (CDW), and others. Sections 2.3 and 2.4 will focus on the remaining regions labeled in the cuprate phase 35 2.3. dx2y2 Superconductivity diagram of Fig. 2.4. The discussion will be largely guided by experimental results. 2.3 dx2y2 Superconductivity A superconducting ground state emerges when the cuprates are doped away from half lling. The doping p is the number of holes (or electrons) introduced to the CuO2 planes per unit cell per CuO2 plane. The superconducting state exists as a dome in the phase diagram which, on the hole-doped side, peaks at an optimal doping of p  0:16. On the underdoped side, the dome persists down to p  0:05 and, on the overdoped side, up to p  0:27. An empirical relationship between Tc of the superconducting state and hole doping p has been established [44, 45]: 1 Tc Tc;max = 82:6 (p 0:16)2 ; (2.7) where Tc;max is the critical transition temperature at optimal doping. This expression appears to apply universally to all of the hole-doped cuprates except near p = 1=8 where a proposed charge stripe order is believed to be commensurate with the lattice spacing and results in a depression of Tc. A fundamental dierence between conventional and unconventional superconduc- tors is the symmetry of the superconducting gap function  (k). Conventional su- perconductors are characterized by the formation of electron pairs with zero total angular momentum which implies isotropic attractive forces between electrons in all spatial directions. These isotropic superconductors have a full gap surrounding the entire Fermi surface as in Fig. 2.9(a). In contrast, the electron pairing state of un- conventional superconductors has nite angular momentum resulting from electron correlations caused by a large Coulomb repulsion at each Cu site in the CuO2 planes. As for conventional superconductors, the ground state of the HiTc cuprates continues to be an antisymmetric spin singlet state [47]. Table 2.2 lists the four distinct singlet pairing states for a square CuO2 plane allowed by group-theoretic calculations [48]. All but the s+ superconductors have directions in the kxky-plane for which (k)! 0. Because the HiTcs are strongly two-dimensional, little dispersion is expected in the z-direction, and these gap nodes are line nodes that run along kz. The rst evidence for line nodes in the superconducting gap came in 1993 from both penetration depth measurements of Hardy et al. on single crystal YBa2Cu3O6:95 [49] and from NMR studies of Kitaoka et al. [50]. For example, the penetration depth measurements showed that (T ) / T below 20 K, a direct con- 36 2.3. dx2y2 Superconductivity (a) (b) Figure 2.9: The superconducting gap in k-space for (a) isotropic (s-wave) and (b) dx2y2 (d-wave) superconductors. The cylindrical Fermi surface is shown as a bold circle in the kxky-plane. The hatched region represents lled electronic states. For an s-wave superconductor, the energy gap 2 is isotropic. For a d-wave superconductor, the sign and magnitude of the gap is a function of the direction in the kxky-plane. Figure from Ref. [46]. Informal Group-theoretic Representative Nodes name notation state s+ A1g const. none s (g) A2g xy(x2  y2) line dx2y2 B1g x2  y2 line dxy B2g xy line Table 2.2: Allowed singlet pairing states for a square CuO2 plane. Table adapted from tables in Refs [46] and [47]. sequence of line nodes in (k). While these studies showed that the superconducting state of the HiTcs is distinct from that of the familiar isotropic BCS superconductors, they could not distinguish between the three allowed unconventional pairing states listed in Table 2.2, all of which possess line nodes. The remarkable phase-sensitive scanning SQUID measurements of the Kirtley-Tsuei collaboration have unambigu- ously shown that the pairing symmetry in a wide variety of cuprate systems (hole- doped and electron-doped) is predominantly dx2y2 [47]. From here on, the notation s-wave will refer to the s+ state in Table 2.2 and d-wave to the dx2y2 state. As alluded to in Fig. 2.3, YBa2Cu3O6+y has a small orthorhombic distortion. This distortion breaks the symmetry of the lattice and allows for an admixture of 37 2.3. dx2y2 Superconductivity all the possible pairing symmetries. The most astonishing measurements by Kirt- ley et al. have shown that, in optimally doped YBa2Cu3O6+y, the gap symmetry is x2y2 +s where s=x2y2  0:1. As a result, the magnitude of the gap in the b-axis direction is 20% larger than it is in the a-axis direction and the nodes are shifted from the diagonal and occur at 43 and 139. A cartoon representation of the x2y2 +s gap symmetry is shown in Fig. 2.10. It is important to emphasize that Figure 2.10: Cartoon representation of the x2y2 +s gap symmetry. The red and blue lobes are 180 out of phase. The small s-wave component of the gap enhances the lobes in one direction and shifts the nodes slightly o diagonal. the admixture of the s-wave gap symmetry is a technical detail associated with the or- thorhombic symmetry of the lattice. It does not imply that phonon mediated Cooper pairing is playing a role. For simplicity, the symmetry of the cuprate superconducting state will be referred to as dx2y2 . 2.3.1 Delicate Superconducting State The unconventional dx2y2 superconducting state is extremely sensitive to disorder. Even a tiny amount of chemical impurity or structural disorder can mask the intrinsic behaviour of these materials. The conventional BCS superconductors are widely known to obey Anderson's theorem: for an s-wave superconductor, nonmagnetic elastic scattering by impurities has little eect on  or Tc. Physically, the reason is that the scattering tends to average the gap over all directions in k-space which has no eect when the gap is isotropic. In the case of a dx2y2 superconductor, the gap rapidly averages to zero because of the sign changes around the Fermi surface and superconductivity is suppressed. The number one issue that arises when evaluating 38 2.4. The \Normal State" the merit of experimental claims regarding the cuprate superconductors is that of sample quality (or lack thereof). 2.4 The \Normal State" The meaning of the term \normal state" needs clarication. In condensed matter physics, one might interpret \normal state" to mean a state that is adequately de- scribed by a gas of noninteracting Fermions or quasiparticles. By this denition very little of the cuprate phase diagram is \normal". The non-superconducting state of the cuprates exhibits a wide variety of behaviour, much of it exotic, that varies across the doping phase diagram. Very near half lling, the antiferromagnetic insulator state is a spectacular ex- ample of the breakdown of single-electron band theory. Above Tc, the underdoped region of the phase diagram is highly anomalous and continues to evade theoretical understanding. As described in the next section, it is only the overdoped region which can truly be called normal because it appears to be a metallic state that is reasonably well treated using Fermi liquid theory. This section will also serve to emphasize that mapping the entire phase diagram of Fig. 2.4 requires a variety of experimental techniques applied to several dierent cuprate compounds. For example, YBa2Cu3O6+y is by far the cleanest underdoped cuprate making it ideal for quantum oscillation measurements. However, this ma- terial cannot be cleanly overdoped and it does not have a neutral cleavage plane required for the powerful surface techniques such as angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS). The most cited surface measurements of underdoped cuprates in the literature are almost always done on Bi2Sr2CaCu2O8+. The Tl2Ba2CuO6+ compound is unique in that it is naturally overdoped. The overdoped region of the phase diagram can also be accessed with Sr- doped La2CuO4+y or Pb-doped Bi2Sr2CuO6+x. However, in these compounds large concentrations of the dopant cations are required which results in less clean samples. Tl2Ba2CuO6+ has the additional advantage that it has a suitable cleavage plane re- quired for surface measurements. As will be seen in the following section, many of the key conclusions regarding the overdoped side of the cuprate phase diagram are derived from experiments performed on Tl2Ba2CuO6+ crystals. 39 2.4. The \Normal State" 2.4.1 Overdoped Generically, the physical properties of a Landau Fermi liquid [51{53] obey the same temperature power laws derived for a noninteracting ideal Fermi gas. For example, the magnetic susceptibility is temperature independent and the specic heat capacity is proportional to temperature. However, the expected magnitudes of these properties are often strongly modied from the noninteracting Fermi gas values due to electron- electron interactions. These modications are characterized by an experimentally determined set (usually three) of Landau parameters. For the heat capacity, the free electron mass is replaced by an eective mass m? which is determined by the Landau parameter usually denoted F1. An electrical resistivity varying with temperature as T 2 is regarded as a key signature of a Landau Fermi liquid17. Both the in-plane ab and out-of-plane c resistivities of the overdoped cuprates show Fermi liquid-like behaviour [39]. Perhaps more remarkable, are the polar angle magnetoresistance oscillation (AMRO) measurements on overdoped Tl2Ba2CuO6+ that conrm the existence of a highly 2D cylindrical Fermi surface long-predicted by single-electron band the- ory [54]. These measurements are done in high eld ( 45 T) that suppresses the superconducting state such that the Fermi liquid behaviour can be conrmed even at low temperatures. These results were independently conrmed, also using overdoped Tl2Ba2CuO6+ samples, by ARPES measurements [55] and nally by de Haas-van Alphen (dHvA) quantum oscillation measurements [56]. 2.4.2 Underdoped The overdoped normal state is the only region of the cuprate phase diagram that is in any way conventional. By contrast, at optimal doping, the resistivity is highly anomalous in that it is linear over a wide temperature range [57]. As the cuprates are further underdoped, anomalous behaviour abounds. Fermi Arcs ARPES was the rst technique to probe the Fermi surface in this region using single crystals of Bi2Sr2CaCu2O8+ [58]. Below Tc, the data show a superconducting gap and quasiparticle excitation peaks consistent with a dx2y2 order parameter. Above Tc, the data reveal features with which condensed matter physicists continue to grapple 17The electrical resistivity tends to get the initial attention because it is relatively easy to measure, and the raw data requires little to no post-measurement manipulation. 40 2.4. The \Normal State" over a decade later. The quasiparticle peaks vanish, as expected, but the gap persists up to temperatures well above Tc. Even more remarkable, is that the temperature at which the gap closes depends on position in the Brillouin zone. The temperature above which the entire Brillouin zone is gapless (i.e. the Fermi surface is fully formed) is denoted T ? and is strongly doping dependent. The underdoped region between Tc and T ? is known as the pseudogap phase and in the phase diagram of Fig. 2.4 is represented by the area that is shaded grey. The evolution from line nodes to the so-called Fermi arcs and nally to a fully formed Fermi surface is shown in Fig. 2.11. There is now an abundance of experimental evidence for the pseudogap phase across the cuprate family collected from spectroscopy, transport, thermodynamic, and other measurements [60]. Fluctuations Experiments have shown that the magnitude of the gap is continuous as tempera- ture is increased through Tc from the superconducting state and into the pseudogap state. In x1.5.1, it was pointed out that critical 
uctuations of the order parameter jj ei' are likely important in the cuprates. The high-resolution thermal expansion measurements of Meingast et al. on YBa2Cu3O6+y unambiguously show that critical 
uctuations of the gap gains strength as the sample is underdoped [61]18. These and other observations have led many to speculate that T ? marks the onset of phase inco- herent Cooper pairing (often referred to as preformed pairs) and Tc marks the onset of phase coherence. This scenario is dramatically dierent than that in BCS super- conductors where the gap magnitude goes to zero at Tc. An alternative suggestion is that the pseudogap exists independent of, and competes with, superconductivity [40]. Fermi Pockets After nearly a decade of trying to come to grips with Fermi arcs, a startling discovery was made. Quantum oscillations were observed in underdoped YBa2Cu3O6:5 in very high pulsed magnetic elds [62]. Quantum oscillations in this material have now been observed by a number of dierent groups and leave essentially no doubt that there exists a closed Fermi surface in this region of the phase diagram. The oscillations occur as electrons execute quantized orbits as the magnitude of the applied magnetic eld is swept causing dierent Landau levels to cross the Fermi surface. The period 18The thermal expansion measurements are unfortunately limited to dopings that remain quite close to optimal doping. 41 2.4. The \Normal State" (a) (b) (c) Figure 2.11: Cartoon of the evolution from nodes!arcs!complete Fermi surface as a function of temperature in the rst Brillouin zone. The point  is the centre of the Brillouin zone. (a) In the superconducting state (T < Tc), the order parameter k goes to zero only at the four points shown. (b) Between Tc and T ?, gapless Fermi arcs expand out from the nodal points. (c) Above the pseudogap temperature T ?, there is a fully formed Fermi surface. In contrast, overdoped cuprates are more conventional in that the the gap opens at the same temperature for all points on the Fermi surface, namely Tc. Based on gures from Refs. [58] and [59]. of the oscillations in units of inverse eld is related to the cross-sectional area of the Fermi surface perpendicular to the applied eld. Figure 2.12 shows the size of the implied Fermi surface pockets. How to reconcile the dierences seen in the ARPES and quantum oscillation measurements remains an open issue. Undoubtedly the bigger question is, in going from overdoped to underdoped, how does a large cylindrical Fermi surface evolve into 42 2.5. Pnictide Superconductivity Figure 2.12: Cartoon of the Fermi surface electron pocket deduced from quantum oscillations of the resistivity of YBa2Cu3O6:5 (not to scale) [62]. small pockets? 2.5 Pnictide Superconductivity The discovery of the broad family of pnictide superconductors has sparked an avalanche of activity as researchers apply to these materials the theoretical and ex- perimental tools developed and rened over the past two decades of cuprate research. As a result, the cond-mat archives have been inundated with pnictide submissions (surpassing 50/month) leaving behind even the most diligent literature watchers while the rest of us gasp for air. The onslaught began in January 2008 when the layered compound LaO1xFxFeAs was reported to be a 26 K superconductor [63]19. This section attempts to give a qualitative description of the status of the pnictide superconductors as of the end of 2008. Details and precision will be sacriced in favour of brevity for a number of reasons: (1) anything written will almost instantly be out of date, (2) consensus has been reached on very little, and (3) a broad knowledge of the pnictide superconductors is not of central importance to this thesis. 2.5.1 Qualitative Assessment After the initial discovery of superconductivity in LaO1xFxFeAs, the number of members in pnictide superconductor family grew rapidly. This growth includes var- 19As of this writing, the onslaught continues. There were two submissions yesterday, Aug. 16, 2009 (a Sunday). 43 2.5. Pnictide Superconductivity ious substitutions of the elements in the LaO1xFxFeAs 1111 structure, and three other distinct crystal structures labeled 122, 111, and 011 [64]. Tables 2.3 and 2.4 give an indication of the enormous eort that has been devoted to developing the pnictide materials. Figure 2.13 shows two platelet samples of single crystal Ba0:72K0:28Fe2As2 (Tc  29 K). The surface area of 122 crystals are typically  1 mm2 and freshly (a) (b) Figure 2.13: Ba0:72K0:28Fe2As2 platelet single crystals. (a) A mirror-like freshly cleaved ab-plane surface. (b) Edge-on view of a crystal mounted on a 0.5 mm thick sapphire plate. This crystal has a denite bend. cleaved surfaces have a mirror-like nish. The crystals are malleable as shown by the bend in the sample in the gure. This property could turn out to be extremely important for applications. Currently, extraordinary eorts are made to make multi- layer superconducting tapes from thin lm HiTc cuprates that can be bent through a limited radius. The problem is that the ceramic cuprate platelets will survive limited 
exing, but are completely immalleable. Early attempts at superconducting pnictide wires have already shown promise [66]. All pnictide superconductors have quasi-2D square planes containing one pnicto- gen element from group VA (the nitrogen group) of the periodic table. These planes seem to be the essential ingredient for superconductivity. The intervening \stu" be- tween the planes is structural, ensures charge neutrality, and provides opportunities for cation substitution, allowing the planes to be doped. The vast majority of the known pnictide superconductors contain FePn-planes (Pn: pnictogen atom) and are called iron-pnictide superconductors. Of those, a large majority contain FeAs-planes and make up the iron-arsenide superconductors. One can't help but to notice the similarities between the pnictides and cuprates: both contain doped 2-dimensional planes that give rise to superconductivity. How- 44 2.5. Pnictide Superconductivity Structure Formula Tmaxc (K) 1111 e-doped LaFeAsO1xFx 28 CeFeAsO1xFx 41 PrFeAsO1xFx 52 NdFeAsO1xFx 52 SmFeAsO1xFx 55 SmFeAsO1x 55 GdFeAsO1xFx 36 GdFeAsO1x 54 Gd1xThxFeAsO 56 TbFeAsO1xFx 46 Tb1xThxFeAsO 52 DyFeAsO1xFx 45 Ca(Fe1xCox)AsF 22 Sr(Fe1xCox)AsF 4 h-doped La1xSrxFeAsO 25 122 e-doped Ba(Fe1xCox)2As2 24 Ba(Fe1xRhx)2As2 24 Ba(Fe1xNix)2As2 18 Ba(Fe1xPbx)2As2 18 Sr(Fe1xCox)2As2 20 Ca(Fe1xCox)2As2 17 h-doped Ba1xKxFe2As2 38 Sr1xKxFe2As2 38 Sr1xNaxFe2As2 35 Eu1xKxFe2As2 34 Ca1xNaxFe2As2 20 Table 2.3: Electron- and hole-doped 1111 and 122 iron-arsenide superconductors. This list is almost certainly incomplete. Much of the information presented in this table was complied, and generously shared, by Jenny Homan [64, 65]. ever, there are also important dierences that have signicant consequences. Fig- ure 2.14 compares the CuO2 planes of the cuprates to the FeAs planes of the iron- arsenides and Table 2.5 summarizes the ionization states and electron congurations 45 2.5. Pnictide Superconductivity Structure Pnictogen Formula Tmaxc (K) 1111 As LaONiAs 3 P LaO1xFxFeP 7 LaONiP 3 Bi LaONiBi 4 LaOCuBi 6 122 As SrNi2As2 0.6 p BaNi2P2 1.9 111 As Li1xFeAs 18 Na1xFeAs 23 Structure Chalcogen Formula Tmaxc (K) 011 Se FeSe 6 FeSe0:5Te0:5 16 Table 2.4: Other pnictide superconductors. The FeSe materials are not pnictides, but the simple structure is appealing, with FeSe planes that are equivalent to FeAs planes. This list is almost certainly incomplete. Much of the information presented in this table was complied, and generously shared, by Jenny Homan [64, 65]. ion conguration BaFe2As2 Ba 2+ [Kr]4d105s25p6 ([Xe]) Fe2+ [Ar]3d6 As3 [Ar]4s23d104p6 ([Kr]) Table 2.5: Valencies and electron congurations of the ions in the parent compound BaFe2As2. All of the electron shells are completely lled with the exception of the 3d6 shell of the Fe atoms. of undoped BaFe2As2. In the cuprates, only the [Ar]3d 9 planar Cu2+ cations have incomplete electron shells. In the iron-pnictides, it is the [Ar]3d6 Fe2+ shells that are incomplete. In both cases, the essential physics is determined by electrons in 3d-orbitals and there is likely to be signicant overlap with neighbouring lled p- orbitals. In the cuprates, strong coulomb repulsion localizes the single 2-dimensional 46 2.5. Pnictide Superconductivity (a) (b) Figure 2.14: An ab-plane view of the cuprate CuO2 plane and iron-arsenide FeAs plane (the tetragonal phases). (a) Neighbouring Cu2+ sites are separated by inter- vening O2 anions (identical to Fig. 2.6). (b) Neighbouring Fe2+ sites are separated by intervening As3 anions. dx2y2 hole leading to an antiferromagnet insulator at zero doping. In contrast, the Fe-pnictide planes have four d holes (6 d electrons) requiring a multiband description and leading to a more 3D nature. To what extent all ve Fe d orbitals must be con- sidered is not a settled issue. From an even number of d electrons, one would expect the normal state undoped parent compounds to be semiconducting or semimetal- lic [67, 68]. Most of the experimental eorts have focused on mapping out the doping phase diagram, determining the Fermi surface, or probing the symmetry of the supercon- ducting gap. At this early stage, digesting all of the results is dicult. One may even have to face the ugly possibility that dierent members of the pnictide family play by dierent rules. What follows are some of the recurring themes seen from experimental reports. The Undoped Normal State Electrical resistivity measurements of the undoped parent compounds show metallic- like behaviour. These measurements also reveal a tetragonal-orthorhombic structural transition typically of the order TS  150 K. This transition either coincides with, or just precedes, an antiferromagnetic ordering transition at TN / TS. The Neel temper- ature is suppressed as the materials are either electron- or hole-doped. On both sides of the phase diagram, it was reported that antiferromagnetism and superconductiv- ity coexisted at some intermediate dopings. However, SR measurements suggest that these samples were likely inhomogeneous and phase separated into patches of superconductivity and patches of antiferromagnetism [64]. 47 2.5. Pnictide Superconductivity The Fermi Surface and Gap Symmetry The most widely cited experimental determination of the Fermi surface is an ARPES study by Ding et al. of single crystal Ba0:6K0:4Fe2As2 [69]. These authors report two concentric cylindrical hole-like Fermi surfaces at the Brillouin zone centre ( point) and smaller electron-like pockets centred on the zone edges (M points). ARPES mea- surements by other groups sometimes dier in the ne details, but are qualitatively consistent. Moreover, these results are in reasonable agreement with band struc- ture predictions [64]. The electron Fermi surface pocket is connected to one of the hole Fermi surface sheets by a momentum space vector (; 0), so-called Fermi surface nesting. This is also the characteristic wave vector of the antiferromagnetic state as determined by neutron diraction measurements [70]. The ARPES measurements by Ding et al. in the superconducting state yield information about the energy gap symmetry. They interpret their data as showing two large gaps on the hole and electron pockets connected by the nesting vector and a smaller gap on the remaining hole pocket. All three gaps are isotropic and close simultaneously at the bulk transition temperature Tc. There are numerous other experimental probes of the superconducting gap, and the debate regarding its symmetry is far from settled. An intriguing suggestion is that the superconducting gaps are indeed isotropic but that the sign of the gap found on hole-like Fermi surface is the opposite of that on the electron-like Fermi surface. This unconventional s-wave scenario is called the s state. The s suggestion is certainly fashionable, but may ultimately be dicult to unambiguously prove (or disprove). The gap symmetry in the pnictides will be discussed more thoroughly in chapter 8. 48 Chapter 3 Superconductor Electrodynamics This short chapter will review the relevant electrodynamics of conduc- tors/superconductors in electromagnetic elds. The treatment is relatively simple, but essential to the experimental techniques employed in this thesis. Moreover, ex- perimental data will be interpreted by appealing to results found within these three sections. The chapter is organized as follows: x3.1 introduces the complex surface impedance for conductors, x3.2 develops the two-
uid model for the electrical con- ductivity of superconductors, and x3.3 calculates the power absorbed by a conductor in an electromagnetic eld. 3.1 Surface Impedance The experimentally accessible quantity in microwave experiments is the complex sur- face impedance ZS(!; T ) = RS(!; T )+ iXS(!; T ), where RS(!; T ) is the surface resis- tance andXS(!; T ) the surface reactance. Imagine a conductor that lls the half-space z > 0, the surface impedance is dened to be the ratio of tangential electric eld E and the tangential magnetic eld H at the surface of the conductor [5]: ZS(!; T ) = RS(!; T ) + iXS(!; T )  Ex(z; t) Hy(z; t)  z=0 : (3.1) RS is responsible for power absorption by a conductor in an electromagnetic eld and the screening of electromagnetic elds from the interior of the conductor is charac- terized by XS. For a nonmagnetic conductor, the electric and magnetic elds are related through Maxwell's equation: rE = 0@H @t ; (3.2) such that: 0@Hy(z; t) @t = @Ex(z; t) @z : (3.3) In the limit of local electrodynamics (   for a superconductor) and for oscillatory 49 3.1. Surface Impedance applied elds, the tangential elds inside the conductor are given by: Ex(z; t) = Ex0e z+i!t; (3.4a) Hy(z; t) = Hy0e z+i!t: (3.4b) For a good conductor, the propagation constant is  = p i0! and  is the complex conductivity such that J = E [71]. Substitution into Eq. 3.3 produces the desired result for the surface impedance: ZS(!) = i0!  = r i0!  : (3.5) Before developing the surface impedance further by separating the real and imaginary parts, the special case of a thin conductor is considered. 3.1.1 The Thin Limit Figure 3.1 shows the thin limit geometry. A thin slab conductor of thickness t placed in a uniform magnetic eld Hy = H0e i!t. The spatial extent of the slab in the x and y directions is much greater than the thickness t. Inside the conductor, the magnetic Figure 3.1: A conductor of thickness t with propagation constant  is placed in a uniform magnetic eld with amplitude H0. Fields propagate to the left and right within the conductor. 50 3.1. Surface Impedance eld is given by: Hy(z) = Hy10e z Hy20ez; (3.6) where the oscillatory time dependence ei!t has been omitted. The boundary condi- tions Hy(z) = H0 must be satised at both z = 0 and z = t. These conditions are met when: Hy(z) = H0  1 et et  et e z  1 e t et  et e z  : (3.7) Similar to Eq. 3.6, the electric eld inside the conductor is given by: Ex(z) = Ex10e z + Ex20ez: (3.8) Using Eq. 3.3 to relate the coecients of the electric and magnetic elds, the electric eld can be rewritten as: Ex(z) = i0!  H0  1 et et  et e z + 1 et et  et e z  ; (3.9) the important dierence being the plus sign between the two terms inside the braces. The eective surface impedance is then: ZeS (!) = Ex(z; t) Hy(z; t)  z=0 = i0!   2 et  et et + et  : (3.10) It's not too much trouble to show that the term in the braces is tanh (t=2), such that: ZeS (!) = ZS(!) tanh  t 2  = ZS(!) tanh  i0!t 2ZS(!)  ; (3.11) where ZS(!) is given by Eq. 3.5. For a very thick conductor, the hyperbolic tangent approaches one and ZeS ! ZS as expected. 3.1.2 RS(!) and XS(!) Good normal conductor For a good conductor in the normal state and at not too high frequencies,   !" (" being the permittivity of the conductor), and the conductivity is purely real [71]. Under these conditions: RS(!) = XS(!) = r 0! 2 : (3.12) 51 3.1. Surface Impedance The classical skin depth  is dened to be:  = r 2 0! ; (3.13) such that: RS(!) =   ; (3.14) where  = 1= is the resistivity of the metal. Normal conductor thin limit Returning to the thin limit expression Eq. 3.11 and inserting ZS = (1 + i)= for a good metal, one has: ZeS (!) = (1 + i)   tanh  (1 + i) t 2  : (normal metal) (3.15) To obtain the general expression for the eective surface resistance requires some algebra and a number of trigonometric identities. The result is: ReS (!) =    sinh t   sin t  cosh t  + cos t   : (normal metal) (3.16) The limiting t=  1 behaviour is found using the power series expansions of the trigonometric functions and in each case keeping the rst two nonzero terms. In this limit, the eective surface resistance is: ReS (!)  t3 64 = 20! 2t3 24 : (t=  1) (3.17) The frequency dependence of ReS (!) is shown in Fig. 3.2. General expressions In the general case,  = 1  i2 is complex and when substituted into Eq 3.5, the real and imaginary parts can be separated to yield: RS(!) = s 0!( p 21 +  2 2  2) 2(21 +  2 2) ; (3.18a) XS(!) = r 0! 2(21 +  2 2) 1qp 21 +  2 2  2 : (3.18b) 52 3.1. Surface Impedance 0 10 20 30 40 50 60 0.00 0.04 0.08 0.12    R S eff( )  R S eff( )~ 2    (t/ <<1)  R S ( )~ 1/2 Su rf ac e R es is ta nc e ( ) Frequency (GHz) Figure 3.2: The solid red line is ReS (!) using  = 5:28  cm and t = 2 m. The low frequency limit t=  1 is shown as the dashed black line. The usual ReS (!)! RS(!) limit for t=  1 is shown as the dashed blue line. These expressions are completely general and hold for any ohmic conductor. Approx- imate forms are typically generated in an attempt to gain physical insight. In the next section it will be shown that in the superconducting state, the approximation 2  1 is typically valid except very near Tc, in which case: RS(!)  1 22 r 0! 2 ; (3.19a) XS(!)  r 0! 2 : (3.19b) Note that in the approximation   i2 the propagation constant becomes   p0!2  1=, which when substituted back into Eqs. 3.4 lead to solutions that exponentially decay over a length scale given by . Written in terms of  rather than 2, Eqs. 3.19 become: RS (!; T )  1 2 20! 23 (T )1 (!; T ) ; (3.20a) XS (!; T )  0! (T ) ; (3.20b) 53 3.1. Surface Impedance where the explicit frequency and temperature dependencies have been put in. These are very useful approximate forms that one often refers to when doing simple data analysis. Note that independent measurements of both RS (!; T ) and  (T ) are re- quired to extract 1 (!; T ). It is perhaps surprising that the surface resistance of a superconductor should be proportional to the real part of the conductivity 1(!; T ). The next section will clarify that the 1 in Eq. 3.20a is really the quasiparticle conductivity, that is, the conductivity due to the unpaired charge carriers excited out of the Cooper pair con- densate. The screening of electromagnetic elds is predominantly provided by the Cooper pair supercurrent and any residual unscreened eld will cause a quasiparticle normal current to 
ow. A large quasiparticle conductivity will produce a large nor- mal current and lead to a large power dissipation (Pdiss / qpE2). Section 3.3 will show that the power dissipated by a conductor in an electromagnetic eld is directly proportional the to the conductor's surface resistance, and hence RS / 1. It may be instructive to consider the circuit analogy of Fig. 3.3 which also serves as a nice introduction to the the two-
uid model. The lossy quasiparticle normal Figure 3.3: The normal, or quasiparticle, current is represented by IR and the su- percurrent by IL. In the normal state R  !L and in the superconducting state !L R. current in a superconductor is represented by IR and the inductive screening of the supercurrent by IL. To see how the surface resistance relates to the conductivity, one need only to consider how the dissipated power (/ RS) relates to R (/ 1=1). For a normal metal, IR  I and the power dissipated Pdiss  I2R is proportional to R (or 1=1) as expected. On the other hand, deep in the superconducting state XL = !L  R. The power dissipated by the quasiparticle current is still I2RR, but the magnitude of IR is greatly reduced since the majority of the current is shorted 54 3.2. Two-
uid Model through the inductor: Pdiss = I 2 RR = I 2R  XL R +XL 2  I2X 2 L R : (3.21) In this case, the dissipated power (i.e. RS) is inversely proportional to the normal state resistance/resistivity (i.e. proportional to quasiparticle conductivity). 3.2 Two-
uid Model The starting point for this section is the Drude model that was rst introduced in x1.3 when discussing the London equations. The Drude equation of motion is repeated here in a dierent, but equivalent, form: dJ dt =  ne2 m?  1   J : (3.22) Solving for  = 1  i2 and assuming a harmonic (/ ei!t) time dependence for the applied electric eld yields: 1 = 0 1 1 + !2 2 ; (3.23a) 2 = 0 ! 1 + !2 2 ; (3.23b) 0 = ne2 m? : (3.23c) Integrating 1 (!) over all frequencies gives a result that is independent of both  and !: Z 1 0 1 (!) d! =  2 ne2 m? : (3.24) Eq. 3.24 is a very general oscillator strength sum rule and is derived from a Kramers- Kronig transformation relating the real and imaginary parts of the causal response function  (!) in Appendix A. In the generalized two 
uid model, the total conductivity is described as coming from two parallel and independent sources, one being due to normal electrons, and the other due to the superconducting electrons:  (!) = [1n (!) i2n (!)] + [1s (!) i2s (!)] : (3.25) 55 3.2. Two-
uid Model The total electron density ne2=m? is temperature independent and equal to the sum of the normal 
uid density nne 2=m? and the super
uid density nse 2=m?. In the clean limit, all of the charge carriers are assumed to condense into the superconducting ground state at T = 0 such that: ne2 m? = nse 2 m? (T = 0) = nne 2 m? (T ) + nse 2 m? (T ) : (3.26) Allowing  in the superconducting state to go to innity, one nds from Eq. 3.23a that 1s becomes a -function at zero frequency. This -function represents the perfect dc conductivity of a superconductor and its coecient is set by Eq. 3.24: 1s (!) =  2 nse 2 m?  (!) : (3.27) At nonzero frequency, s is strictly determined by 2s. In the same  !1 limit, 2s becomes: 2s(!) = nse 2 m?! = 1 0!2L (T ) ; (3.28) which is the same London penetration depth obtained from the second London equa- tion, but now related to s2. Recalling the denition of the magnetic penetration depth 2 = 1=0!2 reveals that a measurement of  will dier from  2 L = 1=0!2s if there is signicant contribution to 2 from 2n. When this is the case, the normal 
uid plays a role in screening electromagnetic elds and cannot be experimentally distinguished from the super
uid. The total integrated quasiparticle conductivity 1n is a measure of the normal 
uid density nn: nne 2 m? (T ) = 2  Z 1 0 1n (!; T ) d!; (3.29) and provided 2s  2n, a measurement of the magnetic penetration depth yields the super
uid density: nse 2 m? (T )  1 0 1 2 (T ) : (3.30) The two preceding equations together with Eq. 3.26 require that any oscillator strength lost from the super
uid density must appear in the integrated quasiparticle conductivity. Figure 3.4 summarizes the expected conductivities of a superconductor in the normal state and the superconducting state at T = 0 and at nite temperature. 56 3.3. Power Absorption Figure 3.4: Cartoon summary of the conductivity of a superconductor. For T > Tc, the conductivities are given by the Drude model, Eqns. 3.23. At T = 0, there are very few quasiparticles except at high frequency, most of the spectral weight from 1 appears in the zero-frequency -function. The imaginary part of the conductivity is completely dominated by the super
uid contribution. At nite temperature, 2(!) is still dominated by 2s(!), but some of the super
uid density is transferred to the quasiparticle normal 
uid. Figure taken from Ref. [72]. 3.3 Power Absorption The broadband spectrometer presented in this thesis exposes conductors to microwave magnetic elds. In this section, an expression for the power absorbed by the samples 57 3.3. Power Absorption from the electromagnetic eld is calculated. The power P absorbed per unit area A is given by: dP dA = 1 2 < [n  S] ; (3.31) where S = E H? is the Poynting vector and n is an outward unit vector normal to the surface of the conductor [73]. For the geometry adopted in x3.1 in which the conductor lls the half space z > 0 with the magnetic eld applied along the y-direction, Eq. 3.31 reduces to: dP dA = 1 2 < [Ex0Hy0] = 1 2 < ZSH2y0 = 12RSH2y0: (3.32) Provided Hy0 is uniform over the surface of the conductor, the power absorbed is: P = 1 2 RSH 2 surfA; (3.33) where Hsurf is the applied microwave magnetic eld at the surface of the conductor. 58 Chapter 4 Experimental Techniques: Penetration Depth In the experimental Superconductivity Group at the University of British Columbia (UBC)20 customized experimental techniques are applied to study advanced con- densed matter systems. Each project in the superconductivity group aims to: 1. Identify a crucial issue in the eld that needs to be addressed. 2. Build a custom experimental apparatus designed specically to resolve the issue with maximum sensitivity. Of particular interest is HiTc superconductivity which remains one of the greatest unsolved problems in condensed matter physics. The approach outlined above applies to both cryogenic measurements and to the crystal growth program. The next two chapters will deal consecutively with measurements of the temperature dependence of the magnetic penetration depth (Ch. 4) and spectroscopic measurements of the microwave surface resistance (Ch. 5). The discussion of the latter will be far more detailed because it is the primary focus of this dissertation. 4.1 Magnetic Penetration Depth The magnetic penetration depth is a fundamental property of a superconductor. Its absolute value is directly related to the super
uid density and its temperature de- pendence is sensitive to the symmetry of the superconducting gap. Figure 4.1 shows the density of states (DOS) N(E) as a function of energy for (i) a BCS s-wave su- perconductor, (ii) a d-wave superconductor, and (iii) a dirty d-wave superconductor. The super
uid fraction xs(T )  2(0)=2(T ) is related to the DOS through [76]: 2(0) 2(T ) = 1 2 kBT Z 1 0 d! N(!) N0 [f(!)(1 f(!))] : (4.1) 20UBC Superconductivity Group, 59 4.1. Magnetic Penetration Depth Figure 4.1: For an s-wave superconductor the quasiparticle DOS is zero below the gap energy . The DOS peaks at  and asymptotically approaches N0. At T = 0, for a pure d-wave superconductor, the quasiparticle DOS is nonzero at all energies and there is a weaker singularity at E   = 0. 0 is the superconducting gap maximum and  is the normal state Fermi energy. Impurities and/or disorder destroy Cooper pairs leading to an enhanced DOS at the Fermi level and the peak at E   = 0 is further suppressed [74]. Figure provided courtesy of Saeid Kamal [75]. Here N(!) is the DOS in the superconducting state, N0 is the normal state DOS at the Fermi surface, and f(!) is the Fermi function. The s-wave BCS quasiparticle DOS leads to an exponentially activated penetra- tion depth. In contrast, a d-wave superconductor has a nite quasiparticle DOS at all energies. For a clean d-wave superconductor, the DOS varies linearly with E  , which leads to a penetration depth that grows linearly with temperature at low-T . As mentioned in x2.3.1, Cooper pairing in a d-wave superconductor is very sensitive to impurities and disorder. These imperfections lead to an enhancement of the quasi- particle DOS at the Fermi level and a T 2 temperature dependence for the penetration depth. Figure 4.2 shows the measured temperature dependence of (T ) for a BCS superconductor. This measurement by Hardy et al. was made on Pb0:95Sn0:05 alloy with a critical transition temperature of Tc = 7:2 K [49]. The penetration depth is nearly temperature independent up to  40% of Tc. In the same work, these authors measured the temperature dependence of (T ) of a single crystal of high-purity YBa2Cu3O6:95. The data are shown in Fig. 4.3. This sample has Tc = 93 K and a transition width Tc < 0:25 K. In stark contrast to the the Pb0:95Sn0:05 sample, the 60 4.1. Magnetic Penetration Depth   Figure 4.2: Measured temperature dependence of the penetration depth of a Pb0:95Sn0:05 sample. The observed exponential temperature dependence is charac- teristic of an s-wave BCS superconductor. Figure taken from Ref. [49].   Figure 4.3: Measured temperature dependence of the penetration depth of a YBa2Cu3O6:95 single crystal. The observed T -linear behaviour at low temperatures is characteristic of a clean d-wave superconductor. Figure taken from Ref. [49]. 61 4.1. Magnetic Penetration Depth penetration depth shows a strong linear term up to  25 K. This behaviour is ex- pected for a clean d-wave superconductor, and indeed this measurement was taken as the rst serious experimental evidence for line nodes in the cuprate superconducting gap. The same group then systematically studied the eect of adding low concentra- tions of impurities to otherwise pure YBa2Cu3O6:95 samples. In particular, Zn which substitutes for Cu in the CuO2 plane, was shown to cause the penetration depth to evolve from T -linear to T 2 as shown in Fig. 4.4. Zinc is a nonmagnetic impurity and is   Figure 4.4: Measured temperature dependence of the penetration depth of a YBa2 (Cu1xZnx)3O6:95 single crystal with x = 0; 0:0015; and 0:003. The observed temperature dependence shows a crossover from linear to quadratic which is expected for a order parameter with line nodes in the presence of a strong scattering impurity. Figure taken from Ref. [77]. not expected to signicantly aect a BCS s-wave superconductor. The sensitivity of the cuprate superconducting state to small concentrations of a nonmagnetic impurity was a clear signature of a gap with nodes. This section established that the temperature dependence of the magnetic pen- etration depth can provide insight into the symmetry of the superconducting order parameter which is one of the most fundamental properties of the superconducting state. The next two sections describe two related methods that can be used to ac- curately measure (T ). Both techniques probe the magnetic moment caused by 62 4.1. Magnetic Penetration Depth the expulsion of magnetic elds from the interior of the sample. The techniques are most reliable when a platelet sample is in a homogeneous magnetic eld in a ge- ometry with low demagnetization factors. Figure 4.5 shows the geometry used in the measurements presented in this thesis. A platelet sample which is thin in the Figure 4.5: A platelet sample placed in a homogeneous external magnetic eld. The broad face of the sample is parallel to the applied eld. Figure taken from Ref. [46]. c-axis direction is inserted into a homogeneous magnetic eld applied parallel to the ab-plane. In this case, the magnetic eld at the sample surface is nearly equal to the applied eld. The magnetic moment m generated by the sample in the Meissner state is: m  Happlied  (sample volume eective penetrated volume): (4.2) In particular, it will be shown that: m / (c 2)A; (4.3) where A is the area of the ab-plane of the sample. In a typical measurement the platelet samples are about 20 m thick in the c-axis direction and   0:1 m at 1 K. Since the experimental signal is proportional to the dierence in c2 one would need to know the sample thickness to four signicant gures (that is, to within tens of nanometers) to extract the penetration depth with 10% accuracy. It is, therefore, generally impractical to measure the absolute penetration depth by excluded volume techniques. Rather, these techniques measure the change in the sample penetration depth (T ) = (T ) (T0) away from an experimental base temperature T0. It is in this way that the temperature dependence of the magnetic penetration depth is extracted such as in gures 4.2 through 4.4 [46, 72]. 63 4.2. Ac Susceptometry 4.2 Ac Susceptometry An ac susceptometer consists of three co-linear coils. There is one large primary (drive) coil and two identical counter-wound secondary (pickup) coils. The coil ar- rangement is shown in Fig. 4.6. The primary coil creates identical ac magnetic elds Figure 4.6: When both of the secondary coils are empty, the net induced voltage across the secondary coils is zero (left). When a superconductor, such that T < Tc, is placed inside one of the secondary coils, the magnetic elds through the top and bottom secondary coils are not equal and there is a net induced voltage (right). in the top and bottom secondary coils. When the coils are empty as in the left hand side of the gure, the net voltage across the secondary coils is zero. Now consider a superconducting platelet at T u 0 inserted into the bottom secondary coil. Because of screening by the supercurrents, the magnetic eld in the bottom coil is reduced and there is a net voltage vsn induced across the secondary coils. The magnitude of the in-phase induced voltage is proportional to the volume from which the magnetic 64 4.2. Ac Susceptometry Figure 4.7: Below Tc, magnetic eld lines are expelled from the sample to within a penetration depth (T ) from the sample surface. With the magnetic eld parallel to the crystallographic b-axis, the volume of the eld-free region (red line) is independent of b. eld is excluded21. Figure 4.7 is a cartoon showing the expulsion of a magnetic eld from a platelet sample. This platelet has width a in the a-axis direction, length b in the b-axis direction, and thickness c in the c-axis direction. At T u 0 and for a magnetic eld applied parallel to the b-axis, the volume of the eld free re- gion is V  (c  20a)(a  20c)b22. For a platelet sample with a  0c, one has 21The out-of-phase component of the induced voltage corresponds to losses (energy dissipation) in the sample. 22This result ignores eects due to thermal expansion which are typically negligible for thin platelet samples with the eld applied parallel to the broad face of the sample [46]. 65 4.2. Ac Susceptometry V  (c  20a)A where A = ab is the area of the broad face of the platelet. At T u 0, the induced voltage across the secondary coils is denoted X(T u 0)  vsn. Now imagine changing the sample temperature such that 0 < T < Tc. The change in temperature increases the penetration depth above the base-temperature value by an amount (T ) and results in a decrease in the volume of the eld free region V = 2(T )A. Similarly, the voltage across the secondary coils will decrease by an amount X(T ). Collecting these results one has: X(T u 0) X(T ) = vsn X(T ) = c 20 2(T ) ; (4.4) or: (T ) = c 20 2vsn X(T ): (4.5) Typically, (0) is in the range 1000 2000 A and the sample thickness is c  20 m, so that the approximation (T )  (c=2vsn)X(T ) is often valid. The sensitivity of the (T ) measurement increases as the area of the sample since the prefactor c=vsn is proportional to A1. The ac susceptometer used by the UBC superconductivity group was designed and built by Chris Bidinosti and operates at 12 kHz [78, 79]. At this frequency, the magnetic elds completely penetrate the platelet sample when it is in the normal state. For this reason, increasing the temperature of the sample above Tc is equivalent to extracting the sample from the bottom secondary coil. Thus the voltage vsn, which calibrates the (T ) measurement, is simply equal to the dierence of the in-phase voltage of the lock-in detector when the sample is at the base temperature (typically 1.2 K) and at a temperature above Tc. The method used to control the sample temperature is the same for all of the cryogenic measurements discussed in this thesis. The method uses a sapphire hot- nger which allows the sample temperature to be changed independently from the rest of the apparatus [80, 81]. With this arrangement, the body of the experiment remains at a xed temperature and only a single calibration factor is needed. As shown in Fig. 4.8, the sample is mounted on a sapphire plate which in turn is attached to a temperature regulated sapphire block. A thin-walled tube made of amorphous quartz is used to thermally isolate the sapphire block from a copper block which is at the experimental base temperature. When the apparatus is fully assembled, only the sample and the tip of the sapphire plate are loaded into the bottom secondary coil. Sapphire was chosen to support the sample because crystalline sapphire has a very 66 4.2. Ac Susceptometry Figure 4.8: Cartoon of the typical design of the sample thermal stage. The sample is mounted on the tip of a temperature controlled sapphire plate which is thermally isolated from the low-temperature reservoir that sets the experimental base temper- ature. high thermal diusivity so that the sample, sapphire plate, and sapphire block are isothermal. The sample is axed to the tip of the sapphire plate using a thin layer of Dow Corning high vacuum silicone grease. The grease provides good thermal contact and allows for easy installation and removal of the samples. The heater is a 1500  chip resistor with a low temperature coecient of resis- tance [82]. For the ac susceptometer, the temperature sensor is a Lake Shore Cernox 1050 resistive thermometer [83]. The heater and bolometer are axed to the sapphire block using a thin layer of GE (General Electric Company) varnish which provides good thermal contact and sucient mechanical strength. The sapphire plate is at- tached to the sapphire block using vacuum grease, GE varnish, or a combination of the two. One end of the quartz tube is glued to the sapphire block and the op- posite end to a sapphire disc using Stycast 1266 clear epoxy. The sapphire disc is glued into a close-tting recess in a copper block that is in direct contact with the 67 4.3. Cavity Perturbation low-temperature reservoir that sets the experimental base temperature. For the ac susceptometer, the low-temperature reservoir is a dewar of pumped liquid 4He with a typical base temperature of 1.2 K. Twisted pairs of manganin wire are thermally anchored along the length of the quartz tube using GE varnish. The resistivity and dimensions of these wires were chosen such that their thermal conductance is much less than that of the quartz tube. Fine (25 m diameter) gold wires are used as a bridge between the heater/bolometer and the manganin wires. A detailed analysis of the thermal properties of the sample stage is presented in x5.10. 4.3 Cavity Perturbation The technique of cavity perturbation measures how the introduction of a sample changes the resonant properties of a cavity. It is simplest to imagine a high-Q cylin- drical cavity as in Fig. 4.9. When operated in the TE011 mode, there is an electric eld node and a magnetic eld antinode along the central axis of the cavity. The blue lines inside the cavities of Fig. 4.9 represent microwave magnetic elds. Small holes are drilled through the sides of the cavity to couple microwaves into (TX) and out of (RX) the cylindrical cavity. The resonance frequency f of an empty cavity is determined by the cavity dimensions. The width of the resonance w is a measure of the cavity quality factor (Q-factor): Q = 2 peak energy stored energy dissipated per cycle = f w ; (4.6) where, for a transmission geometry, w is the full width at half maximum of the trans- mitted power. For an empty cavity, the energy dissipated per cycle is primarily due to absorption of energy of the currents in the cavity walls. To maximize the cavity Q, the copper cavity walls are coated with Pb0:95Sn0:05. This alloy is a conventional superconductor with a transition temperature Tc  7 K. The cavity is operated in a pumped liquid 4He bath and its temperature is regulated at 1.2 K. The superconduct- ing coating of the cavity allows very large Q factors to be obtained. For cylindrical cavities, Q & 107 is routine. Now consider a superconducting platelet sample at the experimental base temper- ature that is introduced into the cavity. Because of screening eects, the microwave magnetic eld is expelled from the interior of the sample to within a penetration depth 0. The geometries of the sample and magnetic eld are the same as shown in Fig. 4.7. The expulsion of magnetic eld from the sample reduces the eective volume 68 4.3. Cavity Perturbation     Figure 4.9: Cartoon of the experimental technique of the cavity perturbation. With an empty cavity, the cavity resonant frequency is determined by the volume of the cavity and the particular microwave mode being used. The width of the resonance, determined by absorption in the cavity walls, is narrow. When a superconducting sample is inserted, the eective volume of the cavity is decreased, shifting the reso- nance to higher frequency. Additional absorption of the microwave magnetic eld by the sample broadens the resonance. Figure provided courtesy of Richard Harris [84]. of the cavity and shifts the resonance to a higher frequency f + f . The sample also absorbs some of the microwave magnetic eld causing a broadening of the resonance 69 4.3. Cavity Perturbation width from w0 to w > w0. For a platelet conductor of thickness c, the change in the resonance frequency and Q of the cavity are given by [85]: f f = 1 2 Vs Vr  1<  tanh (c=2) c=2  ; (4.7a)   1 Q  = Vs Vr =  tanh (c=2) c=2  : (4.7b) Here,  is the propagation constant of the electromagnetic eld inside the sample dened in x3.1, Vs is the sample volume, Vr is the eective volume of the resonator (eective cavity volume - sample volume), and < and = denote the real and imag- inary parts of the quantities in brackets. This thesis will focus on frequency shift measurements. However, measurements of the quality factor (at low T and not too high !) can also be used to extract the surface resistance of the sample. For a normal metal  = (1+i)= and for a superconductor  = 1=. Furthermore, in a superconductor c   except very close to Tc and for most normal metals at microwave frequencies c  . When c  ; , the hyperbolic tangents of Eqns. 4.7 are to a good approximation equal to one, so that: f f = 1 2 Vs Vr  1  c  metallic; (4.8a) f f = 1 2 Vs Vr  1 2 c  superconducting: (4.8b) Using the same sapphire hot nger scheme shown in Fig. 4.8, the sample temperature can be changed independently of the cavity temperature. Hence one can track changes in f caused by changes in  as a function of temperature. The frequency shift f from a base temperature T0 is dened as: f  f(T ) f(T0) = fA Vr (T ); (4.9) where A = Vs=c is the sample area and (T )  (T )(T0). A measurement of the cavity frequency shift as a function of temperature is therefore directly proportional to the change in the magnetic penetration depth. Furthermore, the sensitivity of the measurement is determined by the cavity lling factor Vs=Vr [46, 49, 72, 75]. The above analysis has ignored (1) the eects of thermal expansion and (2) the contribution to f from changes in the c-axis penetration depth. Changing the tem- perature of the sample will cause slight changes in the sample dimensions. As a result, 70 4.4. Sub-kelvin Cavity Perturbation all of Vs, Vr, and c change during a temperature scan. The thermal expansion correc- tions are small for thin (c < 20 m) crystals and at temperatures below 100 K [46]. In this thesis, penetration depth measurements are reported on thin samples for temper- atures below 50 K and the eects of thermal expansion are ignored. In the geometry of Fig. 4.7, the volume of the eld-free region of the sample is determined by both a(T ) and c(T ). As a result, the net frequency shift f of the cavity is determined by [75]: f = fA Vr h ab(T ) c a c(T ) i ; (4.10) where ab(T ) ! a(T ) when the applied eld is aligned with the sample b-axis as in Fig. 4.7 and ab(T ) ! b(T ) when the applied eld is aligned with the sample a-axis. By measuring the f(T ) due to a platelet sample, then cleaving that sample into a set of narrow needles to increase the c(T ) contribution and measuring f(T ) again, it is possible to separately determine (T ) in all three crystallographic directions. For example, this procedure was successfully applied to a single crystal of YBa2Cu3O6:95 [86]. 4.3.1 Calibration Calibrating the resonator is equivalent to determining the eective volume Vr of the cavity when loaded with a sample of volume Vs. The calibration is done by comparing fs due to a Pb0:95Sn0:05 sample of area A in the superconducting state at T0 (Eq. 4.8b) to fn due to the same sample in the normal state (Eq. 4.8a). For calibration purposes (T0) is assumed to be suciently close to zero (compared to the normal state skin depth ) such that: Vr = fA 2 (fs  fn) ; (4.11) where  is the normal state skin depth of the Pb0:95Sn0:05 calibration sample which can be determined from dc resistivity measurements. The temperature dependence of the magnetic penetration depth can then be determined from shifts in cavity resonance frequency via: (T ) =  Vr Af f: (4.12) 4.4 Sub-kelvin Cavity Perturbation Up until recently, the microwave resonators operated by the UBC superconductivity group have been operated in a glass dewar of pumped liquid 4He and have been limited 71 4.4. Sub-kelvin Cavity Perturbation to base temperatures of 1.2 K. At 1.2 K, the super
uid 4He still has a relatively high thermal conductivity and as a result the temperature of the entire super
uid bath can be regulated precisely using a single heater and thermometer. A stable resonator temperature is necessary to measure small changes in the cavity Q and resonance frequency. The left-hand side of Fig. 4.10 is a cartoon representation of the standard UBC cavity perturbation setup using a cylindrical cavity. To access lower sample Figure 4.10: Left: The standard experimental setup. The cavity is cooled to 1.2 K in a pumped bath of super
uid 4He. The sample base temperature is also 1.2 K. Right: In this conguration, the cylindrical cavity is still cooled to 1.2 K, however, the sample stage is mounted on a pumped 3He pot which in principle allows for base temperatures of  300 mK. base temperatures, the sample stage has been placed onto a pumped 3He pot. This pot is used only to cool the sample stage; the resonator temperature is still set by the temperature regulated super
uid 4He bath. This arrangement, shown in the right- hand side of Fig. 4.10, allows for sub-kelvin measurements without sacricing the high stability of the cavity Q and resonance frequency. The 3He pot is pumped by an activated charcoal pump cooled to 4.2 K in a liquid 4He storage dewar. The 3He can be recovered by warming the charcoal above 30 K. 72 4.5. Loop-Gap Resonator 4.5 Loop-Gap Resonator The main advantages of cylindrical microwave cavities are (1) a simple design and (2) high Q-factors are easily obtained. There are, however, drawbacks. The rst is that low-frequency measurements require relatively large cavities ( 103 cm3 at 1 GHz). This volume is much larger than the typical platelet sample (1 mm1 mm20 m = 2  105 cm3) and would result in a very low lling factor thereby sacricing the sensitivity of the measurement. A second drawback is that, even in the TE011 mode, it is not possible to position the sample/sample holder in a region that is completely free of electric elds. The dielectric response of the sample/sample holder can contribute non-negligible background signals. The desire for precision (T ) measurements motivated the design of a 1 GHz loop-gap resonator shown in Fig. 4.1123 [49, 75, 87, 88]. Figure 4.12 shows the res- onator assembly along with the positions of the coupling loops and the sample. The loop-gap resonator is the microwave equivalent of an LC resonant circuit. In this de- sign the electric elds are predominantly conned to the region of the gap, which acts as a parallel plate capacitor. The capacitance is approximately C = "0`w=t  70 pF, where   10 is the dielectric constant of sapphire. Approximated as a single turn inductor, the inductance of the loop is L = 0aa1=`  1:4 nH. These estimates give a resonance frequency: f = 1 2 p LC  0:5 GHz: (4.13) The nite length ` of the loop decreases the inductance and the presence of the cylindrical vacuum can surrounding the resonator further increases the resonance frequency. The measured resonance frequency of the loop-gap apparatus is 940 MHz. It is critical that the electric joint between the loop-gap and the support base be essentially lossless. This joint, labeled \Superconducting joint" in Fig. 4.12, must carry current to complete the LC circuit. A weak link in this joint will dramatically reduce the Q-factor of the resonator. To create a superconducting joint a wide groove has been milled into the bottom-left face of the loop-gap (Fig. 4.11). Screws are used to tightly secure the loop-gap to the support base. The narrow edges of the bottom- left face dig into the support base causing the Pb0:95Sn0:05 coatings of the opposing surfaces to fuse and form a superconducting joint. With this design, Q-factors in excess of 106 can be obtained. 23RS(T ) measurements are also possible using the 1 GHz loop-gap resonator. These measurements are, however, challenging because RS(!; T ) is small at low frequency. The cavity perturbation measurements presented in this thesis will primarily focus on (T ) data. 73 4.6. Robinson Oscillator 1 GapLoop Figure 4.11: Top: Digital photograph of the 1 GHz loop-gap resonator. The loop-gap apparatus is machined from a single copper block and electroplated with Pb0:95Sn0:05. A high-purity sapphire plate lls the gap region. Bottom: Dimensioned drawing of the loop-gap resonator. All dimensions are given in millimeters. The thickness of the sapphire plate is exaggerated for clarity. All other dimensions are drawn to scale. The length of the resonator is ` = 11:43 mm. The rounded regions in the actual resonator serve to improve the quality of the electroplating in the corner regions. Figure provided courtesy of Saeid Kamal [75]. 4.6 Robinson Oscillator In the most basic data acquisition technique, the cavity is driven by injecting mi- crowave power using one of the coupling loops. The microwave frequency is swept through the cavity resonance and the second coupling loop is used to detect the trans- mitted microwave power. On resonance, there is a peak in the transmitted power. 74 4.6. Robinson Oscillator Figure 4.12: Top: Cartoon of the loop-gap resonator assembly with a sample loaded in the resonator. There are coupling loops on either side of the resonator. Bottom: Equivalent circuit of the loop-gap resonator. The gap provides a high-capacitance region and the loop can be modeled as a single turn inductor. Figure provided courtesy of Geo Mullins [89]. Using this technique one obtains signals like those depicted in Fig. 4.9, which can be t to a Lorentzian line shape to extract the centre of the resonance. This resonance frequency can then be tracked as a function of the sample temperature. To obtain the highest resolution data, however, the cavity is used as a component of a circuit which, under suitable conditions, undergoes stable oscillations at the resonant frequency of the cavity [90]. Figure 4.13 shows a schematic diagram and digital photograph of the oscillator circuit used. In this scheme, the cavity is not driven by a microwave source, rather, the signal extracted from the cavity is amplied and fed back into the cavity using a feedback loop. There are two necessary conditions for oscillations. The rst is that there must be sucient gain to compensate for signal losses encountered over one complete loop around the feedback circuit. The second is that the phase change around the loop must be a multiple of 2. The total phase change is set by the length of coaxial cable used and is ne tuned using a commercial phase shifter. The oscillations are initiated by noise in the feedback loop. The high-Q cavity essentially acts as a narrow bandpass lter and transmits only the frequencies 75 4.6. Robinson Oscillator (a) (b) '
 60 ) & 
 //B7&
&  "#
&  "#
 Figure 4.13: (a) Schematic diagram of the oscillator circuit. The signal extracted from the cavity is amplied and then injected back into the cavity. Stable oscillations are achieved when the total phase shift around the feedback loop is a multiple of 2. The signal from the feedback loop is mixed with that from a local oscillator set to be 100 kHz o of the cavity resonance. The down-converted signal is continuously monitored by a counter. Figure provided courtesy of Geo Mullins [89]. (b) Digi- tal photograph of the acutal oscillator circuit. The room temperature elements are mounted on a temperature controlled aluminum plate. 76 4.6. Robinson Oscillator that are within fr=2Q of the resonant frequency fr of the cavity. As a result of the cavity, each of the transmitted frequencies undergoes a dierent phase shift that varies from =2 to +=2. This transmitted signal is then amplied, phase shifted, and then injected back into the cavity. Only the frequency that experiences a total phase shift of 2 will constructively interfere and result in stable oscillations. A directional coupler is used to direct part of the signal to a mixer where the signal is mixed with an o-resonance signal from a local oscillator (microwave source). The local oscillator is set to be 100 kHz o-resonance and the down-converted signal is continuously monitored by a counter. The frequency shifts of the cavity due to changes in sample temperature will cause identical frequency shifts in the down- converted signal. The stability of the oscillator frequency is determined by the stability of the total phase shift around the feedback loop. The room temperature electronics are mounted on a temperature controlled aluminum plate to limit variations in the eective path length due to temperature gradients. Since the phase shift of the cavity is given by: tan = Q  f fr  fr f  ; (4.14) changes in the phase around the loop will result in changes in the oscillation frequency f . From Eq. 4.14, one sees that the magnitude of these frequency changes are inversely proportional to the cavity Q. The stability of the frequency shift measurement is therefore directly proportional to Q. Figure 4.14 shows the stability of the oscillation frequency over a period of more than 6 hours. The cavity temperature is 1.2 K and the sample temperature is 420 mK. The maximum drift is less than 0.1 Hz/minute which corresponds to a change in  of order 0:2 A/minute for a 1 mm2 sample. Figure 4.15 shows down-converted oscillator frequency from a K-doped BaFe2As2 single crystal as a function of time. The frequency at the 420 mK base temperature is shown by the black points. Interwoven between the base temperature points are frequencies measured at elevated sample temperatures from 1.4 to 5.0 K shown as blue points. The frequency shift f is found by subtracting the average of the base temperature frequencies on either side of each blue point. This process is designed to remove the eects of the background drift from the f data. The results of applying this procedure to the data of Fig. 4.15 are presented in Fig. 4.16. In chapter 8, this procedure, combined with substantial averaging, is used to obtain (T ) data with sub-angstrom resolution at sub-kelvin temperatures. 77 4.6. Robinson Oscillator 0 100 200 300 400 0 5 10 15 20 25 30   Fr eq ue nc y (H z) Time (minutes) Figure 4.14: Measured drift of the oscillator frequency. The maximum drift is less than 0.1 Hz/minute. The cavity resonant frequency is 940 MHz and the cavity Q is 6 105. 78 4.6. Robinson Oscillator 0 50 100 150 200 106740 106750 106760 106770 106780   O sc ill at or  F re qu en cy Time (minutes) Figure 4.15: The down-converted oscillator frequency as a function of time. Interwo- ven between base temperature frequencies (black points) are frequencies obtained for elevated sample temperature (blue points) ranging from 1.4 to 5.0 K. 0 1 2 3 4 5 0 10 20 30 0 20 40 60  (Å )  f ( H z) Temperature (K) Figure 4.16: (f) (left axis) and  (right axis) as a function of temperature. This plot was generated using the data of Fig. 4.15. There are three data points at each temperature. The time elapsed between measurements at any given temperature is approximately one hour (see Fig. 4.15). 79 Chapter 5 Experimental Techniques: Microwave Spectroscopy This chapter describes a novel experimental technique developed to measure the ab- solute surface resistance of high-quality superconducting single crystals over a broad range of microwave frequencies. Because the platelet samples are small and have such low loss, probing their surface impedance is technically challenging. At microwave frequencies, the cavity-perturbation techniques described in the previous chapter have been successful in measuring both components of Zs [5, 72, 91]. This technique, while extremely useful, is limited in the sense that each resonator works only at one xed frequency, thus requiring multiple apparatuses to construct a coarse spectrum of data such as that produced by the UBC group in 1999 [92]. Here, a non-resonant bolometric technique is described that is capable of measur- ing the low-temperature absorption of unconventional superconductors continuously over a broad range of microwave frequencies. The samples are weakly connected to a base temperature and exposed to microwave magnetic eld whose frequency can be continuously varied. As the sample absorbs microwave power its temperature in- creases and this change in temperature is measured as a function of frequency using a resistance bolometer. This measurement is then used to extract the frequency de- pendent surface resistance RS (!; T ). This experimental technique has already been employed by Patrick Turner and the UBC group with enormous success [93]. In particular, two impressive results have recently been published. The rst was an investigation of the frequency dependence of the a- and b-axis quasiparticle conduc- tivity 1 (!; T ) of YBa2Cu3O6+y from 200 MHz to 22 GHz [94, 95], and second was the development of a novel method for measuring the absolute penetration depth of YBa2Cu3O6+y in the T ! 0 limit at several dopings [96]. This chapter will describe a second-generation design that is modular, allowing for measurements down to 1.2 K in a pumped liquid 4He cryostat or to temperatures be- low 100 mK using a dilution refrigerator [97]. An improved self-calibrating detection system that makes use of software lock-in detection is also described in detail. 80 5.1. Design 5.1 Design The spectrometer is a 6.6 mm long rectangular coaxial line made of copper and kept at low temperature inside a cryostat. The exterior of the spectrometer body is a rectangular chamber that forms the outer conductor of the transmission line. Centred in the chamber is a broad rectangular inner conductor, also made of copper. The centre conductor is shorted to the outer conductor at an end-wall. Fig. 5.1 is a schematic cross-section of the rectangular transmission line. Figure 5.1: Schematic cross-section of the terminated coaxial line showing the probe and reference samples suspended on sapphire plates in symmetric locations in an rf magnetic eld. Each sapphire plate is equipped with a heater and a temperature sensor and is isolated from the low-temperature bath via a thermal weak link (the quartz tube). Figure provided courtesy of Patrick Turner [93]. A dimensioned scale drawing of the rectangular coaxial line used in this thesis is shown in Fig. 5.2. The outer and inner conductors of the experiment body mate with the outer and inner conductors of a standard 0.141 in. cylindrical semirigid coaxial line24 which runs from room temperature to the waveguide body. The inner conductor of the cylindrical transmission line is inserted into a small hole drilled into the centre conductor of the rectangular coaxial line. The outer conductor of the cylindrical transmission line is stainless steel, chosen for its low thermal conductivity, and the inner conductor is silver plated copper weld, chosen for its low loss. Microwaves are supplied using a Hewlett Packard 83630A 10 MHz to 26.5 GHz synthesized sweeper and propagate down the semirigid coaxial cable and into the rectangular waveguide structure. When operated below the cuto frequency of the lowest transverse electric (TE) modes, only the transverse electromagnetic (TEM) mode propagates, producing magnetic elds that loop symmetrically around the inner conductor 24The coaxial cable used was UT-141-SS from Micro-Coax. 81 5.1. Design .063 .222 .128 .030 .1483.76 mm 0.76 mm 5.64 mm 3.25 mm 1.60 mm 3.76 m 0.7 3.25 . m .60 mm Figure 5.2: Scale drawing of a cross-section of the rectangular waveguide structure. Each of the two sapphire plates supports a sample centred between the inner and outer conductors of the transmission line. The red line represents the TEM magnetic eld looping around and centre conductor in the plane of the page. An estimate of the magnitude of the magnetic eld H at the sample sites can be made using Ampere's Law: I H  ds = iencl; (5.1) where iencl is the current enclosed by the Amperean loop. Choosing the red path shown in Fig. 5.2 and assuming that H is constant everywhere on that path gives H = iencl=s, where s is the circumference of the loop. An estimate of the power P delivered to the waveguide is needed to calculate the current ienlc = p P=zo. This estimate is dicult because, although the impedance of the rectangular coaxial line is designed to match that of the cylindrical coaxial cable (zo = 50 ), a non-negligible re
ection is still likely at that connection. Furthermore, some of the incident power will be dissipated due to losses in the cylindrical transmission line used to deliver microwaves to the waveguide. The cylindrical coaxial line goes from room tempera- ture to  1:2 K so that the actual loss per foot is dicult to determine despite the specications supplied by the manufacturer. Nevertheless, the insertion loss speci- cations at 20C are 0.25 dB/ft. at 1 GHz and 1.26 dB/ft. at 20 GHz. A continuous 4 ft. length of cable was used in the experiment so that a typical incident power of 22 dBm is reduced to 21 dBm = 126 mW and 17 dBm = 50 mW at the sample site for 82 5.2. Rectangular Coaxial Line Assembly frequencies of 1 GHz and 20 GHz respectively. These powers correspond to currents of i1 GHzencl = 50 mA and i 20 GHz encl = 32 mA resulting in H 1 GHz = 3:7 Am1 = 0:046 Oe and H20 GHz = 2:4 Am1 = 0:030 Oe. Experimentally determined values of H are presented in x5.4 and agree reasonably well with these estimates. The samples are supported by 22 mm long sapphire plates that are 500 m wide and 100 m thick and whose presence has little perturbing eect on the microwave eld. The samples are held very near the end-wall of the experiment body where there is an electric eld node and a magnetic eld anti-node and thus are exposed to predominately magnetic elds. The sapphire plates are fed through 1.6 mm (1/16 in.) diameter cuto holes in the waveguide body that do not allow any modes below 145 GHz to propagate out of the cavity. When loaded, the sapphire plates are located approximately 400 m from the conducting end-wall. The last 15 cm of the cylindrical coaxial cable leading to the waveguide has been made superconducting in order to reduce heating near the experiment body due to resistive losses. Any excess heat can change the sensitivity of the measurement or provide spurious signals. To make the superconducting coaxial line, a 15 cm section was disassembled and then the interior of the outer conductor was coated with 50/50 PbSn solder, which is superconducting below 7 K. The cable was then reassembled with a new stainless steel inner conductor which was also PbSn coated. The inner surfaces of the rectangular coaxial structure were also coated with 50/50 PbSn solder. 5.2 Rectangular Coaxial Line Assembly In this section, the design and assembly of the rectangular coaxial line is described. The dimensions were chosen such that the impedance of the rectangular coaxial line is 50  to match that of the semirigid cylindrical coaxial line and so that the relevant TE01 and TE10 cuto frequencies were as high as possible while leaving adequate space to measure samples up to 1:5  1:5 mm2 in area. A detailed analysis of the design criteria for this apparatus can be found in Ref. [97]. The coaxial structure is assembled from three separate copper pieces. The design was chosen such that the important dimensions (see Fig. 5.2) could be machined as accurately and symmetri- cally as possible. Figure 5.3 is a scale CAD25 drawing of the three pieces that form the electrically shorted rectangular coaxial line. Precision machining was carried out by the UBC Physics machine shop. Prior to assembly, surfaces that mate to form the rectangular coaxial line were pre-tinned with 50/50 PbSn solder. Tinned surfaces are 25Computer-Aided Design done using IronCAD 10.0. 83 5.2. Rectangular Coaxial Line Assembly inner conductor (septum) cutoff holes cutoff holes endwall for semi-rigid coax Figure 5.3: Scale drawing of the three mating pieces that form the rectangular coaxial line shorted by an end-wall. All of the screws shown are #4-40. coloured grey in Fig. 5.3. Figure 5.4 is a closeup digital photograph before and after PbSn tinning. Figure 5.4: Top: Bare copper surface of the septum and end-wall. The hole in the septum accepts the centre conductor of the semirigid UT-141 coaxial cable. Bottom: The same surfaces after PbSn tinning. 84 5.3. Sample Stage Design/Sample Loading Next, the cylindrical semirigid coaxial cable is soldered to the leftmost piece shown in Fig. 5.3. The design is such that the centre conductor of the semirigid cable passes through a small hole allowing it to mate with the broad septum of the rectangular coax as shown in the sequence of photographs in Fig. 5.5. A strength of this design Figure 5.5: Left: The centre conductor of the semirigid cable protrudes from a small hole in the outer conductor of the rectangular coaxial cable. Middle: The septum mates with the centre conductor. Right: The connection is complete. is that electrical connection between the centre conductors at the transition from cylindrical to rectangular coaxial lines is not blind. By visual inspection, one can be sure of a quality joint. At this stage a liberal amount of water soluble acid 
ux was applied and the assembly was heated above the melting temperature of the solder. After cooling, the excess 
ux was removed using an ultrasonic water bath and then by scrubbing with ethyl alcohol. Next, a conservative amount of 
ux was applied and the nal (right piece in Fig. 5.3) piece of the assembly was tightly bolted in place. The assembly was once again taken above the melting temperature of the solder and then cleaned. Figure 5.6 highlights the details of the transition from cylindrical to rectangular coaxial line. 5.3 Sample Stage Design/Sample Loading Both the sample of interest and the reference samples are supported by thermal stages described in the previous chapter and shown in Fig. 4.8. However, rather than using a sapphire block to connect the sapphire plate to the quartz tube, one end of the sapphire plate is inserted 1 mm into the quartz tube and held in place using a small amount of Emerson & Cuming Stycast 1266 epoxy. The sapphire plate is equipped with two bolometers and one heater. One bolometer is suitable for measurements from 1 to 10 K and the other from 0.1 to 1 K. The quartz tube is 12.1 mm long with 85 5.3. Sample Stage Design/Sample Loading A Semirigid Centre Conductor Teflon Dielectric Semirigid Outer Conductor Septum Sapphire Plate Tip Figure 5.6: Top: Scale drawing of the cross-section of the assembled rectangular coaxial line. Bottom: Detail view of the transition from cylindrical to rectangular coaxial line. inner and outer diameters of 0.635 and 1.2 mm respectively. Copper clad NbTi wires, 50 m in diameter, serve as electrical leads. The copper was etched away from the midsection of the wire and left only on the tips for the purposes of soldering. The etched wires are glued along the length of the quartz tube using Stycast 1266 epoxy. These leads were chosen for their low thermal conductance. Four wire resistance measurements are made from the free ends of the superconducting wires (one for each of the two bolometers and one for the heater). Figure 5.7 is a digital photo of 86 5.3. Sample Stage Design/Sample Loading Quartz Tube Sapphire Plate Figure 5.7: Digital photograph of the sample thermal stage. The NbTi wires are in place, but the bolometers and heaters were not yet installed at the time of the photograph. the actual sample-side thermal stage. The bare (uninsulated) NbTi wires must pass through small holes in the copper block. Figure 5.8 shows the technique used to avoid electrical shorts. The reference stage is similar in construction to the sample stage, except that in Figure 5.8: Wiring of the sample stage. The bare NbTi wires are kept separate within a coil of masking tape. The interior walls of the copper block are coated with electrically insulating Stycast 2850-FT epoxy. 87 5.3. Sample Stage Design/Sample Loading place of the quartz, a rectangular bar of a copper alloy is used. Because the reference sample is a normal metal with a very large surface resistance compared to a typical superconducting sample, much less sensitivity is required from the reference stage to acquire reasonable signal levels. The design criterion was to make the thermal conductance of the reference stage ten times that of the quartz tube at 1 K. The resistivity, specic heat, and thermal conductivity of the alloy were estimated to be   1:1  107 m,   0:22 Wm1K1, and cV  171 JK1m3. The details of these determinations can be found in Appendix B. The nal dimensions of the alloy bar are L W  H = 11:8 mm  1:35 mm  0:76 mm with an active length26 of 9.25 mm. The electrical leads used for the reference stage are 75 m diameter insulated manganin wires, whose thermal conductance and heat capacity are negligible compared to those of the copper alloy bar. Figure 5.9 shows the wiring of the bolometers and heater. Visible on the topside of Figure 5.9: Digital photograph of the sample the stage wiring. The red bar represents a length of  500 m. the sapphire plate are two bolometers. The bolometers are attached to a second sap- phire plate using Stycast 1266 epoxy which is in turn xed to the main sapphire plate using silicone vacuum grease. Gold leads are attached to the sides of the bolometers using silver epoxy. The gold leads extend towards the quartz tube where they are joined to the copper-clad NbTi wires by indium solder. To load samples into the rectangular coaxial line, the side plate shown in Fig. 5.10 was designed. The thermal stage passes through a Vespel SP-2227 tube attached to 26The active length is the length of the bar that is not shorted by copper or sapphire, and is the length that determines the sensitivity of the thermal stage. 27The Vespel SP-22 material was obtained from DuPont, Newark, DE, USA. 88 5.3. Sample Stage Design/Sample Loading Electrical Connector Sapphire Plate Vespel Tube Copper Blocks CopperSide Plate Quartz Tube Figure 5.10: Scale CAD drawing of the side plate that allows samples to be easily loaded into the rectangular waveguide. The labeled Vespel tube and copper blocks provide thermal ltering to isolate the bolomters from temperature oscillations of the low temperature bath and/or the main body of the experiment. a copper side plate. The Vespel tube and its copper blocks serve as a thermal lter as described in x5.5. The side plate mates with the main body of the experiment by means of a pair of guide pins as shown in Fig. 5.11. A cross-section of the fully assembled apparatus is shown in Fig. 5.12. 89 5.3. Sample Stage Design/Sample Loading Figure 5.11: Top: The side plate slides onto guide pins in the main body of the apparatus. The tip of the sapphire plate will enter into one of the cuto holes. Bottom: A digital photograph of the actual apparatus. The tip of the sapphire plate (nearly invisible in the picture) is in the cuto hole. The sample, attached to the sapphire by grease, is visible. 90 5.3. Sample Stage Design/Sample Loading Sample Stage Quartz tube Reference Stage CuGa Alloy Sapphire Plate Cutoff Hole Rectangular Cavity Centre Conductor (Septum) Figure 5.12: Scale drawing of a cross-section of the fully assembled microwave spec- trometer. Visible are sample and reference thermal stages. The ends of the sapphire plates pass completely through the rectangular coaxial line to maintain maximum symmetry. Samples are placed symmetrically on either side of the septum. 91 5.4. Measurement Technique 5.4 Measurement Technique This experiment gives an indirect measurement of the power absorbed by a conducting sample in a microwave magnetic eld. Recall Eq. 3.33 from x3.3 that the absorbed power is given by: P = RSH 2A; (5.2) where A is the area of the broad face of the crystal. The factor of 1=2 is absent because in the measurement both faces of the sample are exposed to the eld thereby doubling the eective area. The experimental geometry has two independent sites on either side of the inner conductor that experience identical magnetic elds28 (see the two sample sites in Fig. 5.1). This important fact allows for a simultaneous measurement of the power absorbed by two dierent conducting samples exposed to the same magnetic eld. One of the samples is a normal metal alloy and is used as a power meter. From here on this sample is referred to as the reference sample. The other sample is the sample of interest, and can be any conducting material (high temperature superconductor, heavy fermion, normal metal, ...). Because the reference sample is a normal metal, its surface resistance is well understood and is given by classical the skin-eect relation introduced in x3.1.2: RrefS =   = r 0! 2 : (5.3) The reference samples used in the measurements presented in this thesis are made from a silver:gold (70:30 atomic %) alloy. This material has a temperature indepen- dent resistivity of 5:28  0:3  cm from 1.2 to 20 K [93]. Note that, in order for Eq. 5.3 to be valid, one must be in the local electrodynamics limit, where the trans- port mean free path is much less than the skin depth. This condition can be violated in pure metals at low temperatures, where the mean free path becomes long. The choice of an alloy for the reference material avoids such a scenario. A measurement of the power absorbed by the reference alloy along with Eqns 5.2 and 5.3 can be used to determine the magnitude of the eld H at the sample sites. Experimentally determined values of H are given in Table 5.1 for microwave frequen- cies of 1 GHz and 20 GHz and an incident power of 22 dBm. These results agree with the estimates made in x5.1 to within an order of magnitude. Section 5.9 will show that, due to standing waves in the microwave circuit, the power delivered to 28This assumption is valid for the low frequency TEM mode of the coaxial transmission line. The assumption breaks down when the asymmetric higher frequency modes start to propagate. 92 5.4. Measurement Technique f (GHz) RS (m ) P (W) H (Am 1) 1 14.4 3.65 20.3 20 64.6 0.0927 1.52 Table 5.1: Experimental determination of the magnitude of H at the sample sites. the spectrometer  H / pP  is strongly frequency dependent. Taking the ratio of the powers absorbed by the two samples results in the following simple expression: P sam P ref = RsamS RrefS Asam Aref ; (5.4) where the sample of interest has been labeled with the superscript \sam" and the factor H2 has dropped out because it is identical for the two sample sites. P sam, P ref , Asam, Aref , and RrefS are all known quantities and hence R sam S is easily determined. All that remains is to describe how the power absorption is measured. When the samples absorb power from the magnetic eld they heat up, and the change in temperature is detected by the bolometer. The temperature change T of the sample is related to the thermal conductance G of the thermal weak link in the following way: T = P G : (5.5) In practice the microwave power is modulated at low frequency ( 1 Hz) causing a low frequency oscillation in the temperature of the samples. The large thermal diusivity  = =cV of the sapphire plate ensures that the bolometer temperature tracks the sample temperature on a time scale that is very short compared to the modulation period. That is, on the scale of the modulation period, the temperature prole of the bolometer matches that of the sample as the microwave power is turned o and on. Lock-in detection is used to synchronously detect the sample temperature oscil- lation as a change in the voltage across the bolometer. The samples studied are typically very high purity high temperature superconducting single crystals that can have a very small surface resistance (for example RS  1   at 1 GHz and 1 K). Therefore, the signals to be detected can be very small and easily swamped by ex- 93 5.5. Thermal Filtering ternal interference. For this reason, the ac voltages from the sample bolometer are amplied (typically by a factor of 104 for a superconducting sample) within a bias box before being transmitted to the lock-in detector through an optical isolation stage. The 1 Hz ac voltage from the bias box is proportional to the amplitude of the variation in the bolometer resistance. A method for converting this voltage to the absolute power absorbed by the sample is needed and a simple way of determining the necessary calibration factor has been developed. With no microwaves present, one can mimic the sample temperature 
uctuations by passing a 1 Hz current IHtr through the chip heater with resistance RHtr. Then the bolometer response can be measured for a known power (P = I2HtrRHtr). Because the sensitivity (i.e. the inverse of the thermal conductance) of the thermal weak link is temperature dependent, a calibration factor needs to be determined separately for all temperatures. For the calibration factor to be meaningful, it is essential that the bolometer re- spond the same for power dissipated by the heater and sample. That is, a nanowatt of power delivered to the heater for a period of one second should be equivalent to a nanowatt of power delivered to the sample over one second. This equivalence is achieved only if both the sample and heater are in good thermal contact with the sapphire plate and the sapphire plate has a suciently high thermal diusivity. Sec- tion 5.10 will experimentally probe the upper limit of suitable modulation frequencies. It will be shown that, in practice, the modulation frequency is limited by the thermal properties of the thermal weak links (quartz tube, alloy bar) and not the contact between the sapphire plate and sample/heater. The temperature of the sapphire stage is set by the bias current ( 1 1000 A) through the bolometer. This allows frequency sweeps to be performed at temperatures ranging from the base temperature up to  30 K. For the semiconductor bolometers that are used, dR=dT decreases with increasing temperature (see Fig. 5.14 in x5.6). This loss in sensitivity is partly compensated for by the larger bias current needed at the higher temperatures. It is important, however, that the bolometers do not exhibit excess noise in the presence of the bias current. The type of bolometer selected and its calibration are discussed in x5.6. 5.5 Thermal Filtering The issue of thermal ltering is important because the measured signal is a periodic temperature modulation that must be carefully isolated from other spurious tempera- ture 
uctuations. A semi-rigid coaxial line is used to transmit high-power microwave 94 5.5. Thermal Filtering elds to the rectangular coaxial line which can result in a modulation of the tem- perature of the main experiment body. These temperature modulations cannot be permitted to reach either the reference or sample bolometers. Although the temper- ature modulation of the coaxial cable has been minimized through heat sinking and by making the last 15 cm of the cable superconducting, further thermal ltering was deemed necessary. Specically, low-pass thermal lters have been placed between the main body and the thermal stages. The thermal stages are connected to the experiment body through a length of Vespel SP-22 tube which is thermally anchored to the low temperature bath using copper blocks and copper wire. Vespel SP-22 is graphite-lled polyimide material that has a low thermal conductivity and is easily machined. The Vespel acts as a thermal resistance whereas the copper blocks have a large heat capacity and very low thermal resistance. To analyze heat transfer in the experiment, an equivalent electrical circuit can be used to model the nal design. In this model, voltage corresponds to temperature, resistance to inverse thermal conductance, capacitance to heat capacity, and current to thermal power. There is, however, an important distinction that must be made between this model and conventional circuit analysis which is most easily seen by recalling the denition of thermal conductance: G = 1 jT2  T1j A ` Z T2 T1 (T )dT; (5.6) where T1 and T2 are the temperatures at opposite ends of a material length ` and cross-sectional area A and (T ) is the temperature dependent thermal conductivity. Clearly, the thermal conductance depends on the temperature dierence across the material, whereas in conventional circuit analysis a material's resistance does not depend on the voltage dierence across it. Provided that the temperature dierence across the material is not too large, the thermal conductance can be approximated as G  (A=`) T where T = (T1 + T2)=2 is the average temperature. When this approximation is valid the thermal conductance of the material no longer directly depends on the temperature dierence across it and analyzing the equivalent thermal circuits proceeds as in conventional circuit analysis. Figure 5.13 shows the equivalent circuit for one of the low-pass copper/vespel lters. The notation eT represents the amplitude of a temperature oscillation. For example, the temperature of the experiment body is TB + eTBej!t, where TB is the average dc temperature. This section will address the following question: Given a 95 5.5. Thermal Filtering Figure 5.13: Equivalent circuit representation of one of the copper/vespel low-pass thermal lters connecting a thermal stage to the experiment body. temperature oscillation of the experiment body of amplitude eTB, what are the ampli- tudes of the oscillations at the points labeled eT1 and eT2? Specically, the quantities eT1.eTB and  eT2.eTB will be calculated29. The driving oscillation of amplitude eTB could arise from microwave power dissipated in the semirigid coaxial cable. This dis- sipated power could be absorbed by the spectrometer body and, since the microwave power is modulated at low-frequency, would lead to temperature oscillation. The spectrometer temperature could also oscillate due to a periodic variation in the ex- perimental base temperature (for example an oscillation of the dilution refrigerator still temperature, see chapter 9). eT1 occurs at the sample thermal stage and is sep- arated from the bolometer only by the quartz tube. It is clear that a substantial eT1 would contaminate the experimental signal. The circuit in Fig. 5.13 is that of two cascaded low-pass lters. It is true in general that: eT1eT2  = 1p1 + !2R21C21 ; (5.7) and that:  eT1eTB  =  eT2eTB   eT1eT2  : (5.8) However,  eT2.eTB is a more complicated quantity because one must deal with the parallel combination of ZC2 and R1 + ZC1 , where ZC = 1=j!C. The full solu- tions for  eT1.eTB and  eT2.eTB are given in Appendix C, however, here the limit !R1C1; !R2C2  1 is considered. It is clear that, in this high-frequency limit, the 29 eT1 and eT2 are complex numbers due to a phase shift of the temperature oscillation with respect to the driving oscillation at TB with amplitude eTB. 96 5.5. Thermal Filtering parallel combination of Ze = ZC2jj(R1 + ZC1)  ZC2 . In this case: eT2eTB   1p1 + !2R22C22 (!R2C2  1); (5.9) and hence:  eT1eTB  =  eT2eTB   eT1eT2   1p1 + !2R21C21 1p1 + !2R22C22 (5.10a)  1 !212 ; (5.10b) where the thermal time constants 1 = R1C1 and 2 = R2C2 have been dened. In this model C is the heat capacity of the copper blocks that are thermally anchored to the experimental base temperate and epoxied to the Vespel tube. Specif- ically, C = V cm where  = 8920 kg/m 3 is the density of copper, V is the copper block volume, and cm = 8:0  103T JK1kg1 is the specic heat per unit mass of copper [3]. The thermal resistance is R = G1  `=A T, where ` is the length of Vespel between the copper blocks and A is the cross-sectional area of the Vespel tube. The thermal conductivity of Vespel SP-22 was measured by Locatelli et al. to be VP = (0:0017 T 2) Wm1K1 valid over the temperature range of 50 mK to 2 K [98]. Table 5.2 gives the physical dimensions of the copper thermal capacitors and Vespel thermal resistors and Table 5.3 gives numerical values for the thermal capac- itors, resistors, and time constants at 1.2 K and 100 mK. The ltering improves as temperature decreases because the Vespel thermal conductivity evolves as a higher power of T than the copper specic heat which goes as T -linear at low temperature. C1 C2 R1 R2 V (107 m3) 9.77 5.42 ` (mm) 3.3 6.7 A (105 m2) 1.13 1.13 Table 5.2: Physical dimensions of the spectrometer low-pass thermal lters. Given are the copper block volumes and the Vespel tube lengths and cross-sectional areas. 97 5.6. Signal Conditioning T = 1:2 K T = 100 mK C1 (J/K) 84 7.0 R1 (K/W) 0.12 17 1 (s) 10. 120 C2 (J/K) 46 3.9 R2 (K/W) 0.24 35 2 (s) 11 140 !212 (103) 4.3 660 Table 5.3: Numerical values of the spectrometer thermal capacitors, resistors, and times constants. Also given is !212 which is the factor by which the oscillation amplitude  eTB is suppressed at T1. This factor is given for ! = 2(1 Hz) which is a typical modulation frequency for the microwave power and satises the high frequency limit !  1. 5.6 Signal Conditioning As described in x5.4, the raw signal in this measurement is due to temperature oscil- lations of the sample in response to the low frequency (typically 1-3 Hz) modulation of the microwave power. The bolometers chosen are Haller-Beeman NTD (Neutron Transmutation Doping) resistive thermistors30 because they show no excess noise above the Johnson noise limit in the presence of a bias current [93, 99]. For compar- ison, the rst generation broadband spectrometer used a Cernox 1050 temperature sensor purchased from Lake Shore Cryotronics, Inc. which showed 40 dB of excess noise in the presence of a 1:3 A bias current. As noted previously, both the sam- ple and reference thermal stages are equipped with two bolometers, one suitable for measurements in the 1-10 K range and one suitable for measurements in the 0.1-1 K range. The calibrations of the two sample stage bolometers are shown in Fig. 5.14. Figure 5.15 shows that at low temperature, a semi-log plot of the bolometer resistance versus T1=2 is approximately linear. This fact can be used to make low-temperature extrapolations of the calibration. The signal is obtained by passing a dc bias current IBIAS through the 30Haller-Beeman Assoc. Inc., 5020 Santa Rita Rd., El Sobrante, CA 94803 USA. Haller-Beeman Assoc. Inc. are no longer operating. 98 5.6. Signal Conditioning 0.1 1 10 100 101 102 103 104 105 106   R es is ta nc e ( ) Temperature (K) Figure 5.14: The resistance versus temperature calibration of the sample stage Haller- Beeman bolometers. At sub-kelvin temperatures, the low-temperature bolometer (blue data) was calibrated against a commercial thermometer on the dilution refrig- erator of David Broun at Simon Fraser University [100]. bolometer and monitoring the voltage change across it as its resistance changes VHB = IBIASRHB. For small amplitude temperature oscillations the bolometer re- sponse VHB is directly proportional to the microwave power absorbed by the sample. The bolometer biasing serves the additional purpose of setting the sample tempera- ture. The dc power dissipated PBIAS = I 2 BIASRHB by the bolometer along with the thermal conductance of the sample stage determines by how much the sample tem- perature is elevated above the base temperature. Figure 5.16 shows how the sample temperature is set and measured. A bias voltage VB is applied across a bias resistor RBIAS and the bolometer RHB using a 1.5 V or 9 V battery. At low-temperature RHB  RL and the lead resistance can be ignored. In this case, IBIAS = VBIAS=RBIAS and can be changed by switching either the bias resistor or bias voltage. The dc volt- age across the bolometer is amplied by a factor of 10 using an operational amplier (op-amp) in the non-inverting conguration. The bolometer resistance, and hence 99 5.6. Signal Conditioning 0.0 0.5 1.0 1.5 2.0 2.5 102 103 104 105 106   R es is ta nc e ( ) T -1/2 (K -1/2) 10 K 1.0 K 0.5 K4.2 K 0.2 K Figure 5.15: Haller-Beeman resistance as a function of T1=2 below 10 K. Figure 5.16: Simple bias circuit used to set and measure the sample temperature. All elements are at room temperature except the bolometer, RHB, and the leads, RL, that run from room temperature to low temperature. 100 5.6. Signal Conditioning the sample temperature is determined by: RHB = VOUT 10IBIAS ; (5.11) where digital voltmeters are used to measure VBIAS and VOUT. After the 10 amplier, the bolometer voltage passes through a series of lters separated by buer ampliers. The 0.1 Hz high-pass lters block the dc component of the signal before the nal high-gain amplier. The complete signal processing circuit is shown in Fig. 5.17. A recovery switch is included so that the RC time constant of the high-pass lters can be temporarily reduced. This function can be useful, for example, when altering the dc bias conditions to change the sample temperature; the transient response (or the time to reach a stable equilibrium) can be signicantly reduced by shorting the 300 k  resistors with the 1 k  resistors. All ampliers are powered using 12 V rechargeable sealed lead-acid batteries. The noise characteristics of the 10 op-amp were carefully considered to achieve optimal performance. Fig- ure 5.18 shows the equivalent noise circuit for a non-inverting amplier [101]. It is particulary important that this amplier be optimized because its output is typically amplied by a another factor of 103  105. In the circuit of Fig. 5.17, the source resistance of the 10 amplier is the parallel combination of RBIAS at room tempera- ture and RHB at low-temperature. Before proceeding, it is necessary to calculate the eective noise temperature Te of this parallel combination. The bias voltage source VB is an alkaline battery and is assumed to be noiseless. Any real resistor R can be represented by an ideal (noiseless) resistor in series with a noise voltage: e2R = 4kBTR; (5.12) or equivalently, an ideal resistor in parallel with a noise current: i2R = 4kBT R : (5.13) This is the Johnson (thermal) noise of a resistor at temperature T and  is the bandwidth over which the noise is measured. Figure 5.19 shows how to nd the eective noise temperature of two resistors at dierent temperatures. In the rst step, the real resistors are replaced by ideal resistors in series with a noise voltage source. Next, the voltage sources are replaced by current sources which puts RBIAS in parallel with RHB. The nal circuit is reduced to a single noise current i2RS and a 101 5.6. Signal Conditioning Figure 5.17: Electronics for conditioning the raw bolometer signals. The electronics: (i) apply a dc bias to the bolometer which sets the sample temperature, (ii) provides both low- and high-pass ltering, and (iii) provides high-gain amplication of the desired signal. 102 5.6. Signal Conditioning Figure 5.18: Non-inverting amplier equivalent noise circuit. The noise sources in- clude: op amp input noise voltage en, op amp input noise currents in and i 0 n, source resistor noise voltage eRS, feedback resistor noise voltage eRF, and the gain-setting resistor noise current iRG. Figure 5.19: Calculating the eective noise temperature of parallel resistors, RBIAS at room temperature T1 and RHB at low-temperature T2. single ideal source resistance RS: RS = RBIASRHB RBIAS +RHB (5.14a) i2RS = i 2 B + i 2 HB (5.14b) i2RS = 4kB  T1 RBIAS + T2 RHB  : (5.14c) 103 5.6. Signal Conditioning One can choose to now revert back to a noise voltage source e2RS = i 2 RSR 2 S such that: e2RS = 4kB  T1 RBIAS + T2 RHB  RBIASRHB RBIAS +RHB 2 (5.15a) = 4kBTeRS; (5.15b) where the eective noise temperature is: Te =  T1 RBIAS + T2 RHB  RBIASRHB RBIAS +RHB  : (5.16) In typical operation RBIAS  1 M  is at room temperature and RHB  100 k  is at 1 K. These values give a source resistance of RS  91 k  at an eective noise temperature Te  28 K. Returning to Fig. 5.18, the individual noise sources have to be combined to nd the total output noise e20. Because the relative phases of the individual noise sources are random, the noise powers must be added to get the total output noise power (as was done in Eq. 5.14b). The following three assumptions are made to calculate the total spot noise e20 at the op amp output: (1) the op amp inputs have innite impedance (Z ! 1), (2) the op amp input current noise sources are equal  i2n = i 0 n 2  , and (3) RF  RG. For the actual circuit RF=RG  9:1. Initially, an analysis of the instantaneous voltages and currents will be considered. To obtain the nal result we will switch back to the appropriate rms values which will be represented by symbols with a bar overtop (i.e. p x2 versus x). To begin, consider the noise voltage at the non-inverting op amp input: V+ = en + eRS + inRS: (5.17) Next, to analyze the noise voltage at the inverting input, apply the assumption that RF  RG such that the currents i0n and iRG go to ground via RG. There is an additional current i due to eRF that has a path to ground through RF and RG. Considering all of the noise sources, one can write the following two expressions for the noise voltage at the inverting input: V = (iRG + i0n + i)RG; (5.18a) V = iRF  eRF + e0: (5.18b) 104 5.6. Signal Conditioning Eliminating i from these equations one arrives at: V = (e0  eRF) 1 G + (iRG + i 0 n)RG; (5.19) where G = 1 + RF=RG  RF=RG is the gain of the amplier and RF=G  RG. In general, the voltage at the output of the op amp is given by: e0 = A(V+  V) = A  en + eRS + inRS  e0  eRF G  (iRG + in)RG  ; (5.20) where A is the open-loop gain of the op amp (typically A  109) and in = i0n has been applied. Isolating e0 and using A G one arrives at: e0 = (en + inRS + eRS)G+ eRF  (iRG + in)RF; (5.21) where now RF  RGG has been used. Finally, recognizing that it is the noise powers that should be added and identifying e2RS = 4kBTeRS, e 2 RF = 4kBTRF, and i2RG = 4kBT=RG the total noise voltage at the op amp output normalized to a 1 Hz bandwidth is:q e20 = r e2nG 2 + i2nR 2 SG 2 + 4kBTeRSG2 + i2nR 2 F + 4kBTRF + 4kBT RG R2F: (5.22) The eective temperature Te is given by Eq. 5.16 and T  300 K. For typical op amps and the RF and RG used in this experiment, the last three terms are small compared to the rst three. Therefore, e20 is independent of RF and RG. Moreover, the third term is independent of the type of op amp used. The criteria for selecting the best op amp is then reduced to minimizing: e2n + i 2 nR 2 S: (5.23) For comparison, Table 5.4 summarizes the resulting e20 for four dierent op amps at 10 Hz. The rst is the Linear Technology's low-noise, precision, JFET input op amp LT1792, the second is Analog Devices' precision, low-power BiFET op amp AD548, the third is Analog Devices' low-noise, precision op amp OP27 and the fourth is National Semiconductor's Quad LM741 op amp. From the data in the table it is clear that there is an optimal balance between the input voltage noise and input current noise. The LT1792 is the best op amp for this application. One way to further reduce the output noise is to cool the bias resistor to 1 K so that the eective 105 5.7. Calibration q e2n q i2n q e2n + i 2 nR 2 S q e20 q e20 nVp Hz   pAp Hz   nVp Hz   nVp Hz   nVp Hz  op amp Te = 28 K Te = 1 K LT1792 8.3 0.01 8.3 130 96 AD548 80 0.002 80 810 800 OP27 3.5 1.7 150 1600 1600 LM741 120 1.2 160 1600 1600 Table 5.4: Noise evaluation of several candidate op amps. The op amp input noise sources e2n and i 2 n were obtained from manufacturer datasheets. Except for i 2 n of the AD548 op amp, the values given for e2n and i 2 n are evaluated at a frequency of 10 Hz. The AD548 i2n value is the input current noise at 1 kHz and it is likely much larger at 10 Hz. Te is the eective noise temperature of the parallel combination of RBIAS and RHB and is given by Eq. 5.16. The quantity e20 is given by Eq. 5.22 and is the output noise of the non-inverting amplier circuit as shown in Fig. 5.18. temperature is reduced from 28 K to 1 K. However, the table shows that there is little gain in the overall performance. Furthermore, because the bias resistor is used to set the temperature, it is convenient to have it at room temperature where it can easily be interchanged. The OP27 op amps were chosen for the buer ampliers in Fig. 5.17 for their low input voltage noise which makes essentially no contribution to the overall noise signal. The nal stage of the signal processing circuit is another non-inverting amplier that can be analyzed in exactly the same way as the initial 10 amplier. The dominant noise signal at its input will be e20 from the rst stage amplier. The expected overall output noise will be approximately given by q e20 times the gain of the second amplier. 5.7 Calibration Calibration signals are processed exactly as described above. However, rather than detecting voltage oscillations VHB due to temperature changes caused by modulated microwave power, the heater on the sapphire is used to mimic the microwave power absorbed by a sample. Specically, a square wave signal (between 0 and some am- plitude VHtr) is applied across the chip heater. This signal has the same frequency (1 Hz) that is used to modulate the microwaves. Figure 5.20 shows the simple cir- 106 5.8. Signal Acquisition and Processing Figure 5.20: The power dissipated by RHtr due to a square wave voltage signal VHtr mimics the sample power absorption due to square-wave-modulated microwaves. cuit used for calibration. The chip heater has a temperature independent resistance RHtr  750  and the power dissipated by the heater is given by: PHtr = I 2 HtrRHtr =  VHtr RD +RL +RHtr 2 RHtr; (5.24) where RD is a power dropping resistor. By measuring the bolometer response at many applied powers, a calibration factor to convert bolometer response to absolute power can be determined. Figure 5.21 shows a typical calibration curve. This calibration must be repeated each time the sample temperature is changed because the sensitivity of the system depends on the temperature dependent thermal conductance G(T ) of the thermal weak link. 5.8 Signal Acquisition and Processing Lock-in detection is used to acquire signals synchronously with the microwave mod- ulation frequency. As discussed in the next section, an upper limit of 10 Hz is imposed on the usable modulation frequency. This limit is determined by the ther- mal properties of the materials used to construct the sample stages. Because such low modulation frequencies are used, it was practical to develop a relatively inex- pensive software lock-in detection system. At the heart of the system is a National Instruments (NI) multifunction Data Acquisition (DAQ) module controlled over a 107 5.8. Signal Acquisition and Processing 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8   Po w er  (n W ) Bolometer Response (V) Figure 5.21: A typical calibration curve for the microwave spectrometer thermal stage. universal serial bus (USB) connection [102]. The NI USB-6216 has 16 analog inputs (16 bit resolution) which can be sampled at a rate of 400 kS/s, 2 analog outputs (16 bit resolution) which can updated at a rate of 250 kS/s, 32 digital inputs/outputs and 2 counters/timers (32 bit resolution) with a maximum frequency of 80 MHz. The NI USB-6216 device is controlled with a NI LabVIEW [102] program which acts as a signal generator, a multichannel oscilloscope, and two lock-in detectors. One counter is used to generate a low frequency square-wave signal. This square-wave is used to modulate the microwave power and also serves as the lock-in reference signal. Three analog inputs are used to simultaneously read in the modulation signal and the sample and reference side bolometer signals after conditioning by the electronics of Fig. 5.17. The two analog outputs are used to generate the reference and sample calibration signals VHtr (see Fig. 5.20). Figure 5.22(a) shows the detected bolometer signal and the modulation signal after it has been scaled to be between 1. As will be seen in the next section, the thermal weak-link (quartz tube/CuGa alloy) between the bolometer and low-temperature 108 5.8. Signal Acquisition and Processing reservoir acts essentially as a low-pass lter causing the bolometer response to become rounded. The low-pass and high-pass lters of the conditioning electronics further modify the signal shape. Figure 5.23 shows the eect of low- and high-pass ltering (a) -1.0 -0.5 0.0 0.5 1.0 Si gn al  (V ) Time (s) N orm alized M odulation 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3  (b) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.0 -0.5 0.0 0.5 1.0 Fi lte re d Si gn al  (V ) N orm alized M odulation   Time (s) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 (c) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.0 -0.5 0.0 0.5 1.0 In  P ha se  (V ) N orm alized M odulation   Time (s) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 (d) -1.0 -0.5 0.0 0.5 1.0 O ut  o f P ha se  (V ) Shifted M odulation Time (s) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3  Figure 5.22: (a) Raw sample signal (blue points) detected by the NI USB-6216. The grey line is the microwave modulation signal normalized to be between 1. (b) The signal after ltering to isolate the fundamental frequency. A phase shift relative to the modulation signal is apparent. (c) The in-phase component vx(t) obtained by multiplying the ltered signal by the normalized modulation signal. (d) The out-of- phase component vy(t) obtained by multiplying the ltered signal by the normalized modulation signal shifted by =2 (grey line). a square-wave. Figure 5.24 is a schematic diagram of the complete detection system. There is no direct electrical connection between the computer and the experimental apparatus. A Hewlett-Packard HP83630A 10 MHz to 26.5 GHz microwave source is used to deliver microwave power to the apparatus. The microwave source is controlled by the LabVIEW data acquisition program and is electrically isolated from the computer 109 5.8. Signal Acquisition and Processing (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.0 -0.5 0.0 0.5 1.0   Sq ua re  W av e Time (s) (b) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.0 -0.5 0.0 0.5 1.0   Sq ua re  W av e Time (s) (c) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.0 -0.5 0.0 0.5 1.0   Sq ua re  W av e Time (s) (d) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.0 -0.5 0.0 0.5 1.0   Sq ua re  W av e Time (s) Figure 5.23: (a) A 1 Hz square-wave. (b) Low-pass (LP = 3 Hz) ltering rounds the corners on the rising and falling slopes. (c) A high-pass lter, with corner frequency below the square-wave fundamental frequency (HP = 0:05 Hz), causes the 
at part of the square-wave to slope. (d) Square-wave after low- and high-pass ltering. The shape is very similar to that of Fig. 5.22(a). using a USB optical link. The modulation signal from the NI USB-6216 DAQ counter is optically isolated from the microwave source using a UBC-built optical link. The analog inputs and outputs are isolated using AFL-300 commercial bre-optical links from A. A. Lab Systems [103]. The electronics of Fig. 5.17 are contained within the \Bias Box Amplier". This box contains two copies of the same circuit, one to condition the sample side bolometer signal and the other for the reference side. 110 5.8. Signal Acquisition and Processing Figure 5.24: Schematic diagram of the detection system. The data acquisition com- puter is electrically isolated from the experiment using bre optic links. 5.8.1 Lock-in Detection This section describes the software lock-in detection. Because the detected signal (Fig. 5.22(a)) originates from a square-wave modulation of the microwave power, its frequency spectrum contains all the odd harmonics of the fundamental frequency. By applying a discrete fourier transform (DFT), one obtains the frequency components 111 5.9. Power Ratio and RS(!) of the signal. The acquired signals contain an integer number of periods and applying a rectangular window function does not introduce frequency leakage into the DFT results [104, 105]. Digital ltering is achieved by multiplying the transformed data by a narrow Gaussian centred on the fundamental frequency: H(f) = e(f)=2 2 ; (5.25) where  is the fundamental frequency and  is the width of the Gaussian lter. Then applying an inverse discrete fourier transform (IDFT), one selects only the fundamental frequency component of the original signal, see Fig. 5.22(b) resulting in a single sine wave. From the gure, the phase dierence between the ltered signal and the modulation signal is apparent. The in-phase part vx(t) of the sine wave is obtained by multiplying the ltered signal by the modulation signal scaled to be between -1 and +1. The result is shown in gure 5.22(c). The rst harmonic of a square-wave with a peak-to-peak voltage swing A is v(t) = 2A= sin(2f0t). Averaging jv(t)j over an integer number of periods T gives: jv(t)j = 4A 2 ; (5.26) such that A = 2jv(t)j=4. In the same way, the in-phase amplitude is obtained using X = 2jvx(t)j=4. As shown in Fig. 5.22(d), the out-of-phase part vy(t) is found by multiplying the ltered signal by the modulation signal after it has been shifted in phase by =2 rad and the out-of-phase amplitude is given by Y = 2jvy(t)j=4. For the plots of Fig. 5.22, X = 0:325 V and Y = 0:233 V such that R = p X2 + Y 2 = 0:400 V and  = tan1(Y=X) = 35:6. In this way, the bolometer response R can be obtained for any given incident power (microwave power absorbed by the sample or power dissipated by the heater). In practice, the phase is selected such that X is a maximum and Y  0. When detecting small signals buried in noise, it is advantageous to detect X since it can average to zero, whereas R is restricted to be positive quantity. 5.9 Power Ratio and RS(!) This section evaluates the performance of the microwave spectrometer and, in par- ticular, the ratio technique. Figure 5.25 shows typical measured power spectrums for both the reference and the test sample. The measured power absorption of both samples varies wildly as a function of the microwave frequency. This structure is 112 5.9. Power Ratio and RS(!) 0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8   R S  ra tio Frequency (GHz) 0 1 2 3 4   R ef er en ce  P ow er  (n W ) 0 2 4 6 8 10 0 5 10 15 20 25    Sa m pl e Po w er  (n W ) Figure 5.25: Top: The absolute power absorption spectrum of the test sample from 0.2 to 26 GHz. The test sample is a pure tin platelet coated with 1 m of a AgAu alloy. Middle: The measured power spectrum of the AgAu alloy reference sample. For this measurement, both samples were at a temperature of 1.2 K. Both power spectra show a complicated structure which is due to standing waves in the circuit delivering microwave power to the spectrometer. Bottom: A smooth curve is obtained when the RS(!) ratio of the two samples is taken. 113 5.9. Power Ratio and RS(!) due to standing waves that exist in the circuit used to deliver microwaves to the spectrometer. However, below the cuto frequencies of the lowest TE modes in the shorted transmission line, the magnetic elds at the two sample sites are identical and the ratio of the absorbed powers (RS / power) results in a smooth spectrum as shown in the bottom panel of the gure. Next, Fig. 5.26 shows the RS(!) ratio when AgAu alloy samples are installed on both thermal stages. As expected, the ratio is smooth and near one over a broad range of frequencies. The bottom panel of the gure shows that there is some curvature to the ratio. This curvature is reproducible, although its origin is not understood. 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 TE 01,10  cut-off 24.4 GHz TE 01,10  1/2-wave 33 GHz R S  ra tio   R S  ra tio TE 01,10  1/4-wave 27 GHz 0 10 20 30 40 0.90 0.95 1.00   Frequency (GHz) Figure 5.26: Surface resistance ratio with AgAu alloy platelets installed on both the reference and sample thermal stages. The ratio is close to one over a wide range of microwave frequencies. The top and bottom frames show the same data on two dif- ferent scales. The arrows mark the locations of the TE01 and TE10 cuto frequencies and resonances, the values of which are given in Table 5.6. 114 5.9. Power Ratio and RS(!) Before considering the high frequency range of the data where the ratio technique breaks down, the very low frequency limit is investigated in detail. A zoomed-in view that emphasizes the low-frequency range of the AgAu ratio data is given in Fig. 5.27. Below 0.1 GHz the ratio deviates from 1.0 showing rst a dip  !  !"  !#  !$  !% "! "!" "!#   R e ff S (ω ) R at io f = ω/2pi (GHz) Figure 5.27: Measured surface resistance ratio of two AgAu alloy samples in the low- frequency thin-limit regime. The dashed red line is a t to the data using Eq. 3.16 for each sample. The resistivity was xed to be  = 5:28  cm obtained from dc resistivity measurements and the sample thicknesses t1 and t2 were t parameters. before a rapid increase. This structure is due to the AgAu alloy samples entering into the thin limit regime as described in x3.1.1. Figure 5.28 shows the two AgAu samples mounted on the sapphire plates. Since the thickness t2 of the reference platelet is less than t1, the thickness of the sample-side platelet, it is the rst to enter the thin-limit. The ratio data in Fig. 5.27 is modeled by using Eq. 3.16 for the thin-limit ReS (!) for both samples. The data are t using  = 5:28  cm from dc resistivity measurements and allowing the sample thicknesses t1 and t2 to be t parameters. The results of the t are summarized in Table 5.5. The results for t1 and t2 obtained from the ts are in good agreement with the expected values. Next, the high frequency end of the ratio data in Fig. 5.26 is considered. As stated previously, the ratio technique works provided that only the symmetric transverse electromagnetic (TEM) mode propagates in the rectangular coaxial line. Figure 5.29 115 5.9. Power Ratio and RS(!) Figure 5.28: Left: The sample on the reference side is 65 m thick. Right: the sample-side AgAu platelet is  78 m thick. In addition to the samples, some of the silicone grease used to mount the platelets to the sapphire is visible. t1 (m) t2 (m) measured 78 3 65 2 t 75:5 0:4 66:7 0:3 Table 5.5: Comparison of the AgAu platelet thicknesses determined from optical measurements and from ts to the RS(!) ratio data. shows a cross-section of the rectangular coax with the sample positions and the electric and magnetic eld congurations for the TEM modes and the two lowest-frequency TE modes. The magnetic eld of the TEM mode loops symmetrically around the inner conductor in the plane of the page and its magnitude is identical at each of the sample sites. The TE modes propagate only above their respective cuto frequencies and the magnetic elds form closed loops that are not restricted to the plane of the page. Notice that the elds from either of the TE modes add antisymmetrically to the TEM mode elds so that the total magnetic elds at the two sample sites are no longer equivalent causing a breakdown of the ratio technique. The TE01 mode is particularly damaging because the magnetic eld strength at the sample sites is large, whereas the magnetic eld from the TE10 mode is largely screened from the samples by the broad inner conductor. To estimate the cuto frequencies of the lowest-frequency TE modes the work of Pyle [106] is followed. In this work a method for calculating the cuto frequency for the TE10 mode in a ridged waveguide of any aspect ratio to within a few percent 116 5.9. Power Ratio and RS(!) TE10 TE01 A B C D (i) TEM(ii) (iii) (iv) Lines of electric flux Lines of magnetic flux Figure 5.29: (i) Schematic of the cross-section of a rectangular coaxial line. Samples are positioned symmetrically on the two sides of the broad inner conductor. (ii) Field conguration for the TEM mode. The magnetic eld loops around the inner conductor in the plane of the page. The magnitude of the magnetic eld is identical at the two sample sites. (iii) and (iv) Field congurations for the TE10 and TE01 modes respectively. For these modes the elds are not restricted to the plane of the page. Notice that when either of the TE10 or TE01 modes are added to the TEM mode the eld strengths at the two sample sites dier causing a breakdown of the ratio technique. A, B, C, and D denote the dimensions of the line segment to which they are closest. is described. Figure 5.30 shows how the TE01 and TE10 modes of a rectangular transmission line can be mapped onto the TE10 mode of a ridged waveguide by strategically inserting conducting walls that do not alter the eld conguration in the rectangular transmission line. Specically, conducting walls are inserted in regions where the magnetic eld is weak and such that electric eld lines are perpendicular to the walls. Applying Pyle's method for calculating cuto frequencies of a rectangular coaxial line is valid provided the septum is not too thin (i.e. provided D 6 B D). The results of this method, when the dimensions given in Fig. 5.2 are used, are 117 5.9. Power Ratio and RS(!) d s a b TE10 TE01 d sa b Figure 5.30: Top: Placing an imaginary horizontal conducting wall halfway up the rectangular coaxial line maps the electromagnetic elds of the TE10 mode onto elds of the TE10 mode of a ridged waveguide. Bottom: Placing an imaginary vertical conducting wall centred on the rectangular coaxial line maps the electromagnetic elds of the TE01 mode onto elds of the TE10 mode of a ridged waveguide. summarized in Table 5.6. The dimensions were chosen such that the spectrometer could easily accommodate 1  1 mm2 platelet samples while maximizing the cuto frequencies of the TE01 and TE10 modes. Even above the cuto frequencies, the TE modes may not be immediately gener- ated if the rectangular coaxial transmission line is made very symmetric. The mode must be excited by either asymmetries in the construction of the coaxial line or by Mode Cuto (GHz) 1=4-wave (GHz) 1=2-wave (GHz) TE01 24.47 26.96 33.33 TE10 24.37 26.89 33.25 Table 5.6: Cuto frequencies and 1=4- and 1=2-wave resonances of the TE01 and TE10 modes. 118 5.9. Power Ratio and RS(!) the discontinuity encountered when transitioning from the cylindrical to the rectan- gular coaxial line. There are mode resonances that can occur when re
ections at the cylindrical-to-rectangular coaxial line transition lead to constructive interference conditions within the microwave spectrometer. For guided waves the !(k) dispersion relation can be written in terms of the mode cuto frequencies !c [71]: !2 = !2c + c 2k2; (5.27) where c is the speed of light in vacuum and k = 2=. The rst possible resonance occurs when the length ` of the rectangular coaxial line is equal to =4 and leads to k = =2`. This 1=4-wave resonance corresponds to an open circuit termination31 of the rectangular coaxial line. At the opposite extreme, a short circuit termination would give rise to a 1=2-wave resonance in which case ` = =2 and k = =`. The 1=4- and 1=2-wave resonances of the TE01 and TE10 modes are then calculated using: !21=4 = ! 2 c + c 2  2 4L2 ; (5.28a) !21=2 = ! 2 c + c 2  2 L2 : (5.28b) The length of the rectangular coaxial line is ` = 6:63 mm and the resonance frequen- cies of the lowest TE modes are given in Table 5.6. The location of the cuto and resonance frequencies are marked in Fig. 5.26. The ratio technique is rst seen to breakdown at a frequency that lies between the 1=4- and 1=2-wave resonances. Getting a resonance that does not occur precisely at either the 1=4- or 1=2-wave conditions is not surprising: while one boundary is a well dened short, the other discontinuity is not, and the electromagnetic eld patterns in the region of this discontinuity will not correspond exactly to the assumed modes. This section is concluded by showing the power absorption spectrum measured in the absence of a sample. That is, the power absorption of the sapphire plate with a small dab of silicone grease used to mount the samples. This measurement was performed with the sapphire plate at 1.2 K and the output power of the microwave source set to +20 dBm (100 mW)32. The data are shown in Fig. 5.31. The sapphire 31An open circuit termination corresponds to a low capacitance joint at the transition from cylin- drical to rectangular coaxial line, whereas a high capacitance joint would favour short circuit termi- nation. 32Note that dBm is a logarithmic scale of power that is referenced to 1 mW (0 dBm = 1 mW). Power measured in milliWatts can be converted to dBm using: P (dBm) = 10 log10  P (mW) 1 mW  . 119 5.9. Power Ratio and RS(!) 0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 Sapphire Pow er A bsorption (nW )  Sa m pl e B ol om et er  S ig na l ( vo lts ) Frquency (GHz) Figure 5.31: Un-normalized background power absorption of the sapphire plate with an incident microwave power of 100 mW. The left vertical axis shows the detected bolometer voltage after amplication. The total gain of the signal conditioning circuit was set to 105. The vertical axis on the right shows the absolute measured power absorption. The red data is the average of the black data. used in the microwave spectrometer is HEMEX grade sapphire from Crystal Sys- tems33. This is the highest quality commercially available sapphire. It was chosen to limit background power absorption of the sample holder. It is clear from Fig. 5.31 that there is a nonzero background signal in the absence of a sample. This signal is reproducible and independent of the amount of grease applied to the sapphire. Moreover, this signal is independent of the presence of the AgAu alloy sample on the reference-side sapphire plate (i.e. the background signal is not a result of thermal crosstalk between the sample and reference thermal stages). The background signal can only be due to power absorption by the sapphire plate. It should be noted that the spectrum in Fig. 5.31 is not normalized and the peak in the spectrum at 10 GHz is not due to enhanced absorption by the sapphire, but because more microwave power is transmitted to the spectrometer at this frequency (compare to Fig. 5.25). The data of Fig. 5.31 demonstrate that the sensitivity of the microwave spec- trometer is currently limited by background absorption of the sapphire plate and not noise in the signal conditioning electronics (Fig. 5.17). Since it is strongly frequency dependent, the un-normalized background power absorption itself is not particulary 33Crystal Systems, 27 Congress Street, Salem, MA 01970 USA. 120 5.10. Frequency Response meaningful or revealing. It is much more important to know to what extent does the background power absorption aect the surface resistance measurements. Fig- ure 5.32 shows the background power data converted to an eective surface resistance RS(!) corresponding to a sample with a planar area of 1 mm 2. The eective sur- 0 5 10 15 20 25 0 2 4 6 8 10   R S  ( ) Frequency (GHz) Figure 5.32: Eective background surface resistance due to microwave power ab- sorption by the sapphire plate at 1.2 K. The blue data is the average of the black data. face resistance steadily increases with microwave frequency reaching 8   at 25 GHz. This background eective surface resistance has been measured to be temperature independent from 1 to 10 K. By 10 K, the sensitivity of the thermal stage is greatly reduced and the background absorption signal cannot be extracted from the noise. 5.10 Frequency Response In order to avoid 1=f noise, one wishes to modulate the microwaves at as high a frequency as possible. The maximum modulation frequency is, however, limited by the dimensions and the thermal properties of the materials used to make the hot nger. Figure 5.33(a) is a cartoon sketch of the hot nger (corresponds to Fig. 4.8). A thermal weak link with thermal conductivity B(T ) and specic heat cB(T ) connects 121 5.10. Frequency Response Figure 5.33: (a) Cartoon sketch of the hot nger. The thermal weak link has thermal conductivity B and specic heat capacity cB and connects the sample stage at x = 0 and temperature T to the base at x = ` and temperature T0. The lumped heat capacity of the sample stage (sapphire plate, glue, bolometers, heater, sample, . . . ) onto which power _Q is incident is represented by CA. (b) This problem can be modeled numerically by representing the distributed thermal conductance and heat capacity of the weak link as a series of cascaded RC lters and the lumped heat capacity of the sample stage as a large single capacitor CA at the end of the chain. the sapphire sample stage at temperature T with heat capacity CA(T ) to a base temperature T0. The goal is to calculate the temperature response of the sample stage to a sinusoidal input power _Q(!; t) = _eQ1ei!t. First, it is assumed that the ac part of the temperature response takes the form T (t; x) = eT1(x)ei!t and the one-dimensional heat equation is applied: @T (t; x) @t = @2T (t; x) @x2 ; (5.29) where   =cV is the thermal diusivity which, for simplicity, is assumed to be temperature independent. One is then left with the following second order dierential 122 5.10. Frequency Response equation: i! eT1(x) = Bd2 eT1(x) dx2 : (5.30) Subject to the boundary conditions: eT1(0) = eT! and eT1(`) = 0; (5.31) the solution of Eq. 5.30 is given by: eT1(x) = eT! sinh 1 + i B x  coth  1 + i B x   coth  1 + i B `  ; (5.32) where a thermal penetration depth  p2=! has been dened. If the heat capacity CA of the sample stage is negligible, then conservation of energy at x = 0 requires _Q = BAB[@T (x; t)=@x]jx=0, where A is the cross-sectional area of the thermal weak link. Evaluating the derivative and canceling the time dependence one nds: eT! = _eQ1 BAB  1 i 2  B tanh  1 + i B `  = _eQ1 GB `eff ` ; (5.33) where the substitution GB  (AB=`)B has been made and a complex eective length `e has been dened. The quantity of interest is the magnitude of the temperature oscillations:  eT! = _eQ1 GB j`eff j ` ; (5.34) where: j`e j = Bp 2 0@cosh h 2 ` B i  cos h 2 ` B i cosh h 2 ` B i + cos h 2 ` B i 1A 1 2 : (5.35) Expanding the trigonometric functions of j`eff j and keeping the rst three nonzero terms leads to a frequency response of a single pole low-pass lter: j`eff j ` = s 1 1 + (!=!C) 2 ; (5.36) with a corner frequency !C = p 6B=` 2. 123 5.10. Frequency Response When CA cannot be ignored, conservation of energy at x = 0 requires that: _eQ1 = CA@T (x; t)@t  BAB@T (x; t)@x  x=0 ; (5.37) and Eq. 5.33 is modied such that: eT! = _eQ1 i!CA + BAB  1+i B  coth h 1+i B ` i : (5.38) In this case, the amplitude of the temperature oscillation is:  eT! = _eQ1GB j`e j`r 1 ! !0 `0e ` +  ! !0 2 j`e j2 `2 ; (5.39) where a frequency !0 and a second eective length ` 0 e have been dened as: !0 = GB CA ; (5.40a) `0e = B 0@ sin h 2 ` B i  sinh h 2 ` B i cos h 2 ` B i + cosh h 2 ` B i 1A : (5.40b) The frequency dependence of  eT! is shown in Fig. 5.34 for CA both non-negligible and negligible. In both cases, the high frequency response falls o slower than that of the single pole lter response. An increase of the sample stage heat capacity CA lowers the corner frequency and increases the slope of the high frequency roll o. The measured normalized frequency response  eT! of the quartz tube sample stage is shown in Fig. 5.35. Data was acquired in two ways. In the rst method, a AgAu sample is exposed to modulated microwaves and the bolometer signal is measured as a function of the modulation frequency. In the second method, an ac square wave voltage is applied to the 750  chip heater adjacent to the bolometer. Again, the bolometer response is measured as a function of the ac frequency. The measured frequency response of the sample stage using microwaves and heater power is very similar. Above 30 Hz the microwave response falls below the heater response and is likely an indication that, at these frequencies, the sample is losing thermal contact with the sapphire plate. Recall that the sample is xed to the sapphire plate using silicone grease. This contact will have its own frequency response and characteristic 124 5.10. Frequency Response 0.1 1 10 0.5 1  Intermediate Response  Full Response  Single Pole Response   N or m al iz ed  R es po ns e Frequency (Hz) Figure 5.34: Plot of the calculated normalized sample stage ac temperature response eT! ( _eQ1=GB)1 to an applied ac heat 
ux. These curves were generated using ` = 10 mm and  = 1 103 m2/s. The full response given by Eq. 5.39 is shown in blue for !0 = 220 s 1. The response given by Eq. 5.34 for negligible CA (!0 !1), called the intermediate response is shown in red. The single pole response of Eq. 5.36 is shown in black. corner frequency. Figure 5.36 shows the measured frequency response of the sample stage using microwaves t to the full (Eq. 5.39), intermediate (Eq. 5.34), and single pole (Eq. 5.36) models discussed above. The best t to the data is clearly the full model which includes the lumped specic heat CA in Fig. 5.33. For ` = 13 mm the t to the full response model gives B = 2:6 103 m2/s and f0 = !0=2 = 19 Hz. Figure 5.37 shows the frequency response of the sample stage at several temper- atures below 10 K t to Eq. 5.39. The corner frequency of the response drops with increasing temperature re
ecting the temperature dependence of both B and !0. The t to the data becomes progressively worse at higher temperatures and could, in part, be due to the temperature dependence of B. Recall that Eq. 5.39 was derived assuming that B was independent of temperature. At the low temperatures, the gradient across the thermal weak link T = T  T0 is small and hence the thermal diusivity does not change much along the length of the weak link. However, at higher 125 5.10. Frequency Response 0.1 1 10 0.2 0.4 0.6 0.8 1  -wave Power  Heater Power   N or m al iz ed  R es po ns e Frequency (Hz) Figure 5.35: Measured frequency response of the sample stage at 1.2 K. The blue points were obtained by exposing a AgAu sample to modulated microwaves. The grey points were obtained by applying ac power to a 750  heater adjacent to the bolometer. temperatures T becomes large and there will be a signicant change in B along the length of the quartz tube. In the next section, a numerical method for solving the frequency response of the thermal stage is discussed. Using this method one can take into account the change in B along the weak link due to a dc temperature gra- dient across its length. Table 5.7 summarizes the parameters obtained from the ts of Eq. 5.39 to the sample stage frequency response at all temperatures. The quartz thermal diusivity can be compared to the low-temperature thermal conductivity and specic heat of vitrious quartz measured by Stephens [107]:  = 0:024T 1:8 W m K ; (5.41a) cV = 2:4 J m3K2 T + 4:4 J m3K4 T 3: (5.41b) Figure 5.38 compares B from Table 5.7 to that calculated using the measurements of Stephens. There is overall good qualitative agreement between the two measure- ments. At high temperatures, the deviation of the points from the line may be due to the fact that the temperature dependence of the quartz thermal conductivity was 126 5.10. Frequency Response 0.1 1 10 1  Measured Response  Full response fit  Intermediate response fit  Single pole fit   N or m al iz ed  R es po ns e Frequency (Hz) Figure 5.36: Measured microwave frequency response of the sample stage t to the full (Eq. 5.39), intermediate (Eq. 5.34), and single pole (Eq. 5.36) models. In each case ` = 13 mm was xed. Temperature (K) B (103 m2/s) f0 = !0=2 (Hz) 1.2 2:62 0:06 18:8 1:0 1.8 2:45 0:06 24 2 2.6 1:29 0:06 14 2 3.9 0:71 0:02 5:4 0:3 5.9 0:209 0:007 1:51 0:12 7.0 0:118 0:004 0:88 0:08 Numerical result 7.0 0.40 0.88 Table 5.7: Parameters obtained from ts of the sample stage frequency response to Eq. 5.39. For all ts ` = 13 mm was xed. Also given is the result of the numerical calculation which includes the temperature dependence of B as discussed in x5.10.1. not taken into account in the derivation of Eq. 5.39. Notice, also, that the quartz thermal diusivity drops rapidly below 0.5 K. This issue will be addressed again in chapter 9 where preliminary sub-kelvin spectroscopic measurements made using a 127 5.10. Frequency Response 0.1 1 10 0.1 1  1.2 K  1.8 K  2.6 K  3.9 K  5.9 K  7.0 K   N or m al iz ed  R es po ns e Frequency (Hz) Figure 5.37: Measured microwave frequency response of the sample stage t to the full response model for several temperatures below 10 K. dilution refrigerator are presented. To conclude this section, the frequency response of the reference stage is shown at 1.2 K and 11 K. The reference stage thermal weak link is made of an alloy for which both the thermal conductivity and the specic heat are expected to evolve linearly with temperature. The thermal diusivity should therefore be temperature inde- pendent resulting in an approximately temperature independent frequency response. This is conrmed by the data shown in Fig. 5.39. 128 5.10. Frequency Response 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0  Fits to data  Numerical  Stephans   B  (1 0- 3  m 2 /s ) Temperature (K) Figure 5.38: The quartz tube thermal diusivity B as a function temperature. The points represent the data obtained from Table 5.7. The line represents the calculated B(T ) using the expressions derived from the measurements of Stephens [107]. The single star-shaped point was determined from the numerical analysis of x5.10.1 and is much closer to the expected value of B at 7.0 K. 129 5.10. Frequency Response 0.1 1 10 0.4 0.6 0.8 1  1.2 K  11 K   N or m al iz ed  R es po ns e Frequency (Hz) Figure 5.39: The frequency response of the alloy reference stage at 1.2 and 11 K. The frequency response is seen to be nearly temperature independent, as expected for a metallic alloy where both (T ) and cV (T ) have the same (linear) temperature dependence. The dierence between the two curves is likely due to the temperature dependence of heat capacity of the AgAu reference sample and that of the sapphire plate. 130 5.10. Frequency Response 5.10.1 Numerical Analysis The calculated frequency response assumed a temperature independent thermal dif- fusivity of the thermal weak-link. This approximation is not valid for the case of a quartz tube weak-link and a sample temperature that is signicantly greater than the experimental base temperature. The numerical model alluded to in Fig. 5.33b allows one to include the temperature dependence of the thermal diusivity of the quartz tube in a relatively straightforward way. It will be shown that including this tem- perature dependence allows the measured high-temperature frequency response to be modeled using much more sensible values for the quartz tube thermal conductivity and specic heat than would otherwise be possible. Furthermore, the development of this \equivalent circuit" model is generally useful when analyzing heat transfer in complicated thermal circuits. One can use the model to predict the transient behav- ior of a system as it evolves from some initial state to a state of thermal equilibrium. Also, as is done in this section, one can examine how various elements in the system respond to an oscillating power input at some specied location within the system. Specically, when designing the dilution fridge setup presented in chapter 9, this model was used to reliably predict the sample site base temperatures and the time required to cool the sample stages to those temperatures. In addition, by assuming an oscillating mixing chamber temperature, a set of low-pass thermal lters was designed to suppress the temperature oscillation at the sample sites to within a predetermined tolerance [97]. First, it is conrmed that the model successfully reproduces the line shape gen- erated by Eq. 5.39. For a sample temperature very close to the experimental base temperature, there is very little change in B along the length of the quartz tube. In this case, each of the thermal resistors and capacitors CB in Fig. 5.33b are identical and the results of the numerical model should match the calculated response. Specif- ically, modeling the quartz tube as n + 1 cascaded RC lters, each resistor R and each capacitor CB is given by: R = `=(n+ 1) A ; (5.42a) CB =  ` n  AcV; (5.42b) where  is the thermal conductivity of quartz, A is the cross-sectional area of the quartz tube, ` is the quartz tube length, and cV is the specic heat of quartz. The nal capacitor CA in the chain is included to take into account the lumped heat 131 5.10. Frequency Response capacity of the sample stage (sample, sapphire plate, heater, bolometer, epoxy, . . . ). The powers into each resistor and each capacitor are: P = T (t) R(T (t)) ; (5.43a) P = d dt [T (t)CB(T (t))] : (5.43b) To model the low-temperature T = 1:2 K response, the measurements of Stephens (Eq. 5.41(a)) are used to set  = 3:33  102 W/mK. Then, B from Table 5.7 is used to set cV = 12:7 J=m 3K. The quartz tube length is 13.0 mm with inner and outer diameters of 0.635 and 1.2 mm, respectively giving a cross-sectional area 0:814  106 m2. The frequency response is modeled by introducing an oscillating power Pin = _Q(t) / cos(!t) at the sample end and then numerically evaluating for the amplitude of temperature oscillation at each branch of the RC ladder. For n = 10, one must evaluate the following set of coupled dierential equations: P0;CA = d dt [TS(t)CA(TS(t))] (5.44a) P0;R = TS(t) T1(t) R [1=2 (TS(t) + T1(t))] (5.44b) P1;C = d dt [T1(t)CB(T1(t))] (5.44c) P1;R = T1(t) T2(t) R [1=2 (T1(t) + T2(t))] (5.44d) P2;C = d dt [T2(t)CB(T2(t))] (5.44e) P2;R = T2(t) T3(t) R [1=2 (T2(t) + T3(t))] (5.44f) : : : P9;C = d dt [T9(t)CB(T9(t))] (5.44g) P9;R = T10(t) T9(t) R [1=2 (T10(t) + T9(t))] (5.44h) P10;C = d dt [T10(t)CB(T10(t))] (5.44i) P10;R = Tbase  T10(t) R [1=2 (Tbase + T10(t))] (5.44j) 132 5.10. Frequency Response subject to the constraints: Pin = P0;CA + P0;R (5.45a) P0;R = P1;C + P1;R (5.45b) P1;R = P2;C + P2;R (5.45c) : : : P9;R = P10;C + P10;R (5.45d) Equations 5.45 can be solved for the eleven unknowns TS(t), T1(t), : : : , T10(t) which are the temperatures at eleven equally spaced points along the quartz tube. The solution will include both the transient and ac response. The results of this analysis are shown in Fig. 5.40. As shown by the blue line, there is good agreement between the numerical calculation and Eq. 5.39. Clearly, one expects the agreement to improve as n is increased. The next goal will be to apply the numerical model to the case where there is a large temperature gradient along the length of the quartz tube. Specically, the case 0.01 0.1 1 10 0.01 0.1 1 Analytical Expression  T S Numerical  T S  T 1  T 2  T 3  T 4  T 5  T 6  T 7  T 8  T 9  T 10   N or m al iz ed  R es po ns e Frequency (Hz) Figure 5.40: The numerically calculated frequency response of the quartz sample stage. The dierent colours represent the calculated response at eleven equally spaced points along the quartz tube. The frequency response at the sample site TS is in blue. The calculated response using Eq. 5.39 is shown as the blue line. 133 5.10. Frequency Response with the sample site at 7 K and the opposite end of the quartz tube at Tbase = 1:2 K will be considered. First, the response to a dc power applied at the sample site will be analyzed. Starting with the entire quartz tube uniformly at 1.2 K (that is, TS(0) = T1(0) =    = T10(0) = 1:2 K), a constant dc power Pin is applied at TS. This input power represents the bias current in the bolometer that is used to set the sample temperature. For small temperature dierences, the thermal conductance can be approximated as: G(T )  1jT2  T1j A ` Z T2 T1 (T )dT  A `   T  (5.46) where T = (T2 + T1)=2 is the average temperature. The temperature gradient across each resistor R in Fig. 5.33 is then given by: T = P G(T )  ` A P (T ) : (5.47) Using Eq. 5.41(a), the calculated T along the quartz is shown by the grey line in Figure 5.41(b) and has a slope of 1.8. Note that the heat capacity aects only the transient response to the dc power and not the nal equilibrium temperatures. Figure 5.41(a) shows the numerically calculated quartz tube transient response to a dc power at eleven evenly spaced points along the tube. Within 2-3 seconds a new equilibrium temperature prole is established along the length of the quartz tube. The numerically calculated T across each resistor (see Fig. 5.33(b)) agrees with the expected T given by Eq. 5.47. The calculated equilibrium temperatures were used to set the temperatures of the n+1 resistors and capacitors. As shown in Eqns 5.44, the resistor temperature was set to be the average of the temperatures at the two ends of the resistor. Equations 5.44 and 5.45 were then used to numerically model the quartz tube frequency response assuming a sample temperature of 7 K. The results are shown in Fig. 5.42. Although the numerically calculated line is not a better t to the data, it was generated using a value of B(T ) that is much closer to the expected value for quartz. The numerically determined B at 7 K (red star) is compared to the expected value in Fig. 5.38. 134 5.10. Frequency Response 0 1 2 3 4 0 1 2 3 4 5 6 7 (b)  TS  T1  T2  T3  T4  T5  T6  T7  T8  T9  T10   Te m pe ra tu re  (K ) Time (s) (a) 2 3 4 5 6 7 0.1 1  Numerical  expected   T= T n +1 - T n (K ) (T n+1 +T n )/2  (K) Figure 5.41: (a) Numerically calculated response of the quartz tube to a dc power Pin applied at the sample site. Initially, the quartz tube is at a uniform temperature of 1.2 K. Within a few seconds of applying the dc power a new equilibrium temperature distribution is reached. (b) The expected temperature prole along the length of the quartz tube is shown as the grey line given by Eq. 5.47. The blue points are the numerically calculated equilibrium temperatures at eleven equally spaced points along the length of the quartz tube. 135 5.10. Frequency Response 0.1 1 10 0.1 1  7.0 K Data  Full Response Fit  Numerical Analysis   N or m al iz ed  R es po ns e Frequency (Hz) Figure 5.42: The grey points are the measured frequency response of the quartz tube sample stage for a sample temperature of 7 K. The t to the data using Eq. 5.39 is shown as the grey line. These are the same 7 K data and t as shown in Fig. 5.37. The numerically calculated response, which includes the temperature dependence of B(T ) is shown by the open points connected by the black line. 136 Chapter 6 Normal State Microwave Spectroscopy This short chapter considers the microwave surface resistance spectra of a number of conductors in the normal state. Each of the materials considered will lead to a RS(!) line shape that is distinct from the others. The dierent phenomena leading to these line shapes will be brie
y explored. 6.1 Frequency-Independent Conductivity In x3.1.2 the surface resistance of a good normal conductor in the low frequency limit was found to be: RS(!) = r 0! 2 ; (6.1) where the conductivity  is a purely real number (see Eq. 3.12). If the scattering rate of the conduction electrons is suciently high,  will remain constant (i.e. have a frequency independent conductivity spectrum) over the bandwidth of the microwave spectrometer (0-26 GHz) that was discussed in the preceding chapter. Figure 6.1 shows the measured normal state surface resistance of a single crys- tal of Sr2RuO4, an unconventional superconductor with a maximum Tc of 1.5 K for the purest samples [34]. For the data shown in the gure, the sample temperature was 2.6 K which is well into the normal (i.e. non-superconducting) state. The data agree fairly well with a !1=2 frequency dependence, but a slight systematic devia- tion suggests a conductivity that is not precisely 
at over the entire measurement bandwidth. Section 6.3 will consider frequency-dependent conductivities in more detail. It is worth noting that the normal state conductivity of Sr2RuO4 is highly anisotropic [34] with the out-of-plane conductivity orders of magnitude smaller than the in-plane conductivity. Thus, although the measured crystal is a platelet sample whose thickness in the c-axis direction is much less than either of the planar dimen- sions, a t to the RS(!) data in Fig. 6.1 to extract  would result in an admixture of the planar conductivity ab and c-axis conductivity c. Measurements of samples 137 6.2. RS(!) in the Thin-Limit 0 5 10 15 20 25 0 5 10 15 20 25 30    2.6 K  1/2 R S  (m ) Frequency (GHz) Figure 6.1: The normal state RS(!) of Sr2RuO4. The data exhibit a frequency dependence that is !1=2 indicating that  (or  = 1=) is 
at over the measurement bandwidth. with several dierent aspect ratios (achieved, for example, by repeatedly cutting the sample) would be needed to separately determine ab and c. 6.2 RS(!) in the Thin-Limit When the frequency-dependent skin depth (!) of the sample becomes of order the sample thickness t, the simple model for the surface impedance breaks down and one must consider an eective surface impedance. The solution to this problem was given in x3.1.1 and for a good conductor, at not too high frequency, ReS (!) is given by Eq. 3.16: ReS (!) =    sinh t   sin t  cosh t  + cos t   ; (6.2) where the limiting t=  1 behaviour is given by: ReS (!)  t3 64 = 20! 2t3 24 : (6.3) Figure 6.2 shows a T = 30 K normal state surface resistance spectrum of the 138 6.3. Drude Conductivity 0 5 10 15 20 0.00 0.05 0.10 0.15 0.20 ~ 1/2   R S  ( ) Frequency (GHz) ~ 2 Figure 6.2: The thin-limit normal state surface resistance spectrum of Ba0:72K0:28Fe2As2. There is a crossover from the thin-limit ! 2 behaviour to the thick- limit !1=2 behaviour. iron-arsenide superconductor Ba0:72K0:28Fe2As2 with Tc = 28 K. The measured sam- ple is a single crystal platelet with an estimated thickness of about 10 m, obtained using digital images from a calibrated optical microscope. At high frequency  < t and the data are reasonably well represented by an !1=2 frequency dependence. How- ever, as the frequency is lowered,  increases until eventually  & t at which point the frequency dependence of RS(!) undergoes a dramatic change. At very low fre- quencies, the data exhibit the expected !2 frequency dependence. The full frequency range ts very well to Eq. 6.2 with two t parameters  = 28:6 0:2  cm and t = 7:79 0:05 m. In fact, the t to the data is a more reliable determination of the sample thickness than is a direct measurement of it. 6.3 Drude Conductivity In this section the constraint that  be real and independent of frequency is relaxed. In this case RS(!) must be derived starting from the general expression for the complex 139 6.3. Drude Conductivity surface impedance: ZS(!) = RS(!) + iXS(!) = s i0! (!) ; (6.4) where (!) is the frequency dependent complex conductivity (see Eq. 3.5). To proceed further, a model for the normal state conductivity must be adopted. Consider, for example, the Drude conductivity given by Eqns. 3.22 and 3.23: (!) = 1(!) i2(!); (6.5a) 1(!) = 0 1 + (!)2 ; (6.5b) 2(!) = 0! 1 + (!)2 ; (6.5c) where 0 is the dc conductivity and  is the average time between electron scattering events. The surface resistance can be obtained by noting that <[Z2S(!)] + jZS(!)j2 = 2R2S(!). For the Drude model, the result is: RS(!) = r 0! 20 hp 1 + (!)2  ! i1=2 : (6.6) Notice that in the low frequency limit !  1, (!)  1  0 and RS(!)  !1=2 is recovered. Figure 6.3 shows the measured surface resistance of a single crystal of SrTi1xNbxO3 for x = 0:002. Pure SrTiO3 is an insulator but when doped with Nb has a superconducting phase when 0:0005  x  0:02 with a typical tran- sition temperature of Tc  300 mK [108]. The normal state RS(!) data in the gure are t to Eq. 6.6. The data and model agree very well with t parameters 0 = (2:39  0:04)  106 1m1 and 1=2 = 62  4 GHz. Note that there is a signicant deviation from the !1=2 frequency dependence. For comparison, the top frame of Fig. 6.3 shows the corresponding frequency dependence of 1(!) over the bandwidth of the microwave spectrometer. 140 6.3. Drude Conductivity 0 5 10 15 20 25 0.00 0.05 0.10 0.15 0.20    1.3 K  Drude  1/2 R S  ( ) Frequency (GHz) 0 1 2    1 ( 10 6 -1 m -1 ) Figure 6.3: Bottom: The measured normal state RS(!) of SrTi1xNbxO3 with x = 0:002. The surface resistance line shape can be modeled by assuming a Drude conductivity (solid line). There is signicant deviation from the !1=2 frequency de- pendence (dashed line) expected for a frequency independent real conductivity. Top: A plot of the frequency dependence of 1(!) for 1=2 = 62 GHz. 141 6.4. The Anomalous Skin Eect 6.4 The Anomalous Skin Eect Another normal state surface resistance spectrum occurs in very pure samples where only a fraction of the conduction electrons participate in the screening of electromag- netic elds at the sample surface. At low temperature (where the resistivity is low) and high frequencies one can enter into a regime, known as the anomalous skin eect, where the conduction electron mean free path ` can become greater than the skin depth  [109, 110]. In this regime the electric eld can vary signicantly over length scales of order the mean free path and the approximation of local electrodynamics (J = E) breaks down. Here the original insights of Pippard are followed to motivate an expression for the surface resistance when ` , often referred to as the extreme anomalous skin eect. Pippard suggested that only conduction electrons that travel approximately parallel to the sample surface contribute signicantly to the current density. The other conduction electrons eectively do not interact with the electric eld since they spend very little time in the region where the electric eld strength is appreciable. Figure 6.4 is a cartoon picture of the extreme anomalous skin eect. Pippard argued that only electrons that remain within  of the surface of the sample Figure 6.4: The anomalous skin eect. The horizontal line represents the sample surface. The shaded blue region of thickness  represents the sample skin depth and the length of the arrow represents the electron mean free path `. The electron shown makes an angle   sin  = =` with respect to the sample surface. Electrons that follow paths with angles greater than  do not signicantly interact with the electric eld. long enough to scatter, will interact with the electric eld. The fraction of conduc- tion electrons that satisfy this condition is given by =`. A careful treatment of these details leads to an eective conductivity given by34: e = 0 ` ; (6.7) where 0 is the usual dc conductivity,  is the propagation constant, and  is a nu- merical factor close to one. Inserting this eective conductivity into the propagation 34See, for example, the discussions given in Refs. [110] and [111]. 142 6.4. The Anomalous Skin Eect constant results in:  = p i0!e (6.8a) = r i0!0 ` (6.8b) such that:  = i+ p 3 2  0!0 ` 1=3 ; (6.9) where the identity 3 p i =  i+ p 3  =2 has been used. Finally, solving for the surface impedance gives: ZS(!) = i0!  = 1 2  20! 2` 0 1=3  1 + p 3i  ; (6.10) such that: RS(!) = 1 2  20! 2` 0 1=3 : (6.11) Note that 0 and ` are related through the expression: ` = 0mvF ne2 ; (6.12) where m is the electron mass, vF is the Fermi velocity, n is the density of conduction electrons, and e is the electron charge. One therefore has: RS(!) = 1 2  20! 2  mvF ne2 1=3 ; (6.13) in which  is the only free parameter and is expected to be close to one. Figure 6.5 shows the measured penetration of magnetic elds into a pure tin (Sn) sample35 from 1.2 to 10 K using the 12 kHz ac susceptometer described in x4.2. Tin is a BCS superconductor with a 3.7 K transition temperature. Below Tc, the ac susceptometer measures (T ) and above Tc the temperature dependence of the skin depth is measured (since (0) is negligible compared to (T ) one is able to extract the absolute skin depth). For Tc < T . 6 K, the skin depth is nearly temperature independent indicating that, in this temperature range, one is sensitive to the residual resistivity 0 of the sample. Using the measured skin depth  = 18 m (T = 4 K and ! = 2(12 kHz)), the residual resistivity is found to be 0 = 1=0 = 35The measured sample was 95 m thick and has a planar area of 1.19 mm2. 143 6.4. The Anomalous Skin Eect 0 2 4 6 8 10 0 5 10 15 20 25 30   ,  ( m ) Temperature (K) 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2    ( m ) Temperature (K) Figure 6.5: Top: Measured (T ) and (T ) of pure Sn from 1.2 to 10 K. There is a narrow superconducting transition at 3.7 K. Bottom: For completeness, the high- resolution (T ) data for T < Tc are included. Below 2 K ( 50% of Tc), (T ) has only a very weak temperature dependence, typical of BCS superconductors. 144 6.4. The Anomalous Skin Eect  (12 kHz) 18 m  (1 GHz) 62 nm  (20 GHz) 14 nm 0 1:5 103  cm n 14:48 1022 cm3 vF 1:88 108 cm=s ` 31 m ` 0 4:62 1016 m2 Table 6.1: Summary of the important anomalous skin eect parameters for pure tin. Over the entire measurement bandwidth of the microwave spectrometer `  (!) and one is in the extreme anomalous skin eect regime. 1:5  103  1cm1. Equation 6.12 can then be used to calculate `. The material properties n and vF were taken from Ref. [14]. Some of the relevant quantities used in the remainder of this section are summarized in Table 6.1 Figure 6.6 presents the measured normal state surface resistance spectrum of the pure Sn sample. It is immediately clear that the data do not follow a !1=2 dependence (dashed line). The solid line is a t to Eq. 6.13 with  the only free parameter. The measured data follow the expected !2=3 behaviour very closely. The t results in  = 0:81. 145 6.4. The Anomalous Skin Eect 0 5 10 15 20 25 0 4 8 12 16  4.0 K  5.1 K  Anomalous Skin Effect  1/2   R S  (m ) Frequency (GHz) Figure 6.6: Normal state microwave surface resistance spectrum of pure Sn. With ` (!) one is in the extreme anomalous skin eect regime and the data should follow !2=3 rather than !1=2. The scale of the RS(!) data is set by temperature independent constants. The incomplete 4.0 K spectrum overlaps with the 5.1 K spectrum. Note that, in order to obtain the best t to the expected !2=3 behaviour, it is essential that the RS(!) spectrum be corrected by removing the systematic frequency dependence in the ratio technique caused by magnetic eld strength asymmetries at the two sample sites. This correction is achieved by dividing the measured data by the AgAu:AgAu RS(!) ratio shown in Fig. 5.26 which changes by approximately 2% over the range 0.1 to 20 GHz. 146 6.5. RS(!; T ) of a Superconductor 6.5 RS(!; T ) of a Superconductor The surface resistance spectra shown thus far have all been in the normal state where the magnitude of RS(!; T ) is relatively high  103101 . However, the microwave spectrometer was specically designed to measure the ultra-low surface resistance of superconductors which can have RS(!; T ) values that are orders of magnitude lower. To emphasize the sensitivity and range of the spectrometer, this chapter is closed by showing the surface resistance data of the pure Sn sample in the superconducting state. Figure 6.7 shows RS(!; T ) at several temperatures that span the range from 1.2 K to just above Tc. The top panel of the gure highlights the range of the microwave spectrometer with surface resistances spanning four orders of magnitude from 106 to 102 . The bottom panel emphasizes that, even in the micro-ohm range, the surface resistance signal-to-noise ratio is very high. In the next two chapters the broadband microwave spectrometer will be used to study the electrodynamics of unconventional superconductors deep in the superconducting state. 147 6.5. RS(!; T ) of a Superconductor 0 2 4 6 8 10 12  5.1 K  4.0 K  3.7 K  3.5 K  3.3 K  3.1 K  2.9  K  2.5 K  1.2 K   R S  (m ) 0 5 10 15 20 0 10 20 . . . . . . 4.0 . . .1 K   R S  ( ) Frequency (GHz) Figure 6.7: Surface resistance spectra of pure tin for T . Tc. The blue spectra in the top and bottom frames are the same data. The top surface resistance scale is in m  and the bottom is in  . 148 Chapter 7 Cu-O Chain Defects in YBa2Cu3O6+y Disorder and inhomogeneity play critical roles in the behaviour of many measurable properties of the cuprates. Early on, cation substitution in the CuO2 planes was found to have striking eects on low temperature properties such as the magnetic penetration depth [49, 77], and scanning tunneling spectroscopy (STS) has provided detailed tests of the in
uence of point-like defects on d-wave superconductors [112]. More recently, attention has been focused upon the puzzling eects of o-plane dis- order. Fujita et al. have shown that o-plane cation disorder has a substantial im- pact on the critical temperature Tc [113]. STS measurements by McElroy et al. on Bi2Sr2CaCu2O8+ (BSCCO) have provided evidence that interstitial dopant oxygen atoms are correlated with mesoscale variation in the electronic spectrum [10]. In this context YBa2Cu3O6+y (YBCO) oers a unique opportunity to study the in
uence of defects, since high purity crystals of YBCO can be grown with extremely low cation disorder both on- and o-plane [114]. As is the case with BSCCO, YBCO's doping can be controlled via o-plane oxygen atoms, but in YBCO these dopants can be or- ganized into Cu-Oy chains whose lling and degree of disorder can be systematically manipulated [115]. It is therefore possible to work with YBCO single crystals with very little cation disorder (since the crystals are not doped by cation substitution) and with varying amounts of Cu-O chain disorder. The ability to separately control both cation and oxygen disorder is unique to YBCO. For instance, YBa2Cu3O7 has lled Cu-O chains that provide a slightly overdoped 86 K superconductor with very low disorder. At y = 0:5 one can prepare a stable phase with alternating lled and empty Cu-O chains (ortho-II superstructure) that also can have very little disorder [116]. Above 115C, the ortho-II phase gives way to the ortho-I phase in which all Cu-O chains are equally occupied on average. Because changes in the total oxygen content of YBCO is negligibly slow below 250C, a low-temperature anneal at 200C followed by a rapid quench can change the level of disorder in the Cu-O chains without chang- ing the total oxygen content [115]. This process is entirely reversible, since very long 149 7.1. Charge Carrier Doping in YBa2Cu3O6+y anneals below 115C will restore the Cu-O chain ordering and recover the original Tc of the ordered phase. In this chapter, oxygen disorder in the Cu-O chains will be identied as the dominant source of quasiparticle scattering at low temperature in the very pure YBa2Cu3O6:5 crystals used by our group [114, 116, 117]. The chapter is organized as follows: x7.1 describes the mechanism by which the charge carrier density of the CuO2 planes of YBa2Cu3O6+y is manipulated. The emphasis will be on YBa2Cu3O6:5 in which, despite the doping process, the crys- talline order of the sample is to a large extent preserved. In x7.2 the microwave spectroscopy of ortho-II ordered YBa2Cu3O6:5 is reviewed. In particular, it is shown that the conductivity spectra of these samples exhibit the essential features expected for a clean d-wave superconductor. Moreover, by comparing data from four dierent single crystals, it is shown that these results are remarkably robust. Section 7.3 will examine how reducing the order of the Cu-O chains modies the microwave spec- troscopy of YBa2Cu3O6:5. The experimental evidence strongly suggests that in high purity YBCO, even for samples with highly ordered Cu-O chains, the width of the quasiparticle conductivity spectrum is largely determined by residual disorder in the Cu-O chains. A discussion of some of the implications of the experimental results appears in x7.4. Finally, x7.5 identies two possible Cu-O chain defects that may be limiting the perfection of the chain ordering. Current and ongoing work to suppress these types of chain defects is discussed. 7.1 Charge Carrier Doping in YBa2Cu3O6+y As previously discussed in x2.1.1, CuO2 planes of YBa2Cu3O6+y can be doped by substituting Ca2+ for the native Y3+ cations. However, this doping mechanism neces- sarily introduces disorder into the sample since the foreign Ca2+ cations are randomly dispersed throughout the crystal structure on a lattice site right between the CuO2 bilayers: a charged impurity at a site so close to the CuO2 layers would certainly lead to strong scattering of the in-plane quasiparticles. A convenient way to dope charge carriers into the CuO2 planes, which is unique to YBa2Cu3O6+y, is to vary the oxygen content (0  y  1) in the so-called Cu-O chains. The Cu-O chains run along the b-direction with all oxygen sites lled in YBa2Cu3O7 and all oxygen sites empty in YBa2Cu3O6. To achieve an ionization state of O2, an isolated oxygen atom in the chain layer removes one electron from its two neighbouring Cu1+ ions, thus changing the copper ionization state to Cu2+ (see Fig. 7.1a). When two consecutive oxygen sites are lled, a total of four electrons are 150 7.1. Charge Carrier Doping in YBa2Cu3O6+y transferred to the oxygen atoms. Triply ionizing Cu is energetically unfavourable and two consecutively lled oxygen sites leads to a single hole formed in the Cu-O chain layer. As the chainlet lengths (segments of consecutively lled oxygen sites) increase, more holes are introduced. These chain holes are partially lled with electrons from the CuO2 planes resulting in hole carriers being introduced into the planes. See Fig. 7.1. Although this is an oversimplied picture, the essential point is that the Figure 7.1: (a) An isolated oxygen atom in the Cu-O chain layer extracts a single electron from each of the neighbouring Cu atoms changing the ionization states from Cu1+ to Cu2+. (b) When two consecutive chain oxygen sites are lled the resulting hole in the Cu-O chain layer is partially lled with electrons from the CuO2 planes. (c) The addition of an isolated oxygen atom does not change the CuO2 plane doping. (d) The doping is only changed when the chainlet lengths grow. Figure provided courtesy of Darren Peets [38]. hole-doping of the CuO2 planes in YBa2Cu3O6+y depends both on the oxygen content and the conguration of the oxygen atoms in the Cu-O chains. Between temperatures of  0C and  250C, although the chain oxygen atoms are mobile, the oxygen content of the sample is eectively xed due to the extremely slow kinetics of oxygen incorporation at the surface. Thus, in this temperature range, using well established procedures [115, 116] the ordering of the chain oxygen atoms can be manipulated without changing the overall chemical composition of the sample. The 151 7.1. Charge Carrier Doping in YBa2Cu3O6+y two ordered phases relevant to this work are the so-called ortho-I and ortho-II phases. (The roman numerals refer to the periodicity of the ordering. For a discussion of the other oxygen orderings and a structural phase diagram see, for example, Andersen et al. in Ref. [115].) In the ortho-I, phase all the Cu-O chains are equally occupied. Thus, the ortho-I phase of YBa2Cu3O6:5 is one with all chains half lled. This phase can be accessed by heating a single crystal to 200C and then rapidly quenching the sample to 0C where the diusion of chain oxygen atoms is negligibly slow. For long anneals between 0C and 115C, the oxygen atoms in the Cu-O chain layers remain mobile and self organize into alternating full and empty chains as de- picted in Fig. 7.2. It is the intervening Cu atoms along the b-axis direction that Figure 7.2: A half-lled Cu-O chain layer in the ortho-II phase. The Cu-O chains alternate between completely unoccupied oxygen sites (light blue) and completely occupied oxygen sites (dark blue). dierentiates a- and b-directions and causes chains to form in one direction and not the other. This arrangement eectively doubles the unit cell dimension in the a-axis direction (hence the nomenclature ortho-II ). This eect is shown in Fig. 7.3 which is a schematic view of multiple unit cells of perfectly ortho-II ordered YBa2Cu3O6:5. 152 7.1. Charge Carrier Doping in YBa2Cu3O6+y Figure 7.3: Multiple unit cells of perfectly ortho-II ordered YBa2Cu3O6:5. The Cu-O chains at the top and bottom of the gure alternate between full (all oxygen sites occupied) and empty (all oxygen sites vacant). Figure provided courtesy of Darren Peets [38]. It is assumed that each oxygen ion in a full Cu-O chain makes the same con- tribution to the doping of the CuO2 planes, independent of the occupation of the neighbouring chains. This assumption is reasonable provided that the chains are strongly one-dimensional with very little coupling between chains as is suggested in Ref. [118]. For oxygen ordered phases of YBCO with innite chain lengths p = p0  y where p0  0:194 is the maximum doping with all Cu-O chains lled and 0  y  1 gives the fraction of Cu-O chains that are full [45]. Thus, for perfect ortho-II order- ing with y = 0:5, p  0:097 is expected. Liang et al. have uniquely determined the doping dependence of Tc for YBa2Cu3O6+y 36. A key result of that work is reproduced in Fig. 7.4. The ortho-II ordered doping of p = 0:097 results in Tc  58 K [45]. By symmetry, in the disordered ortho-I phase each chain is populated equally and y gives the average occupancy of each chain. When equilibrated at nite temperature, there will be clustering of the Cu-O chain oxygen atoms controlled by both intra- chain and inter-chain interactions. (Models for ordering in this type of lattice gas are discussed in Ref. [115].) If one could suddenly quench from a very high-temperature state, in which the oxygen clustering is negligible, each chain site would have an equal probability of being occupied and the doping of the CuO2 planes would be given by p = p0  y2. For example, in YBa2Cu3O6:5 (y = 0:5) a completely random distribution 36Note that Tc is a unique function of p, but not of the oxygen content y. 153 7.1. Charge Carrier Doping in YBa2Cu3O6+y Figure 7.4: YBa2Cu3O6+y Tc as a function of doping p. The red line is the empirical relationship between Tc and p (Eq. 2.7) found to be approximately valid across the cuprate family. The data deviate from the empirical relationship near dopings of 1=8. Inset: Dierence between the empirical Tc and the measured Tc as a function of doping. Figure taken from Ref. [45]. of chain oxygen atoms results in a doping of p = 0:0485 (half that of the ortho-II ordered phase with innite chain lengths). Referring again to Fig. 7.4, one expects a near zero Tc at this doping. In appendix D a simple hole counting argument to show that the doping of the ortho-I phase of YBa2Cu3O0:5 goes as y 2 is presented. This demonstration is followed by two separate proofs that conrm this result in general for 0  y  1. Figure 7.5 shows a half-lled Cu-O chain layer in the ortho-I phase. This gure was generated by writing a very simple program to randomly occupy 50 of 100 available oxygen atom sites in the 10  10 Cu-O chain layer shown. The resulting chainlets that are responsible for the doping of the CuO2 planes are highlighted. A chainlet of ` consecutive oxygen ions will make the same contribution to the doping of the CuO2 plane as ` 1 oxygen ions in a full chain. Since Cu-O chain holes are only partially lled by CuO2 plane electrons, we will say that a chainlet of length ` introduces ` 1 fractional holes into the CuO2 planes. Table 7.1 summarizes the results of this simple exercise. This particular conguration of chain oxygen atoms contributes 24 fractional 154 7.1. Charge Carrier Doping in YBa2Cu3O6+y Figure 7.5: A half-lled 10  10 Cu-O chain layer in the ortho-I phase. There are 100 available chain oxygen sites each with a 50% probability of being lled and 50 chain oxygen atoms. Isolated oxygen atoms do not contribute to the hole-doping of the CuO2 plane. Cu-O chainlets (highlighted in yellow) consisting of ` consecutively occupied oxygen sites results in approximately `  1 fractional holes in the CuO2 plane. holes to the CuO2 plane. A perfect ortho-II ordering of this 1010 Cu-O chain layer would result in 5 lled chains of length 10 and, therefore, 9 5 = 45 fractional holes contributed to the plane. This simple model conrms that the dopings of ortho-I and ortho-II ordered half-lled YBa2Cu3O6:5 dier by nearly a factor of two. If the Cu-O layer size used is increased this factor asymptotically approaches two. This analysis suggests that one can vary the superconducting transition temper- ature of YBa2Cu3O6:5 to be between 0  Tc  58 K by manipulating the Cu-O chain order without any changes to the overall oxygen content of the sample. In practice, neither the ideal ortho-I nor ortho-II states are accessible. Even with very long low- temperature anneals there will be gaps in the lled chains of the ortho-II state and some lled sites in the empty chains. In fact, the best ortho-II ordering appears to occur at an oxygen content of y = 0:52 [119]. Some of the excess oxygen partially lls 155 7.2. Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 Chainlet length fractional holes Count # Contributed ` ` 1 fractional holes 1 0 14 0 2 1 6 6 3 2 4 8 4 3 1 3 5 4 0 0 6 5 0 0 7 6 0 0 8 7 1 7 24 Table 7.1: Chainlet lengths and the number of fractional holes contributed to the CuO2 plane for a 10  10 Cu-O chain layer of ortho-I ordered YBa2Cu3O6:5. The total number of fractional holes contributed by all chainlets is 24. the gaps in the chains, while the empty chains remain suciently depleted of oxygen ions that they do not contribute signicantly to the doping of the CuO2 plane. The ortho-I phase is even more dicult to access. As already alluded to, nite temper- ature correlations will prevent a completely random distribution of oxygen atoms in the chain layer. Moreover, as a practical matter, it is dicult to quench the sample quickly enough to freeze the high-temperature disordered state. As soon as the sam- ple temperature drops below  115C, chainlets will rapidly form and one achieves a quenched state that is undoubtedly some frozen in non-equilibrium state. 7.2 Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 By considering point-like scatterers Hirschfeld et al. have calculated an expression for the microwave conductivity of a d-wave superconductor that is valid for any scattering strength: (!; T ) = ne2 m?  1 i! + 1(")  " ; (7.1) where 1(") is an energy dependent scattering rate and h: : : i" represents a thermal average weighted by the quasiparticle density of states N("). To within logarithmic 156 7.2. Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 corrections, 1(")  u"1 in the strong or unitary scattering limit (large scattering phaseshift) whereas in the weak or Born scattering limit (small scattering phaseshift) 1(")  B". The coecients u and B are determined by the concentration of impurities and/or defects. It is worth noting that Eq. 7.1 is valid for temperatures and frequencies above a characteristic energy scale 
 that is set by the zero-temperature superconducting gap maximum and the normal state impurity scattering rate [120, 121]. Born limit conductivity spectra predicted from Eq. 7.1 have a number of dis- tinguishing features, which include: cusp-like line shapes, temperature independent zero frequency intercepts, tails that fall more slowly than !2, and a !=T scaling of the spectra. Prior to 2003, coarse conductivity spectra pieced together from several xed frequency resonator experiments on fully doped YBa2Cu3O6:99 did not reveal any of these expected features [92]. The desire to better understand the quasiparti- cle conductivity of YBCO was the motivation for the development of the microwave spectrometer described in chapter 5 [93]. YBa2Cu3O6:99 conductivity spectra ob- tained with this apparatus showed increasingly cusp-like line shapes emerging at the lowest measurement temperatures [94]. These data have been interpreted as arising from an intermediate scattering regime that adheres neither to the Born- nor unitary- limits. These ndings have since been supported by thermal conductivity measure- ments [122]. The work presented here, however, will focus solely on YBa2Cu3O6:5 in which the Born-limit scattering features have been clearly observed [94, 95]. The surface resistance spectra of ortho-II ordered YBa2Cu3O6:52 with currents along the a-axis direction measured by Turner et al. are shown in Fig. 7.6 for four temperatures well below Tc [94]. The RS(!; T ) measurements can be converted to quasiparticle conductivity spectra using Eqns 3.20a. This simple analysis neglects the normal 
uid contribution to the screening of electromagnetic elds. At low fre- quency and when T  Tc, this approximation is very good. Appendix E describes how the raw RS(!; T ) spectra can be t to a model that includes normal 
uid screen- ing. This t requires that one adopt a phenomenological model for the quasiparticle conductivity: 1 = 0 1 + (!=)y ; (7.2) where 0, , and y are t parameters; the value y = 2 corresponds to the Drude model of conductivity. Figure 7.7 show the conductivity spectra extracted from the RS(!; T ) measure- ments of Fig. 7.6 following the methods of appendix E. Cusp-like line shapes and a 157 7.2. Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 0 5 10 15 20 0 100 200 300 400 500  6.7 K  4.3 K  2.7 K  1.3 K   R S  ( ) Frequency (GHz) Figure 7.6: Surface resistance spectra of ortho-II ordered YBa2Cu3O6:52 with currents in the a-axis direction. Figure provided courtesy of P. J. Turner [94, 99]. temperature independent zero frequency intercept are clearly visible in the data37. Moreover, the tails of the spectra fall more slowly than !2 as seen from the Drude (1 = 0=[1 + (!=) 2]) t in the inset. Equation 7.1, assuming Born-limit scattering, captures the line shape of the conductivity spectra extremely well as shown by the t to the 6.7 K data in Fig. 7.7. However, using the parameters from this t to make predictions for the lower temperature spectra increasingly underestimates the spectral weight beneath the curves. Rather than scaling as !=T as expected, the data scale as !=(T +T0) with T0 = 2 K suggesting a residual quasiparticle density as T ! 0. The area beneath each conductivity spectrum is a measure of the quasiparticle, or normal 
uid density: nne 2 m? (T ) = 2  Z 1 0 1(!; T )d!; (7.3) whereas the super
uid density is determined from the penetration depth (T ) 37Here, we describe the data as cusp-like if the conductivity spectra show no signs of rolling over or 
attening in the low frequency limit. 158 7.2. Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 0 5 10 15 20 0 10 20 30 0 2 4 6 0 10 20 30   σ 1 (10 6  Ω - 1 m - 1 ) Frequency (GHz)   σ 1 (10 6  Ω - 1 m - 1 ) f/(T+T o ) (GHz K-1) Figure 7.7: Conductivity spectra of ortho-II ordered YBa2Cu3O6:52 with currents in the a-axis direction. The solid red curve is a t to the 6.7 K data using Eq. 7.1 and 1(") = B". The dashed lines are predictions of the conductivity spectra for the lower temperatures using the parameters from the 6.7 K t. Inset: The conductivity spectra collapse onto a single curve when the frequency is scaled as !=(T + T0) with T0 = 2 K. The solid line is a simple Drude t illustrating the inadequacy of the Lorentzian line shape. Figure provided courtesy of P. J. Turner [94, 99]. through: nse 2 m? (T )  1 02(T ) : (7.4) In the two 
uid model the total electron density is the sum of these two contributions: ne2 m? = nse 2 m? (T = 0) = nne 2 m? (T ) + nse 2 m? (T ); (7.5) where it has been assumed that in the clean limit at T = 0 K all of the charge carriers enter into the super
uid condensate. With separate measurements of 1(!; T ) and the absolute (T ), it is possible to quantify to what extent the normal 
uid density tracks the super
uid density. In order to integrate 1(!; T ) over all frequencies each conductivity spectrum is separately t to the phenomenological model given by Eq. 7.2 which captures the cusp-like line shapes extremely well and allows for high frequency extrapolations of 159 7.2. Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 0 5 10 15 20 0 10 20 30 0 5 10 0 10 20 30   σ 1 (10 6  Ω - 1 m - 1 ) Frequency (GHz) ‡ s1 dw from 1/λ2(T) from   n n e2 /m   (10 17   Ω - 1 m - 1 s - 1 ) T (K) Figure 7.8: The same a-axis ortho-II ordered YBa2Cu3O6:52 data as in Fig. 7.7 t to the phenomenological model of Eq. 7.2. The inset compares the normal 
uid density found by integrating the conductivity ts to that found from penetration depth measurements. Figure provided courtesy of P. J. Turner [94, 99]. the data. Figure 7.8 shows the same data as Fig. 7.7 with ts to Eq. 7.2. The inset shows that the slopes of the normal 
uid densities found by integrating 1(!; T ) and from 2(0)  2(T ) agree very well indicating that the bulk of the spectral weight is within the bandwidth of the measurement apparatus. However, while the normal 
uid density calculated from the penetration depth vanishes as T ! 0, as expected theoretically, the normal 
uid density found from the conductivity data clearly extrapolate to a nonzero intercept. The cause of this apparent discrepancy is not currently understood, however, based on the results of the next section, a possible resolution is suggested at the end of x7.4. This section is concluded by comparing the microwave spectroscopy of a number of dierent ortho-II ordered YBa2Cu3O6:5 single crystals. The four crystals studied were grown by Dr. Ruixing Liang using the self-
ux method in BaZrO3 crucibles manufactured in-house [117]. The starting materials for the crystal growth consists of a mixture of Y2O3, CuO, and BaCO3 powders. For samples #1 and #2 all starting powders had an atomic purity of 99:995 ! 99:999%. For samples #3 and #4, the 160 7.2. Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 0 5 10 15 20 0 100 200 300 400 500 600  1.6       4.2  1.2       4.3  1.3       3.1  1.3       4.3  6.9  5.6  6.7   R S  ( ) Frequency (GHz) Figure 7.9: Comparison of RS(!; T ) for four ortho-II order YBa2Cu3O6:5 single crys- tals. Samples #1 (black) and #2 (red) are high-purity single crystals and samples #3 (green) and #4 (blue) are ultra-high-purity single crystals. The low-temperature ( 1 K) surface resistance of the four samples are nearly indistinguishable. The tem- perature evolution of RS(!; T ) is also strikingly similar among these four samples. RS(!; T ) of sample #3 was not measured at T  6 7 K. Y2O3 and BaCO3 powders were unchanged, however, the CuO atomic purity was increased by an order of magnitude to 99.9999%. Samples #1 and #2 will be referred to as high-purity single crystals, whereas samples #3 and #4 will be referred to as ultra-high-purity single crystals. All samples were detwinned at 200C under uniaxial stress. Figure 7.9 compares the measured a-axis surface resistance spectra of samples #1 through #4. For these four samples both the absolute value and the temperature evolution of RS(!; T ) are remarkably similar. This result is particularly impressive considering that the sample growth dates span approximately seven years. Figure 7.9 would not be possible without the consistency and thoroughness of Ruixing Liang's custom crystal growth program. In addition, the measurements shown were taken using two completely independent microwave spectrometers38 and detection systems 38The development of the microwave spectrometer was the result of contributions from numerous researches over a period that spanned more than a decade. Further developments will continue. 161 7.2. Microwave Spectroscopy of Ortho-II YBa2Cu3O6:5 CuO Spectrometer Graduate Sample # atomic purity generation Student Reference 1 99.999% 1 P. Turner [94, 95] 2 99.999% 1 J. Bobowski [123] 3 99.9999% 2 J. Day unpublished J. Baglo J. Bobowski 4 99.9999% 2 J. Baglo unpublished L. Semple Table 7.2: The four YBa2Cu3O6:5 surface resistance data sets in Fig. 7.9 were ob- tained using single crystals with dierent levels of cation purity, dierent microwave spectrometers, and by dierent graduate students. operated by ve dierent graduate students in two dierent cryostats. Table 7.2 summarizes some of the details associated with the spectroscopic measurements of the four samples. Figure 7.10 compares the low-temperature ( 1 K) quasiparticle conductivity of the four single crystals. For clarity the data have been oset along both the frequency and conductivity axes by increments of 2 GHz and 2106 1m1 respectively. When the osets are removed the four conductivity spectra lie along essentially the same curve. Recall that the in the weak-scattering Born limit the energy-dependent scatter- ing rate is given by 1(")  B" where the coecient B is set by the concentration of impurities and/or defects. Since 1(") determines the width of the conductivity spectra, if B is in part set by the chemical purity of the samples, one expects that increasing the atomic purity of the CuO starting powder by an order of magnitude should result in a noticeable decrease in the width of 1(!; T ) of the ultra-high-purity single crystals. To quantitatively compare the widths of the conductivity spectra the phenomeno- logical model Eq. 7.2 is t to each of the data sets in Fig. 7.10. This model captures the conductivity line shapes with a range of dierent parameters and in order to get the most direct comparison of , the parameter y was xed to be 1.40. The results of the ts are shown in Table 7.3. There is no systematic dependence of  on the CuO atomic purity. These results suggest that something other than the chemical purity of the samples is the primary source of quasiparticle scattering in these YBa2Cu3O6:5 single crystals. In the next section, defects in the Cu-O chain ordering are identied as the dominant source of in-plane quasiparticle scattering in YBa2Cu3O6:5. 162 7.3. Microwave Spectroscopy of Disordered YBa2Cu3O6:5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35  Sample #4  Sample #3  Sample #2  Sample #1   1 ( 10 6 -1 m -1 ) Frequency (GHz) Figure 7.10: Low-temperature quasiparticle conductivity spectra of four dierent YBa2Cu3O6:5 single crystals. For clarity neighbouring spectra are oset by 2 GHz along the frequency axis and by 2 106 1m1 along the conductivity axis. Despite the increased chemical purity of samples #3 and #4, there is no noticeable change in the spectral width of 1(!; T ) for the ultra-high-purity samples. Sample #  (GHz) 1 (high-purity) 2.05 2 (high-purity) 2.10 3 (ultra-high-purity) 1.95 4 (ultra-high-purity) 2.40 Table 7.3: Spectral width comparison of the four YBa2Cu3O6:5 single crystals. There is no systematic variation of  when comparing the high-purity and ultra-high-purity samples. 7.3 Microwave Spectroscopy of Disordered YBa2Cu3O6:5 In this section the eect of intentionally reducing the degree of order in the Cu-O chains is systematically studied. All the data presented in this section are from mea- 163 7.3. Microwave Spectroscopy of Disordered YBa2Cu3O6:5 surements of the YBa2Cu3O6:50 single crystal referred to as sample #2 in Figs. 7.9 and 7.10. First, the microwave electrodynamics of the sample when in the highly ordered ortho-II phase are studied and shown to be in good qualitative and quan- titative agreement with previously published data from sample #1 (gures 7.6-7.8) when prepared in the same ortho-II state. The sample #2 measurements were then repeated after intentionally disordering the Cu-O chains, but without changing the oxygen content. The detailed analysis and interpretation of the data that follows suggests that, even in the highly ordered state, the dominant source of quasiparticle scattering in YBa2Cu3O6:5 single crystals is residual disorder in the Cu-O chain layer. With the YBa2Cu3O6:50 crystal prepared in the highly ordered ortho-II state, the temperature dependence of (T )  (T ) (1:5 K) was measured using the 940 MHz resonant cavity [49] and RS(!; T ) was measured at four temperatures below 10 K using the broadband apparatus. The same sample was next heated to 200C in air for one hour before being quenched to 0C to reduce the order of the Cu-O chains, and then the above measurements were repeated. During the disordering process the sample is kept stress free and new twin boundaries are not expected to form. It was conrmed that there was no retwinning of the sample using a polarized optical microscope. After mounting the sample in the broadband apparatus, the sample was kept at or below 77 K for the duration of the disordered measurements so as to suppress room temperature reordering of the Cu-O chains. Following the completion of the disordered measurements, a low-temperature anneal was used to restore the ordering of the Cu-O chains and the magnetization of the sample was measured as a function of temperature in a SQUID magnetometer. To complete the data analysis, a(0) = 202  22 nm and b(0) = 140  14 nm are used for ordered YBa2Cu3O6:50 as obtained from zero-eld ESR measurements on Gd-doped GdxY1xBa2Cu3O6+y [96]. There are no reported measurements of (0) for YBCO with disordered chain oxygen atoms and estimating its value is dicult and uncertain. One way to obtain a reasonable estimate of its value is to require that the low-T slope of 1=2(T ) remain constant upon disordering the Cu-O chains, as suggested by recent Hc1 measurements on underdoped YBCO [124]. This analysis yields a(0) = 238  24 nm and b(0) = 162  16 nm for disordered YBa2Cu3O6:50. These values of (0) are consistent with the relationship between Tc and (0) derived from the ESR measurements on YBCO [96]. It is important to stress that (0) merely sets an overall scale factor for the 1(!; T ) spectra and none of the conclusions of this work rely on its value. In particular, our analysis will focus on the widths of the conductivity spectra which are unaected by the choice of (0). 164 7.3. Microwave Spectroscopy of Disordered YBa2Cu3O6:5 Figure 7.11 shows the measured change in the cavity resonant frequency fr(T ) = fr(1:5 K) fr(T ) due to the YBa2Cu3O6:50 sample both before and after disordering the chain oxygen atoms. The hole doping of the CuO2 planes in YBCO is not a 0 10 20 30 40 50 60 0 5 10 15 20 25  Ordered Cu-O chains  Disordered Cu-O chains   f r (k H z) Temperature (K) 0 2 4 6 8 10 0 5 10 15 20   a ( nm )  Temperature (K) -16 -12 -8 -4 0 m  (10 -6 em u)  Re-ordered Cu-O chains  Figure 7.11: Measured change in the resonant frequency (left axis) of the 940 MHz cavity when loaded with YBa2Cu3O6:50 with ordered (lled black circles) and disor- dered (open black circles) Cu-O chains. The sample was oriented such that currents were directed along the a-axis. After re-ordering the Cu-O chains, the supercon- ducting transition was measured via the magnetization m of the sample (connected blue points, right axis) in a SQUID magnetometer. A one gauss dc magnetic eld was applied perpendicular to the broad face of the platelet sample. Inset: The low- temperature a(T ) after removing the c-axis contribution. unique function of the oxygen content y, but depends both on y and the Cu-O chain ordering [45]. The observed shift in Tc from 55 to 49 K upon disordering the Cu-O chains is due solely to a change in the hole doping of the CuO2 planes. After re- ordering the Cu-O chains into the ortho-II state, magnetization measurements (see Fig. 7.11) showed that the Tc of the sample was restored to 55 K. These data conrm that the oxygen content y of the Cu-O chains was the same for all of the sample #2 measurements reported in this work. For T  Tc the measured fr(T ) is proportional to the change in the penetra- tion depth (T ). The inset of Fig. 7.11 shows that a(T ) / T for both ordered and disordered Cu-O chains. The measured crystal is a platelet with dimensions a b c = 0:482 0:741 0:028 mm3. A small contribution from c(T ) was re- 165 7.3. Microwave Spectroscopy of Disordered YBa2Cu3O6:5 0 5 10 15 20 0 10 20 30 (a)    1  ( 1 0 6  ! -1 m -1 ) Frequency    (GHz) 0 200 400 600 800 1000     YBa 2 Cu 3 O 6.50  - Ordered CuO chains a-axis T c  = 55 K      8.6 K      5.6 K      3.1 K      1.3 K  R S  ( " ! )  0 5 10 15 20 0 4 8 12 16 (b)    1  ( 1 0 6  ! -1 m -1 ) Frequency    (GHz) 0 200 400 600 800 1000     YBa 2 Cu 3 O 6.50  - Disordered CuO chains a-axis T c  = 49 K      8.9 K      5.7 K      3.0 K      1.2 K  R S  ( " ! )   Figure 7.12: (a) Top panel: Measured a-axis surface resistance of YBa2Cu3O6:50 with highly ordered Cu-O chains. Bottom panel: Extracted a-axis quasiparticle conduc- tivity spectra. The solid lines are phenomenological ts to the data. (b) Top panel: Measured a-axis RS(!; T ) of YBa2Cu3O6:50 with disordered chain oxygen. Bottom panel: Extracted a-axis quasiparticle conductivity spectra. The solid lines are phe- nomenological ts to the data. moved by using previous measurements of c(T ) for a YBa2Cu3O6:60 (Tc = 60 K) crystal [125] together with the determination of the doping dependence of c(0) by Homes et al. [126]. The low-T slope of the corrected a-axis data is d(a(T ))=dT = 1:36 0:10 nm/K which is comparable to a previously measured value of 1:05 nm/K found for sample #1 which has a large enough a : c aspect ratio that c-axis corrections were not required [94]. The top panel of Fig. 7.12a shows the measured ortho-II-ordered a-axis surface resistance. The conductivity spectra obtained from the RS(!; T ) data using the 166 7.3. Microwave Spectroscopy of Disordered YBa2Cu3O6:5 analysis of appendix E are shown in the bottom panel of Fig. 7.12a. These spectra are t to the phenomenological model originally given in Eq. 7.2: 1 = 0 1 + (!=)y : (7.6) This model captures the observed line shapes very well and gives a measure of the spectral width (T ) of the data. As previously observed for sample #1, the ordered YBa2Cu3O6:50 conductivity data exhibit features expected for d-wave quasiparticles undergoing weak-limit scat- tering: cusp-like line shapes (no low frequency plateau), temperature independent 1(! ! 0) intercepts, and T -linear spectral widths (T ) [94, 95]. The measured a- axis RS(!; T ) and 1(!; T ) after disordering the chain oxygen are shown in Fig. 7.12b. These conductivity spectra also have cusp-like line shapes and T -linear (T ) charac- teristic of weak-limit scattering, however the widths of the spectra are signicantly broadened. The Tc of the YBa2Cu3O6:50 sample was suppressed by  10% after disorder- ing the Cu-O layer. One expects a comparable suppression of the zero-temperature superconducting gap magnitude 0. As discussed in the next paragraph, in the weak-scattering limit, the scattering rate is inversely proportional to 0. There- fore, a  10% increase in the widths of the conductivity spectra is expected purely from the change in doping. However, the measured spectral widths increased by  300% after disordering the Cu-O chains. A broadening of this magnitude can only be attributed to increased quasiparticle scattering arising from disorder in the Cu-O chain layer. A summary plot of (T ) is given in Fig. 7.14. For completeness, Fig. 7.13 shows RS(!; T ) and 1(!; T ) for currents propagating in the b-direction for the same YBa2Cu3O6:50 sample both before and after disordering the Cu-O chain oxygen atoms. These data exhibit the same qualitative weak-scattering features as the a-axis data. The phenomenological ts to these conductivity spectra include an additional constant oset 1D which accounts for the broad (approximately 
at over the measurement bandwidth) and highly one-dimensional conduction along the chains [94, 95]. In a d-wave superconductor, the linear dispersion of the energy gap sets the avail- able phase space for quasiparticle scattering and results in a strong energy depen- dence of the scattering rate. For point-like defects in the limit of small scattering phase shifts, 1(")  4"=0c2 to within logarithmic corrections [120, 121]. Here c is the cotangent of the scattering phase shift, 0 is the zero temperature super- 167 7.3. Microwave Spectroscopy of Disordered YBa2Cu3O6:5 0 5 10 15 20 0 25 50 75 100 (a)    1  ( 1 0 6  ! -1 m -1 ) Frequency    (GHz) YBa 2 Cu 3 O 6.50  - Ordered CuO Chains b-axis T c  = 55 K      8.7 K      5.6 K      3.0 K      1.3 K 0 200 400 600 800      R S  ( " ! )   0 5 10 15 20 0 8 16 24 32 (b) YBa 2 Cu 3 O 6.50  - Disordered CuO chains b-axis T c  = 49 K      8.8 K      5.6 K      3.0 K      1.3 K    1  ( 1 0 6  ! -1 m -1 ) Frequency    (GHz) 0 200 400 600 800     R S  ( " ! )   Figure 7.13: (a) Top panel: Measured b-axis RS(!; T ) of YBa2Cu3O6:50 with highly ordered Cu-O chains. Bottom panel: Extracted b-axis quasiparticle conductivity spectra. The solid lines are phenomenological ts to the data. (b) Top panel: Mea- sured b-axis RS(!; T ) of YBa2Cu3O6:50 after disordering the chain oxygen. Bottom panel: Extracted b-axis quasiparticle conductivity spectra. The solid lines are phe- nomenological ts to the data. conducting gap maximum, and  = nin=N0 with ni the concentration of defects, n the carrier density, and N0 the density of states at the Fermi energy. In the opposite limit of large scattering phase shift, the scattering rate has a completely dierent energy dependence; 1(")  "1. The T -linear spectral widths shown in Fig. 7.14 indicate that the scattering is closer to the weak limit where 1(")  " [95, 120]. The increased slope of (T ) upon disordering the Cu-O chains indicates that the increase in the number of oxygen chain defects corresponds to an increase in the density of weak scattering defects ni. 168 7.3. Microwave Spectroscopy of Disordered YBa2Cu3O6:5 0 2 4 6 8 10 0 5 10 15 20 25 30 35        a-axis  ordered  disordered        b-axis  ordered  disordered   /2  (G H z) Temperature (K) Figure 7.14: Spectral width (T ) versus temperature for YBa2Cu3O6:50 with ordered chains in the a-axis (solid circles) and b-axis (solid triangles) directions and with disordered chains in the a-axis (open circles) and b-axis (open triangles) directions. Note that the value of (T ) for the ordered b-axis data at 1.3 K is unrealistically small and is likely an artifact of the t to the phenomenological model. These measurements unambiguously show that defects in the Cu-O chain ordering are responsible for the increased weak-limit scattering seen in the disordered measure- ments. By making the reasonable assumption that chain ends are the source of the weak-limit scattering it is argued below that, even with highly ordered Cu-O chains, residual disorder of the chain oxygen atoms is the dominant source of scattering in high purity YBa2Cu3O6:5. The density of oxygen chain defects can be deduced from the relationship between Tc and the hole doping per Cu in the CuO2 plane p. As discussed in x7.1, p = p0  y for the oxygen ordered phases of YBCO where p0  0:194 [45]. Away from perfect order, the number of holes contributed to the CuO2 plane by a Cu-O chain of nite length with ` oxygen atoms is reduced by a factor of (` 1)=`. Thus, if the average chain has ` consecutively occupied oxygen sites, the doping is given by: p  p0  ` 1 `  y: (7.7) 169 7.4. Discussion The simple simulation of a 1010 Cu-O chain layer in x7.1 showed that the doping of ortho-I YBa2Cu3O6:50 is half that of the same sample with perfectly ortho-II ordered Cu-O chains. This corresponds to a doping of p = 0:0485 and a near zero Tc. The fact that the Tc of the sample after the disordering procedure is 49 K indicates that the Cu-O chains are very far away from a fully random ortho-I ordering. Rather, one has an ortho-I phase (all chains equally occupied on average) in which the chain oxygen atoms form chainlets whose average length can be estimated using Eq. 7.7. From Fig. 7.4, Tc = 49 K corresponds to a doping of p = 0:084. Using Eq. 7.7, we estimate that the chainlets have an average length of `  7:51:0. The uncertainty in ` is due to an uncertainty in the oxygen content of the chain layer which is y = 0:50  0:01 for sample #2. At the p = 0:093 doping of the ordered sample the estimated chainlet length is very sensitive to the value of y and lies between the limits 17 < ` < 46. Making the reasonable assumption that each Cu-O chain end acts as a scattering defect gives a defect concentration of ni = 1=`. With disordered Cu-O chains this analysis gives ni = 0:13  0:02, whereas the defect concentration is anywhere from two to six times less when the Cu-O chains are ordered. The spectral widths of the disordered conductivity spectra are undoubtedly largely determined by the concen- tration of defects in the Cu-O chains. If the quasiparticle scattering in the better ordered sample is also dominated by residual defects in the Cu-O chains, one expects the spectral widths of the ordered conductivity spectra to be suppressed by a factor that is consistent with the estimated reduction of ni. Figure 7.14 shows the a- and b-axis spectral widths before and after disordering the Cu-O chain oxygen atoms. The slope of the a-axis (b-axis) disordered data set is 2:9 0:6 (2:4 0:6) times that of the ordered data set, consistent with quasiparticle scattering dominated by Cu-O chain defects at both Cu-O chain orderings. 7.4 Discussion The consistency of the microwave electrodynamics of two generations of YBa2Cu3O6:5 single crystals that were grown over a period that spans at least seven years is a re- markable result on its own. It is clear that, at the levels being considered here, the purity of the starting powders used in the crystal growth is not dominating the quasi- particle dynamics. However, in this type of high-temperature growth one cannot be sure that the samples are not picking up impurities from the crucible which could contribute to the observed scattering rates. Empirically, the variability in the in- house fabricated BaZrO3 crucibles [114] is large enough that, given the extraordinary 170 7.4. Discussion consistency among the dierent samples, it seems improbable that the quasiparticle dynamics are aected by sample contamination from the crucible. One possibility is that the samples may have intrinsically nite ortho-II domains separated by anti- phase boundaries. For example, Andersen et al. found nite size ortho-II domains both experimentally and in simulations that allow for repulsive Coulomb interactions between oxygen atoms next-nearest neighbour chains [115]. A thermodynamically stable substitution within the crystal structure that nucleates these anti-phase bound- aries and therefore sets the domain size could account for the unchanging behaviour that has been observed over such a long period of time [119]. An eort to identify, and the possibly control, such a substitution is currently underway. It is perhaps surprising that in comparing conductivity spectra from the ortho-II state, in which there is a high degree of crystalline order, to those from the disordered state that the spectral widths (T ) are enhanced only by a factor of three. Equa- tion 7.7 implies that in order to achieve a doping of p = 0:0485, as is expected in the random ortho-I phase, one requires an average chainlet length of ` = 239. Such a scenario would lead to a Cu-O chain defect concentration that is more than an or- der of magnitude greater than that when the sample is in the ordered state. Average chainlet lengths of `  2 would result in spectral widths that would be expected to lie outside our measurement bandwidth even at the lowest temperatures. The observed shift in Tc, the estimated average chainlet length `, and the measured spectral widths (T ) all point to a disordered sample in an ortho-I state in which the chain oxygen atoms are not randomly distributed, but form chainlets with an average length esti- mated to be ` = 7:5  1:0. Clustering of the chain oxygen atoms during the 200C anneal and an inability to instantly quench the sample to 0C both contribute to chainlet development. Additionally, during the one to two hours required to mount the sample and then evacuate and cool the experimental apparatus, the sample is at room temperature. Recent attempts by Nunner and Hirschfeld to model the conductivity of BSCCO by considering o-plane extended scatterers have been remarkably successful [127]. These authors were motivated by the fact that interstitial dopant oxygen atoms and cation substitution are known sources of o-plane disorder in BSCCO. The model pre- sented in Ref. [127] led to a plausible understanding of the temperature dependence of the quasiparticle conductivity, using defect densities typical of this material. The data sets presented here are ideal for this treatment since the o-plane disorder dom- 39Note that, an average chainlet length of ` = 2 was obtained in the simulation of Fig. 7.5, the results of which are summarized in Table 7.1. 171 7.5. Chain Defects in YBa2Cu3O6:50 inates the quasiparticle transport, but unlike BSCCO this disorder is considerably smaller and can be easily manipulated in a single sample. It is particularly interesting to note that when Nunner and Hirschfeld allow for a signicant forward-scattering component due to o-plane extended scatterers, they nd that 1(!; T ! 0) calcu- lated is higher than the `universal' limit obtained for point scatterers as T ! 0 and that this enhanced conductivity occurs over a wide frequency range. In other words, there is substantial oscillator strength in the conductivity spectrum that does not condense into super
uid as T ! 0. A similar phenomenon has been observed here in YBCO, where one nds a residual oscillator strength at 1.2 K that, while much smaller than that seen in BSCCO, is still larger than expected for point scatterers. Our measurements now establish that in YBCO o-plane disorder associated with defected Cu-O chains provides the main source of weak quasiparticle scattering and forward scattering by these defects is the likely source of the small residual oscillator strength in YBCO. This material is particularly well suited to settling the controver- sial role that defects play in the physics of the cuprates since there is only one source of disorder; weakly-scattering oxygen defects that lie far from the CuO2 planes. 7.5 Chain Defects in YBa2Cu3O6:50 Having established residual Cu-O chain defects as the main source of quasiparticle scattering in YBa2Cu3O6:5, it is natural to ask if the Cu-O chain ordering can be improved to further reduce the quasiparticle scattering rates in this material. Due to the complex nature of the superconducting order parameter, the cuprates are very sensitive to even small amounts of disorder and defects. This fact makes it very challenging to probe and interpret the intrinsic physics of these materials. YBCO is currently by far the cleanest available cuprate superconductor and ner control of the Cu-O chain ordering could lead to some immediate experimental benets40. For example, a reduced scattering rate (or equivalently an increased quasiparticle mean free path) would lead directly to an enhanced quantum oscillation signal which could help to clarify some of the issues regarding the Fermi surface of YBCO. Thus far, quantum oscillations have been observed in YBa2Cu3O6+y for values of y near 0.5. 40YBCO also has its disadvantages. The presence of the Cu-O chains along the b-axis direction complicates the b-axis transport properties of these materials. Also, YBCO does not have a neutral cleavage plane adding signicant complications when applying the powerful surface probe techniques ARPES and STM. Note, however, that recent and ongoing ARPES studies have demonstrated the ability to control the surface doping of cleaved crystals of YBCO by depositing K onto the surface [128]. 172 7.5. Chain Defects in YBa2Cu3O6:50 Having better control of the oxygen ordering could possibly allow for the observation of oscillations in other oxygen-ordered phases of YBCO which correspond to dierent dopings of the CuO2 planes. In addition to describing a way to monitor the oxygen ordering process, this section will identify two Cu-O plane defects (alluded to in the previous section) that could be limiting the perfection of the chain ordering and a technique to eliminate these defects. The work reported here is ongoing and the technique and results continue to be rened by Lynne Semple. The ordered phases of YBCO and the corresponding temperature versus oxygen content phase diagram are shown in Fig. 7.15. The phases are shown for oxygen con- tents such that the idealized ordering has innite chain lengths. There are additional hard-to-access phases not shown. For example, there is an inverse ortho-III phase that is in principle accessible with the application of high pressure. The idealized phase exists at an oxygen content of 6.33 and has periodic arrangement of full-empty-empty chains. The relevant phase for this thesis is the ortho-II phase in YBa2Cu3O6:5 which occurs below  115C. The rate of chain oxygen ordering is strongly temperature dependent and becomes extremely slow below 0C. Figure 7.16 shows the current temperature program used to order the Cu-O chains. The YBa2Cu3O6:5 crystal is initially heated near the ortho-I to ortho-II transition temperature and then cooled to 80C at a rate of 5C/hour. The sample is the held at 80C for 12 hours. This temperature is chosen because it has been experimentally shown to be the optimal temperature for ortho-II domain growth [115]. The sample is then cooled to 60C at 5C/hour and allowed to anneal for an additional 12 hours after which the power to the furnace heating elements is cut. To further optimize the annealing program one wishes to monitor the Cu-O chain ordering in situ. This can be achieved by measuring the resistance of the sample in realtime throughout the annealing process. The resistance measurement requires that the sample have four low-resistance electrical contacts - two current contacts and two voltage contacts. The contacts are made by masking the sample with aluminum foil everywhere except in the locations of the desired contacts. Gold is then evaporated onto the sample, after which the mask is removed. The sample is next annealed at 390C for a few days, allowing the gold to diuse into the sample and thus reducing the contact resistance. To ensure that the oxygen content of the sample is not altered, the sample is annealed inside a sealed quartz tube along with a large piece of ceramic YBCO having the same oxygen content as the sample [129]. The masking of the sample and the gold evaporation was done by Pinder Dosanjh and the contacts were annealed by Ruixing Liang. 173 7.5. Chain Defects in YBa2Cu3O6:50 Figure 7.15: Top: Idealized structural phases of YBa2Cu3O6+x (0  x  1). The ordered phases are drawn with innite chain lengths. Bottom: Experimentally deter- mined temperature versus x phase diagram. For this thesis, the critical phase is the ortho-II phase. For x = 0:5, the transition from ortho-I ordering to ortho-II ordering occurs at  115C. Figures from Ref. [115]. Figure 7.17 shows the a-axis resistance of a YBa2Cu3O6:33 single crystal annealing over a period of three weeks. The sample temperature was regulated and sample resistance measured using a Cryo-Con 62 temperature controller. The resolution of the resistance measurement is limited by the temperature stability of the sample which was one part in 104 as shown in the inset of the gure. The annealing apparatus 174 7.5. Chain Defects in YBa2Cu3O6:50 0 20 40 60 80 20 40 60 80 100 120 12 hrs 12 hrs 80oC 60oC  Te m pe ra tu re  (o C ) Time (hours) -5oC/hr Figure 7.16: Cu-O chain ordering temperature program (actual sample temperature data). 0 100 200 300 400 500 2.5 3.0 3.5 4.0  Sa m pl e R es is ta nc e ( Time (hours) 0 25 50 75 100 125 Sam ple Tem perature ( oC ) 120 125 130 135 140 145 150 155 160 37.994 37.996 37.998 38.000 38.002 38.004 38.006   Te m pe ra tu re  (o C ) Time (Hours) Figure 7.17: YBa2Cu3O6:33 resistance (blue data) and temperature (red data) as a function of annealing time over a period of three weeks. Inset: The sample tempera- ture is stable to within one part in 104. 175 7.5. Chain Defects in YBa2Cu3O6:50 Figure 7.18: Schematic cross-section of the annealing apparatus. See text for a full description. Not shown are the copper block heater and thermometer, the sample support/loading mechanism, the sample current and voltage leads, and the sample temperature thermocouple. is shown schematically in Fig. 7.18. A thin-walled stainless steel tube thermally isolates a copper block from an aluminum plate. The aluminum plate is attached to a copper sheet which is itself attached to copper tubes through which a mixture of water and antifreeze is circulated. The temperature of the aluminum plate can be set to be anywhere between 5C and room temperature. The copper block is equipped with a calibrated diode thermometer and wrapped with manganin heater wire. The copper block temperature is regulated using the Cryo-Con 62 temperature controller. A hole drilled through the copper block accepts a tight-tting quartz tube through which 4He gas 
ows at a rate of 5 sccm41. The gas 
ow was directed such that He 
ows from the sample towards the electrical leads and the sample mounting hardware. This 
ow orientation minimizes contamination of the sample from the wiring and mounting hardware. Helium gas was chosen for its high thermal conductivity. A thermocouple near the sample site conrmed that the sample temperature was the same as the copper block temperature. The entire setup shown in Fig. 7.18 is insulated from surrounding atmosphere of the room using styrofoam insulation. The majority of the apparatus was designed and assembled by Pinder Dosanjh and Antonio Elias. Figure 7.19 shows both a high-temperature and a lower-temperature detailed view 41The volume 
ow rate unit sccm stands for standard cubic centimeters per minute. 176 7.5. Chain Defects in YBa2Cu3O6:50 15.0 17.5 20.0 22.5 25.0 80 82 84 86 88 Time (hours) 3.40 3.45 3.50 3.55 3.60 Sa m pl e R es is ta nc e ( ) Sam ple Tem perature ( oC ) Sam ple Tem perature ( oC )S am pl e R es is ta nc e ( ) 18 20 22 24 26   300 320 340 360 380 400 2.34 2.36 2.38 2.40 2.42 Time (hours) Figure 7.19: Detailed views of the sample resistance during annealing. Top: At high temperatures the Cu-O chain oxygen atoms are very mobile and an equilibrium resistance is established within a fraction of an hour. Bottom: At lower temperatures the ordering is much slower and an equilibrium resistance is not established even after continuous annealing for several days. of the data of Fig. 7.17. In both detailed views, when the temperature is decreased, there is a sudden increase in the resistance re
ecting the temperature dependence of the sample's resistivity. After the jump in resistance, there is a slow decrease in 177 7.5. Chain Defects in YBa2Cu3O6:50 the resistance due to the ordering of the Cu-O chains. The ordering rate is strongly temperature dependent. At relatively high temperature an equilibrium resistance is reached in a fraction of an hour. At equilibrium the ordering of the Cu-O chains competes with and is balanced by random thermal motion of the mobile oxygen atoms. Further ordering can only be achieved by lowering the temperature. However, at low temperatures equilibrium is not established even after several days of annealing. Although the data presented are from a YBa2Cu3O6:33 with 1=3 of the Cu-O chain oxygen sites occupied, similar data are currently being obtained from YBa2Cu3O6:5 crystals. In principle, these (and similar) data could be used to optimize the eciency of the Cu-O ordering process. However, it is not obvious whether or not the ultimate Cu-O chain ordering can be substantially improved. For example, simply allowing the sample to remain at room temperature in a dry atmosphere for several years (which is not particularly uncommon), although certainly inecient, may leave the Cu-O chains as well ordered as any annealing program that includes a uniform sample tem- perature. There are, however, two types of defects that may be preventable using a more sophisticated annealing scheme. During an anneal ortho-II domains can \seed" from more than one point in the sample. These domains will grow and eventually meet, possibly creating a mismatch in the full-empty-full-empty-. . . ordering at a do- main boundary as pictured in Fig. 7.20. These are clearly large defects that could account for a signicant fraction of the scattering of CuO2 plane quasiparticles. These types of defects have been observed in images obtained from STM measurements on YBa2Cu3O6:5 single crystals [130] and their removal could be the most signicant way to improve the current generation of samples. An annealing scheme to prevent these types of defects from forming is pictured schematically in Fig. 7.21. The average temperature of the sample is set as described in Fig. 7.18, however, the sample is now mounted on a thin ceramic alumina block using silicone grease. The sample is arranged such that its diagonal is aligned along the length of the alumina block. Heat is applied at one edge of the alumina block to establish a temperature gradient along the sample's diagonal. Then, as the average temperature is lowered, the ortho-II phase will develop rst at the cool corner of the sample and then propagate across the sample diagonal and thus avoiding defects from mismatched domains. The gure also shows the sample with four gold contacts at the corners. This conguration of contacts allows one to employ the van der Pauw method to measure both the a- and b-axis resistances. In this method, one uses switches to select dierent combinations of current and voltage contacts. For a thin 178 7.5. Chain Defects in YBa2Cu3O6:50 (a) (b) Figure 7.20: Two possible ortho-II domain boundary defects. The b-axis is along the horizontal direction. platelet sample, one can then independently determine the sample resistance in both the a- and b-axis directions [131]. Jordan Baglo has updated the data collection soft- ware to accommodate the equipment required to employ the van der Pauw method. Measurements on a YBa2Cu3O6:5 single crystal are currently ongoing with early in- dications that a distinct feature in the data marks the ortho-I to ortho-II transition temperature [132]. 179 7.5. Chain Defects in YBa2Cu3O6:50 Figure 7.21: The sample is annealed in a temperature gradient using a ceramic alu- mina block and chip heater. In this scheme, the ortho-II ordering will start at the cool corner and propagate along the sample diagonal. The data presented in this chapter clearly demonstrate that the quasiparticle scat- tering rates in the current generation of ortho-II ordered YBa2Cu3O6:5 single crystals are set, not by cation purity, but by residual disorder in Cu-O chain layer. The most signicant improvement in the samples will likely come from ner control of the oxy- gen ordering process. In particular, if mismatched ortho-II domain boundaries are prevalent in the crystals, their removal will almost certainly lead to a dramatic re- duction in the low-temperature quasiparticle scattering rates. Quantum oscillations, which experimentally probe the Fermi surface of these materials, would immediately benet from the increased mean free paths. 180 Chapter 8 Microwave Electrodynamics of K- and Co-doped BaFe2As2 8.1 Introduction Tremendous interest was generated when the 
uorine-doped layered compound LaFeAsO1xFx was reported to superconduct at 26 K [63]. In remarkably short or- der, the critical temperature of this compound was increased via pressure or chemical substitution to above 55 K, signicantly higher than the highest Tc reported in any s- wave superconductor (i.e. MgB2 with Tc = 39 K [133]). Superconductivity has since been revealed in Ba1xKxFe2As2 with Tc;max = 38 K [134] and in Ba(Fe1xCox)2As2 with Tc;max = 23 K [135]. These so-called 122 compounds are particularly important since, unlike the cuprates or the 1111 iron pnictides, they are not oxides, and there- fore the potentially problematic role of oxygen stoichiometry is downplayed in this family. Moreover, large single crystals with a variety of dierent cation dopings in the 122 pnictides have now been synthesized, which is essential for applying a wide range of measurements to probe their physical properties. The need to understand the pairing mechanism and the origin of the high Tc in any new superconductor drives a need to determine the symmetry of the order parameter, however this usually re- quires several dierent measurements to arrive at a consensus. These measurements need to be performed on single-phase samples with well-characterized stoichiometry and sharp transitions, which imply good homogeneity; otherwise, it is challenging to compare one measurement to the next. A powerful class of measurements that can be thought of as topological include the 
ux quantization measurements (for example, Ref. [136]) that show superconduc- tors are a condensate of pairs and the observation of half 
ux quanta in geometrically frustrated junctions [137] which was decisive in proving the extra broken symmetry in the dx2y2 state of the cuprates. Another group of measurements directly probe the superconducting pairing gap via spectroscopic means. This was famously the case in conventional s-wave superconductivity in which measurements such as tunnel- 181 8.1. Introduction ing [138], infrared [139, 140] and microwave spectroscopy [141] showed a well-dened gap with a sharp threshold energy and essentially no states below the energy gap at low temperatures and in the absence of pair-breaking magnetic impurities. Such measurements were more ambiguous in the cuprates because the presence of nodes in the dx2y2 pairing state gave a characteristic energy gap scale, but without the very sharp threshold and with many states available down to low energies. Angle- resolved photoemission [142, 143] helped resolve this by showing the gap variation as a function of momentum around the Fermi surface. Very many other measurements rely on inferring the presence of a gapped spec- trum of excitations in the system by observing the temperature dependence of a wide range of properties, including thermodynamic, transport, and electrodynamic prop- erties. The electrodynamic properties will be the focus here, but rst an introductory comment on the diculties of making inferences from temperature dependencies in these properties. In conventional s-wave superconductivity, the presence of exponen- tially activated behavior in many properties at low temperatures signals the presence of a nonzero minimum energy gap and hence no nodes. Such measurements [144, 145] even predate BCS theory, but it has always been dicult to do this decisively since it takes high resolution data, preferably over a few decades of the low temperature exponential behavior, to be convincing. This has, for instance, been achieved in high resolution measurements of the temperature dependent microwave loss of high-Q res- onant cavities made of Pb [146]. In the case of a superconducting state with nodes, the temperature dependencies tend towards various power laws. Here there is a signicant challenge in identifying the particular state, or even in being sure that it is not really exponentially activated. One case has proven relatively easy: the line nodes of a dx2y2 pairing state on a cylin- drical Fermi surface gave rise to an unambiguous linear temperature dependence in the London penetration depth [49]. Unfortunately, disorder quickly changes this to a quadratic temperature dependence [120], so that in materials with pair breaking defects, especially cation doping, these techniques again place high demands on res- olution and careful comparison of power laws versus exponentials, plus considerable systematic work on sample dependence and multiple materials within a family. An example of this can be seen in the long eort to understand the penetration depth in the electron-doped cuprates Pr2xCexCuO4 and Nd2xCexCuO4 [147]. Universal consensus regarding the gap symmetry in the iron-based superconduc- tors does not currently exist, but the eld is working hard towards remedying this situation. For example, it is known that the antiferromagnetic ground state in the 182 8.2. Materials and Methods BaFe2As2 parent compound is suppressed through doping, thus allowing superconduc- tivity to emerge [135, 148]; being near to a magnetic state might mean that magnetic 
uctuations are important for pairing and this could be re
ected in the symmetry of the gap. What, then, is the pairing symmetry? Band structure calculations [149, 150] and ARPES experiments [151, 152] demon- strate that multiple bands cross the Fermi surface, making multi-band superconduc- tivity plausible. For the hole-doped Ba1xKxFe2As2 compound, ARPES [69, 153, 154] has reported evidence for at least two dierent superconducting nodeless gaps in the ab-plane. These results are further supported by directional point-contact Andreev- re
ection spectroscopy [155] and microwave surface impedance [156] data which sug- gest fully-, and perhaps multiply-, gapped superconductivity. However, the possibil- ity for a nodal gap has not been completely ruled out: measurements of reversible magnetization [157] and thermal Hall conductivity [158] both yield results consistent with nodes in the gap; 75As nuclear magnetic resonance measurements [159] revealed the spin-lattice relaxation rate 1=T1 to vary close to T 3; and, muon spin-relaxation measurements [160] exhibit a nearly linear variation in temperature of the super
uid density at low temperatures. For the electron-doped Ba(Fe1xCox)2As2 compound, heat transport measurements [161] suggest a nodeless superconducting gap in the ab-plane. However, tunnel diode resonator techniques [162] have revealed that the penetration depth as a function of temperature exhibits a robust power law (instead of exponential), with (T ) / T n and n being between 2 and 2.5, depending on the doping level. The question of pairing symmetry remains open. 8.2 Materials and Methods This work reports measurements of the temperature dependence of the London penetration depth and surface resistance in the hole-doped Ba0:72K0:28Fe2As2 and the electron-doped Ba(Fe0:95Co0:05)2As2 122 compounds. Measurements of three high-quality single crystals are reported (two of Ba0:72K0:28Fe2As2 and one of Ba(Fe0:95Co0:05)2As2), which were grown using an FeAs self-
ux method by Hai-Hu Wen and coworkers at the National Laboratory for Superconductivity in Beijing, China [163]. Note that these crystals dier in that the K-dopant occupies out-of- plane interstitial sites in the crystal lattice, whereas Co substitutes for Fe in the Fe2As2 plane. The sample dimensions and superconducting transition temperatures of the crystals studied are given in Table 8.1. We note that the K-doped samples A and B are in the doping range where the coexistence of a spin density wave and 183 8.2. Materials and Methods Sample Tc ab-surface c-axis aspect (K) (mm2) (m) ratio (Ba,K)Fe2As2 (A) 29.5 0.928 50 20 (Ba,K)Fe2As2 (B) 28 0.508 8 80 Ba(Fe,Co)2As2 (C) 20 1.017 7 140 Table 8.1: The transition temperatures and dimensions of the iron arsenide crystals studied. superconductivity has been reported [164], and sample C has a doping level which is very close to where antiferromagnetism has been observed to coexist with supercon- ductivity [135, 165]. Photographs of the three single crystals are shown in Fig. 8.1. The microwave techniques at UBC are optimized for 1 mm2 platelets and the sam- ples used in these measurements were carefully selected to be the best single crystals available. The sample surfaces are known to degrade from prolonged exposure to ambient atmosphere42. To limit surface degradation, only samples with cleaved ab- surfaces have been measured and between measurements the samples were stored in a vacuum desiccator. Furthermore, these samples have little secondary impurity phase (less than 10%) as checked by specic heat [166]. Sample quality and homogeneity were conrmed via the width of the superconducting transition as a function of an applied dc eld, as shown in Fig. 8.2. The magnetic moment m of sample A was measured in dc magnetic elds applied parallel to the c-axis of the crystal. In low elds (< 10 G), Tc = 30:1 K and Tc < 0:5 K. At 5 tesla, Tc is suppressed by 15% to 25.6 K, but the transition width remains narrow (Tc < 1:5 K): a clear signature of a homogeneously doped sample. Note that the data in the gure have been scaled and oset for clarity. The sample was cooled in zero magnetic eld for the two lowest eld measurements (dashed lines). At higher elds, the sample was cooled in eld (solid lines). For the eld-cooled measurements the sample will trap 
ux when entering into the superconducting state and, as a result, the step in the magnetic moment at Tc is not as pronounced as it is for a zero-eld cooled measurement (compare, for example, the 10 G and 0.1 T curves). As the applied magnetic eld is increased further, the signal-to-noise ratio of the eld-cooled data improves. The 940 MHz loop-gap resonator, described in chapter 4, was used to measure 42Note that the silicone grease used to mount the samples to the sapphire plates eectively coats the sample surfaces and may actually help to mitigate surface degradation. For example, during the time that it takes to mount the sample and hermetically seal the apparatus the samples are exposed to the ambient atmosphere of the room. 184 8.2. Materials and Methods Figure 8.1: Digital photographs of the Ba0:72K0:28Fe2As2 and Ba(Fe0:95Co0:05)2As2 single crystals studied. In the images along the top row the samples are shown mounted on sapphire plates. The photographs of the ab-planes have been scaled to accurately show the relative sizes of the samples. For reference, the sapphire plate is 500 m wide. The edge-view photographs are not on a common scale. For reference, the sapphire plate is approximately 100 m thick. Some of the sample dimensions are given in Table 8.1. The two bottom photographs show that freshly cleaved surfaces are 
at and have mirror-like nishes. The structure seen on the surfaces of the upper images is silicone grease used for mounting the samples. the temperature dependence of the magnetic penetration depth (T ). Microwave magnetic elds are applied parallel to the ab-plane, a geometry in which the magnetic eld at the surface of the sample is almost everywhere equal to the applied eld, and the cavity resonance frequency is measured as a function of the sample temperature. In this geometry, screening currents in the crystal 
ow in both the a- and c-axis di- rections. To limit the eects of c-axis contamination, platelets with large a:c aspect ratios are preferred. When the system is run as an oscillator (see x4.6), an absolute frequency stability of  0:1 Hz/min and sub-angstrom resolution in (T ) are ob- 185 8.3. Penetration Depth 0 10 20 30 40 -0.8 -0.4 0.0 0.4  Field cooled  Zero field cooled 1 G 10 G 0.1 T 1 T   m  (1 0- 4  e m u) Temperature (K) 5 T Figure 8.2: SQUID magnetometer measurement of the magnetic moment m of the thick Ba0:72K0:28Fe2As2 sample (sample A) from low to high magnetic eld. Except for the lowest eld, m has been scaled and oset for clarity. Even at 5 T, where Tc has been suppressed by 15%, the superconducting transition remains narrow (Tc  1:5 K), an indication that the sample is homogeneous. tained. A pumped 4He cryostat was used to reach 1.2 K and measurements down to 400 mK were made using a 3He pot and a charcoal sorption pump. Surface resistance RS(!; T ) measurements were made using the microwave spec- trometer described in chapter 5. As with the penetration depth measurements, mi- crowave magnetic elds are applied parallel to the ab-plane. As previously described in detail, the microwave power is modulated at low frequency and the measured temperature oscillation of the superconducting sample gives a direct measure of the absorbed power. A reference alloy of Ag:Au, placed in an electromagnetically equiv- alent position, is used to calibrate the surface resistance of the FeAs crystal. These spectroscopic measurements were limited to base temperatures of 1.2 K. 8.3 Penetration Depth The measured change in the London penetration depth of all three samples is shown in Fig. 8.3 as a function of temperature. The background frequency shift due to an empty sapphire plate with a small amount of silicone grease (approximately equal 186 8.3. Penetration Depth 0 5 10 15 20 25 30 0.0 0.5 1.0 1.5   (A)  Ba 0.72 K 0.28 Fe 2 As 2 (B)  Ba 0.72 K 0.28 Fe 2 As 2 (C)  Ba(Fe 0.95 Co 0.05 ) 2 As 2  ( m ) Temperature (K) Figure 8.3: The change in the London penetration depth, (T ), as a function of temperature for all three iron-arsenide samples studied. The dashed lines connect the points and are guides to the eye. to the amount used to hold the sample in place) was measured. The background frequency shift data showed no systematic temperature dependence and its magnitude corresponded to less than 1 A in  (for T  2 K). The penetration depth of both K-doped crystals (samples A and B) has also been measured using the 12 kHz ac susceptometer, described in chapter 4. To within calibration uncertainties, the results agree with the 940 MHz cavity perturbation measurements. Specically, the same temperature dependence for  was extracted from both sets of measurements. The quantity 1=2(T ) is directly proportional to the super
uid density and the ratio 2(0)=2(T ) is the super
uid density normalized to one for T ! 0 K43. This ratio is plotted in Fig. 8.4 for all three samples both as a function of T and T=Tc. The absolute penetration depth (0) must be obtained from an independent mea- surement. For the analysis that follows, (0) = 2000 A is used and was obtained from an infrared spectroscopy study on a single crystal of Ba0:6K0:4Fe2As2 [167]. It is important to note, however, that none of the results presented in this chapter depend sensitively on the value of (0). When the super
uid density data are shown with 43Equivalently, as discussed in x4.1, the ratio 2(0)=2(T ) is the super
uid fraction xs(T ). 187 8.3. Penetration Depth 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (A)  Ba0.72K0.28Fe2As2 (B)  Ba0.72K0.28Fe2As2 (C)  Ba(Fe0.95Co0.05)2As2           (0)=2000 Å   2 (0 )/ 2 (T ) T/TC  (0)=2500 Å  (0)=2000 Å  (0)=1500 Å 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0   2 (0 )/ 2 (T ) T (K) Figure 8.4: Top: Extracted super
uid density of all three samples. Curves formed by symbols were generated using (0) = 2000 A. Bottom: When the temperature axis is scaled by Tc, the super
uid density of all three samples collapse onto a single curve. For comparison, the dashed and dotted lines show the super
uid density of the Co-doped sample using (0) = 1500 A and (0) = 2500 A respectively. 188 8.3. Penetration Depth the temperature axis scaled by Tc, all three data sets to collapse onto a single curve. The dashed and dotted lines show the super
uid density of the Co-doped sample when using (0) = 1500 A and (0) = 2500 A respectively. The overall qualitative features of the super
uid density are not aected by the choice of (0). The T=Tc scaling of the 2(0)=2(T ) curves is interesting because the doping of the two types of samples studied (electron-doped and hole-doped) is achieved via two very dierent substitutions. In the Co-doped materials, the Co substitutes for planar Fe cations and one expects strong Cooper-pair-breaking scattering to occur. In contrast, in the K-doped materials, K substitutes for o-plane Ba cations which would presumably result in much weaker (small scattering phaseshift) in-plane scattering. Section 1.5 introduced the Ginzburg-Landau mean eld theory of superconduc- tivity which is valid near Tc. One key prediction of this theory is that as T ! Tc, the super
uid density 1=2(T ) should approach zero linearly. This is the expected behaviour provided that one is not in the temperature regime where 
uctuations of the order parameter become important. As mentioned previously, this regime is ac- cessible in YBa2Cu3O6:95 and critical behaviour has been observed over several orders of magnitude of the reduced temperature 1 T=Tc [16]. Figure 8.5 shows a zoomed in view of 2(0)=2(T ) for all three iron-arsenide samples near Tc. The super
uid density of all three samples approaches zero linearly as a function of temperature, a signature of mean eld behaviour. Extrapolating ts to the linear portions of the data to zero gives a systematic way to determine the Tc of the samples. The region between where the t extrapolates to zero and where the data go to zero is another indication of sample homogeneity. For all three samples measured this region spans 1 K or less. For comparison, in other published measurements of the super
uid den- sity of Ba1xKxFe2As2 single crystals this region is as large as 20-30% of Tc [156, 168]. A large deviation from the linear behaviour is likely an indication of inhomogeneous samples with a relatively wide range of transition temperatures distributed through- out the sample. It is the low-temperature behaviour of (T ) that gives key insights into the structure of the superconducting gap. For example, in a clean d-wave superconductor (T ) / T typically holds up to approximately 20% of Tc. Figure 8.6 shows (T ) for all three samples for T < 0:20Tc. From these data, it is immediately clear that (T ) is not linear in temperature as in clean d-wave superconductors like the HiTc cuprates. Nor does (T ) follow the temperature dependence of a conventional BCS 189 8.3. Penetration Depth 16 20 24 28 0.00 0.05 0.10 0.15 0.20 T C  = 29.4 KT C  = 27.3 KT C  = 20.0 K (A)  Ba 0.72 K 0.28 Fe 2 As 2 (B)  Ba 0.72 K 0.28 Fe 2 As 2 (C)  Ba(Fe 0.95 Co 0.05 ) 2 As 2 (0)=2000 Å   2 (0 )/ 2 (T ) T (K) Figure 8.5: Near Tc, the super
uid density 1= 2(T ) approaches zero linearly indicative of mean eld behaviour. The lines are ts to the linear parts of the data. Extrapo- lating the ts through to the temperature axis is one systematic way to determine Tc of each sample. superconductor, which in the low-temperature limit, is given by [76]: (T ) (0)  r 20 kBT exp 0 kBT  : (8.1) This model generates a poor t to the data and returns unreasonable best t param- eters: (0) = 82 7 A and 2(0)=kBTc = 0:81 0:03. Recall that for a conventional BCS superconductor one typically expects 2(0)=kBTc  3:5 such that (T ) is nearly 
at below 0:4Tc. The dotted line in the gure gives the weak-coupling BCS result for (0) = 2000 A and Tc = 29:4 K. Next, two models that can be used to t the data reasonably well are explored. One is a simple power law model and the second is an s-wave model that allows for two superconducting gaps. Both of these models are motivated by physically realizable scenarios. Power law temperature dependencies of the super
uid density arise when there are nodes in the superconducting gap along certain directions in momentum space. These nodes lead to a nite density of states of quasiparticles 190 8.3. Penetration Depth 0 1 2 3 4 5 6 0 10 20 30 40   (A)  Ba 0.72 K 0.28 Fe 2 As 2        s-wave fit (B)  Ba 0.72 K 0.28 Fe 2 As 2 (C)  Ba(Fe 0.95 Co 0.05 ) 2 As 2  (Å ) Temperature (K) weak-coupling BCS Figure 8.6: (T ) of the three 122-iron-arsenide samples for T < 0:2Tc. The solid red line is an s-wave t to the data of sample A. This model does not adequately describe the data over any temperature range. The weak-coupling BCS result using (0) = 2000 A is shown as the dotted red line. at all temperatures. The precise temperature dependence of the DOS, and hence 2(0)=2(T ), depends on the structure of the nodes and the purity of the sample. For example, see Ref. [169] for a discussion of the power laws associated with dierent theoretical scenarios that compete with or modify a pure dx2y2 gap. As previously discussed, in a clean d-wave superconductor (two-dimensional cylindrical Fermi sur- face with line nodes) (T ) / T . For the same superconducting gap structure in the dirty limit, (T ) / T 2 [120]. There are other exotic nodal gap structures pos- sible which could include point nodes, line nodes, three dimensional Fermi surfaces, and/or multiple Fermi sheets. Each scenario requires a detailed theoretical analysis in the clean and dirty limits to uncover the expected temperature dependence of the super
uid density. The two-gap scenario has been most convincingly demonstrated in the material MgB2. Band structure calculations predict multiple bands: quasi-two-dimensional tubes with their axes parallel to the c-axis in addition to a set of more three- dimensional bands [170]. With only weak scattering between the two types of bands it is possible to have two distinct superconducting energy gaps associated with the 191 8.3. Penetration Depth sets of bands that, however, share a common transition temperature Tc [35]. The total measured super
uid density is then the sum of the contributions from the two gaps [171, 172]. For two s-wave gaps, the low-temperature expansion of 1=2(T ) can be expressed as: 2(0) 2(T )  1 x r 20;S kBT exp 0;S kBT   (1 x) r 20;L kBT exp 0;L kBT  ; (8.2) where x = 2(0)=2S(0) is the fractional contribution of the small gap 0;S to the total super
uid density and: 1 2(0) = 1 2S(0) + 1 2L(0) : (8.3) In any of these more complicated scenarios, penetration depth measurements on their own will not lead to a denite determination of the structure of the superconducting gap. A consensus in the research community will be reached only after a broad range of experimental measurements come together to form a consistent picture of the gap structure. One should be cognisant of the fact that this consistency may only be achieved after single crystals of sucient quality become widely available. There is, however, an enormous eort underway to produce single crystals of all types of pnictide superconductors and the sample quality is improving at an astounding rate. There is, of course, the possibility that the superconducting gap structure could vary between the dierent families of pnictide materials. See, for example, Ref. [173]. Figure 8.7 shows the low-temperature super
uid density of sample C (Co-doped) t to Eq. 8.2 and to the power law model: 2(0) 2(T ) = 1  T T ? n : (8.4) Both of the models t the data well. Fits to the K-doped samples (samples A and B) are equally good and are not shown. The t parameters obtained from all three samples using both models are summarized in Table 8.2. Note that for all three samples, the two-gap s-wave model t parameters (x, 0;S, and 0;L) all systematically increase as the t range is increased. Moreover, x is very small resulting in unbalanced contributions to 1=2(0) from the small and large gaps, and the gap values are always smaller than the BCS weak-coupling result: 20;(L;S)=kBTc < 3:5. These are perhaps all indications that the s-wave two-gap scenario is not the correct picture for the 122 iron-arsenide crystals. Alternatively, if interband coupling is strong, a multiple gap scenario could persist, however, it may not be appropriate to 192 8.3. Penetration Depth 0 1 2 3 4 0.970 0.975 0.980 0.985 0.990 0.995 1.000    Ba(Fe 0.95 Co 0.05 ) 2 As 2  power law fit  2-gap fit 2 (0 )/ 2 (T ) Temperature (K) Figure 8.7: The low temperature super
uid density of the Co-doped sample t to a power law model (Eq. 8.4, dashed line) and to a two-gap s-wave model (Eq. 8.2, dotted line). Both of the models t the data equally well. treat the gaps independently. The power law t, on the other hand, seems to be more robust against changes in the t range (sample B in particular). However, there is some variation in value of the power among the three samples with n ranging between 2.1 and 2.7. These results are consistent with similar, but independent, measurements made on single crystals of the same compounds by the Prozorov group at Ames Laboratory of Iowa State University. See, for example, reference [162]. A detailed explanation of the complete temperature dependence of 1=2(T ) for a system with a complicated Fermi surface, and perhaps with nodes only on certain sections of the Fermi surface, will have to await further developments. However, the consistency of the present results for samples that appear to be very homogeneous, but with dierent doping mechanisms, gives condence that the observed temperature dependence is close to the intrinsic behavior at these doping levels. Finally, attention is now shifted to very low temperatures where an unexpected feature in (T ) emerged in all three samples. Figure 8.8 shows the raw frequency shift f of sample B for T < 2:2 K. In the range from 1-1.3 K, there appears an un- expected feature, or bump, in the measured frequency shift. Initially, it was thought 193 8.3. Penetration Depth Sample A - Tc = 29:4 K two-gap power law t range x 20;S kBTc 20;L kBTc n T ? (K) Tmax (K) Tmax=Tc 4 0.14 1.2% 0.34 1.85 2.22 25.8 5 0.17 1.8% 0.41 2.09 2.25 24.9 6 0.20 2.5% 0.48 2.27 2.31 23.7 7 0.24 3.4% 0.54 2.42 2.35 23.0 Sample B - Tc = 27:3 K two-gap power law t range x 20;S kBTc 20;L kBTc n T ? (K) Tmax (K) Tmax=Tc 4 0.15 1.2% 0.35 1.99 2.11 27.9 5 0.18 1.8% 0.43 2.30 2.11 28.1 6 0.22 2.5% 0.50 2.55 2.09 28.5 7 0.26 3.3% 0.57 2.74 2.12 27.9 Sample C - Tc = 20:0 K two-gap power law t range x 20;S kBTc 20;L kBTc n T ? (K) Tmax (K) Tmax=Tc 3 0.15 1.9% 0.55 2.20 2.73 14.1 4 0.20 2.7% 0.62 2.50 2.60 15.4 5 0.25 4.0% 0.71 2.75 2.53 16.2 Table 8.2: The power law and two-gap low-temperature super
uid density t param- eters for all three samples. The ts were performed over various temperature ranges. For all three samples the two-gap t parameters systematically increase as the t range is increased. that the bump might be due to a contaminant stuck either to sample surface or sap- phire plate. For example, aluminum has a superconducting transition temperature near 1.2 K. Every eort was made to clean the sample and sapphire plate. To en- 194 8.3. Penetration Depth 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (C)  Ba(Fe0.95Co0.05)2As2        unloaded Sapphire   f (H z) Temperature (K) Figure 8.8: The measured frequency shift of sample B for T < 2:2 K (blue triangles). Also shown is the frequency shift due to an unloaded (no sample) sapphire plate (grey squares). The observed feature in the sample frequency shift at 1.0-1.3 K is well above the noise threshold. sure that the observed feature was due to the sample, the frequency shift due to the sapphire plate in the absence of a sample was measured. These data are shown as grey squares in the gure. There is no systematic frequency shift due to the sapphire plate. Note that these data have a lower signal-to-noise ratio than the blue sample data only because of reduced averaging. After cleaning and inspecting the sample the frequency shift data were reacquired and the bump spanning 1-1.3 K remained. Careful low-temperature measurements of the other two samples reveal that a similar feature also exists in these samples, although not as prominent as in sample B. The data are shown in Fig. 8.9. When the temperature axis is scaled by Tc, the feature in all three samples occurs at the same value, namely T=Tc  0:04. This feature appears to be associated with a low-energy anomaly in the superconducting density of states, however, at this time its origin or cause is unknown. 195 8.3. Penetration Depth 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 (A)  Ba0.72K0.28Fe2As2 (B)  Ba0.72K0.28Fe2As2 (C)  Ba(Fe0.95Co0.05)2As2    (Å ) Temperature (K) 0.00 0.02 0.04 0.06 0.08 0.10 0 2 4 6 8 10    (Å ) T/TC Figure 8.9: Top: The low-temperature feature in (T ) appears in all three samples, although it is most prominent in sample B. For clarity, the sample A data has been shifted by 1 A and the sample B data by 2 A. Bottom: When the temperature axis is scaled by Tc, the bumps all occur at the same position: T=Tc  0:04. 196 8.4. Microwave Spectroscopy 8.4 Microwave Spectroscopy This section, and x8.5, present the microwave surface resistance of the three iron- arsenide samples studied. The RS(!; T ) spectra of the two Ba0:72K0:28Fe2As2 samples A and B are shown in Fig. 8.10. There is a striking dierence in the low temperature microwave loss of the two samples; at 20 GHz, the low temperature RS(!; T ) of the thick sample A is roughly 3.5 times larger than that of the thin sample B. How- ever, the absolute temperature evolution of RS(!; T ) is the same for both samples as highlighted by the plot of RS(!; T ) = RS(!; T )  RS(!; T0) in Fig. 8.11. Here T0 is the lowest measurement temperature, T0 = 1:3 K for sample A and T0 = 1:8 K for sample B. This analysis suggests that sample A (the thick sample) has an excess temperature-independent residual loss. The origin of this excess loss is unknown, but several possibilities are discussed in the next section. It is almost a certainty that the measured RS(!; T ) of the thinner crystal (sample B) is closer to the intrinsic loss in these materials, but is not necessarily free of extrinsic eects. In chapter 7, the broadband apparatus was applied to ortho-II YBa2Cu3O6:5 [94]. In that material, the doping of the CuO2 planes is controlled by the oxygen content and ordering of o-plane Cu-Oy chains. The ortho-II ordered half lled system is particularly special because the dopant oxygen atoms are arranged into alternating full and empty chains and have a high degree of crystalline order. As discussed, the conductivity spectra of YBa2Cu3O6:5, extracted from RS(!; T ), exhibit the essential features expected for a clean d-wave superconductor in the limit of dilute weak (small scattering phase shift) point-like scatterers [120, 121]: the spectra have cusp-like line shapes, high frequency tails that fall o slower than !2, and spectral widths that evolve linearly with temperature. Additionally, in this material, residual defects in the ordering of the Cu-Oy chains has been identied as the dominant source of quasiparticle scattering [123]. In the 28% K-doped Ba0:72K0:28Fe2As2 samples it is very likely that the K-dopants are the dominant source of quasiparticle scattering. With such a high concentration of defects, it is reasonable to expect the spectral widths of the quasiparticle conductivity spectra to be well outside of our measurement bandwidth ( 26:5 GHz). In this case, the conductivity, which to a very good approximation is given by: 1(!; T )  2RS(!; T )=20!23(T ); (8.5) would appear frequency independent and thus RS(!; T )  !2. In Fig. 8.10, the highest temperature RS(!; T ) spectra are compared to ! 2. For sample A, the data 197 8.4. Microwave Spectroscopy 0 5 10 15 20 0.0 0.3 0.6 0.9 1.2  Frequency (GHz)  14.5 K  10.1    7.3    5.0    2.9    1.3 R S  (m ) Ba0.72K0.28Fe2As2 Sample A 0 5 10 15 20 0.0 0.3 0.6 0.9 1.2   R S  (m )  Frequency (GHz)  15.0 K    7.4    5.1    3.0    1.8 Ba 0.72 K0.28Fe2As2 Sample B Figure 8.10: Top: RS(!; T ) spectra of Ba0:72K0:28Fe2As2 sample A. The solid line follows !2. Bottom: RS(!; T ) spectra of Ba0:72K0:28Fe2As2 sample B. Again, the solid line follows !2. clearly do not follow the line. Sample B, on the other hand, which is expected to be nearer to the intrinsic loss, follows !2 much more closely. For a temperature independent residual loss, RS(!; T ) / !2 would be expected for both the thick and 198 8.4. Microwave Spectroscopy 1 10 1E-5 1E-4 1E-3 0.01 0.1 1 Sample A    Sample B        R S (T )= R S (T )- R S (T 0)   ( m ) Frequency (GHz) Figure 8.11: RS(!; T ) = RS(!; T )  RS(!; T0) for sample A (solid symbols) and sample B (open symbols). RS(!; T ) is seen to be the same for both samples in- dicating that sample A (the thicker of the two samples) has an extra temperature independent loss. The dashed lines have a slope of 2 indicating that RS / !2. thin samples. This frequency dependence is clearly demonstrated by the dashed lines in Fig. 8.11 for both samples and at all temperatures. Despite falling well outside the measurement bandwidth, an estimate of the spec- tral width (T ) can be obtained by comparing the normal 
uid density nne 2=m? to the loss of super
uid density 1=2(0)  1=2(T ). By making use of the oscillator strength sum rule and the two-
uid model (see x3.2), the super
uid density and the integrated quasiparticle conductivity can be related as follows: 1 0  1 2(0)  1 2(T )  = 2  Z 1 0 1(!; T )dT: (8.6) For a broad conductivity with a 
at region of magnitude 0 that extends from zero frequency out to a spectal width , the right hand side of this expression can be approximated44 as  0. This approximation allows one to estimate the spectral 44In the Drude model, for example, in which 1 = 0=  1 + (!=)2  , the integral in Eq. 8.6 evaluates exactly to 0. Alternatively, consider that the area beneath the conductivity spectrum will be given approximately by the height 0 of the curve multiplied by its width . 199 8.4. Microwave Spectroscopy width of the quasiparticle conductivity using:  2  1 200  1 2(0)  1 2(T )   1 00 (T ) 3(0) ; (8.7) where the division by 2 gives the desired result in frequency units. The second form given (far right hand side of the equation) is valid in the limit that (T ) (0). It is important to note from Eq. 8.5 that since 1(!; T ), and hence 0, are proportional to 3(T ), the estimate of  is, to within a rst order approximation, independent of the value of (0). Since sample B is considered to be the more intrinsic of the two K-doped samples, it will be used to obtain an estimate of (T ). At T = 15 K and !=2 = 20 GHz, RS  0:95 m  and from the penetration depth measurements of the previous section (15 K)  420 A. These measurements together with Eq. 8.5 give an upper bound for 0 . 5:4106 1m1 since the surface resistance likely includes a small extrinsic contribution. A lower bound for 0 can be had by using RS = 0:75 m  from Fig. 8.11 which has likely had some intrinsic loss removed due to the subtraction of RS(!; T0). Taken together one has 4:2 . 0 . 5:4  106 1m1. Finally, using Eq. 8.7, the spectral width of the K-doped samples is estimated to be in the range 250 . =2 . 320 GHz, which lies outside the bandwidth of the microwave spectrometer by an order of magnitude. This analysis was repeated for sample B at all but the lowest measurement temperature and the results are summarized in Table 8.3 and Fig. 8.12. T (T ) RS (m ) RS (m ) 0 =2 (K) (A) @ 20 GHz @ 20 GHz (106 1m1) (GHz) 15.0 422 0.95 0.74 4:2 . 0 . 5:4 250 . =2 . 320 7.4 65 0.31 0.098 0:89 . 0 . 2:8 73 . =2 . 230 5.1 27 0.26 0.044 0:42 . 0 . 2:5 34 . =2 . 200 3.0 8.7 0.23 0.011 0:11 . 0 . 2:3 12 . =2 . 250 Table 8.3: Estimates of the quasiparticle conductivity spectral width of Ba0:72K0:28Fe2As2 sample B at four measurement temperatures. At the low tem- peratures, the raw RS measurements almost certainly dominated by the extrinsic loss and the estimates of 0 are unrealistically large leading to unrealistically small values of . 200 8.4. Microwave Spectroscopy 0 100 200 300   /2  (G H z) 0 4 8 12 16 Temperature (K) Figure 8.12: Estimates of the quasiparticle conductivity spectral width of Ba0:72K0:28Fe2As2 sample B as a function of temperature. The vertical blue lines give the estimated range of =2 at four measurement temperatures. The shaded region is a guide to the eye. Figure 8.13 shows the temperature dependence of RS(!; T ) at 13 GHz for a variety of superconductors. Some of the data sets were acquired using the broadband spectrometer (closed symbols), but the most complete data sets were obtained by tracking the cavity Q of a 13 GHz cylindrical resonator as a function of sample tem- perature (open symbols). With the exception of the YBa2Cu3O6:52 data (sample #1 from chapter 7), which was measured by Ahmad Hosseini [95], the 13 GHz resonator data was obtained by Brad Ramshaw. For each of the conventional BCS s-wave su- perconductors (Pb0:95Sn0:05, Nb, and Sn), RS(!; T ) scales as T=Tc and shows a very weak temperature dependence below T=Tc  0:4. The Ba0:72K0:28Fe2As2 data, on the other hand, show no signs of 
attening and obey a power law RS(!; T ) / T n down to the lowest measurement temperatures. Comparisons to the low-T power law ts to the super
uid density in the previous section, however, are not possible because the RS(!; T ) do not extend to low enough base temperatures and the resolution of the measurement at the lowest temperatures is inadequate. Nevertheless, it is clear that the surface resistance temperature dependence of the Ba0:72K0:28Fe2As2 samples diers signicantly from the s-wave BCS superconductors. Both the quasiparticle conductivity and surface resistance of ortho-II ordered 201 8.4. Microwave Spectroscopy 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.3 0.6 0.9 1.2 1.5     13 GHz  Ba0.72K0.28Fe2As2 (A)  Ba0.72K0.28Fe2As2 (B)          YBa2Cu3O6.52          Pb0.95Sn0.05          Nb          Sn R S  (m ) T/TC Figure 8.13: RS(!; T ) at 13 GHz as a function of T=Tc for several supercon- ductors. Solid symbols were taken using the broadband apparatus and open sym- bols were taken using a 13 GHz resonator. The conventional s-wave superconduc- tors (Pb0:95Sn0:05, Nb, and Sn) exhibit a very weak temperature dependence when T=Tc < 0:4. The Ba0:72K0:28Fe2As2 samples obey a power law RS(!; T ) / T n down to the lowest measurement temperature. The peak in the YBa2Cu3O6:52 data is due to a competition between the loss of normal 
uid density and lowered quasiparticle scattering rates as temperature is decreased. YBa2Cu3O6:52 show a low-temperature peak. This peak is due to the fact that, as temperature is decreased, there is a reduction in the normal 
uid density (i.e. the number of quasiparticles), but simultaneously, the quasiparticle scattering rate de- creases. These two eects compete and at the lowest temperatures the reduced normal 
uid density dominates to suppress the quasiparticle conductivity. At higher tem- peratures, it is the scattering rate that dominates and causes a suppression of the conductivity. Between these two limits, both 1(!; T ) and RS(!; T ) go through a peak [92, 95]. This peak occurs in only the purest cuprates and is very sensitive to any disorder and/or impurities. For example, substituting as little as 0.75% Ni for Cu in YBa2Cu3O6:95 completely suppresses the peak in RS(!; T ) while reducing Tc by only 3% [77]. Given the that spectral width of the quasiparticle conductivity of Ba0:72K0:28Fe2As2 is estimated to be an order of magnitude larger than it is in 202 8.4. Microwave Spectroscopy ortho-II YBa2Cu3O6:52 (compare Fig. 8.12 and Fig. 7.14), the absence of a peak in the temperature dependence of RS(!; T ) is not surprising, even though there may be a modest decrease in the scattering rate below Tc. Because the broadband measurements have revealed an excess extrinsic loss in our Ba0:72K0:28Fe2As2 samples, it is not possible to obtain reliable quasiparticle con- ductivities from our RS(T ) measurements. The exact behavior of 1(T ) necessitates simultaneous measurements of RS(T ) and XS(T ). Limited to what we can extract, an approximated quasiparticle conductivity, we see no evidence of a coherence peak just below Tc. This is in contrast to what has been previously reported [156], but those conclusions were drawn from a broad, nearly 5 K wide enhancement in the quasiparticle conductivity just below Tc in a sample whose superconducting transi- tion temperature width Tc  2:5 K. The behavior of 1(T ) is known to be highly susceptible to inhomogeneity broadening of the superconducting transition. Simi- lar quasiparticle conductivity enhancements below Tc were observed early on in the cuprates [174, 175], but were later attributed to broadened superconducting transi- tions [176{178]. The samples studied in this this work have Tc  0:5 K and we extract a much narrower, sharp cusp (only 0.5 K wide, not shown) that can be at- tributed to superconducting critical 
uctuations, rather than evidence of a coherence peak. We do, however, observe the same lower peak associated with the quasiparticle scattering rate (also not shown). 8.4.1 Sources of the Extrinsic Microwave Loss There exist a number of factors that might account for the observed excess loss in the Ba0:72K0:28Fe2As2 samples. Perhaps the most likely candidate is surface contam- ination. Both samples A and B were cleaved from a larger crystal before they were measured, with one potentially signicant dierence. While one surface of sample A was cleaved just prior to the start of measurements, the opposite surface had been cleaved weeks earlier allowing for the possibility of enhanced degradation of that sur- face. Sample B, on the other hand, had both surfaces freshly cleaved immediately before measurements began. A second possible source of the observed excess loss is the presence of delaminated sample edges. Sample edges that resemble the unbound leafy edge of a book present obvious problems for microwave measurements. In the extreme case, the microwave elds would be allowed to penetrate the regions between loosely connected \sheets" of the sample. Currents that loop around the sample are presented with a tortu- 203 8.4. Microwave Spectroscopy ous path near the edges that increases the eective surface area of the sample. In fact, preliminary measurements of (T ) of an early Ba0:72K0:28Fe2As2 crystal with obvious edge problems gave rise to a linear temperature dependence that had an un- realistic slope of  200 A/K. Scanning electron microscope images, as in Fig 8.14, have conrmed that some crystals do show signs of delaminated edges with sheets  1 2 m thick. This eect would be suppressed for the thinner (sample B) of the Figure 8.14: Scanning electron microscope images of the delaminated Ba0:72K0:28Fe2As2 crystal edges resemble the frayed pages of old telephone books. two Ba0:72K0:28Fe2As2 crystals measured in this work. It is the thinner sample that exhibits what is thought to be the more intrinsic behaviour. A third possible source of extrinsic loss is c-axis contamination of the RS(!; T ) measurements. For very anisotropic materials, like the cuprate superconductors, even a small contribution from c-axis currents can lead to a result that is far from the de- sired measurement of the intrinsic ab-plane behaviour. Samples A and B have aspect ratios that dier by a factor of 4, so it is possible that the dierence in the measured surface resistances is due to an increased c-axis contamination in the thicker sample. The arsenide crystals, however, are believed to be much more three-dimensional with very little anisotropy [64]. The thick crystal (sample A) was intentionally cut into two approximately equal pieces and the surface resistance of these two pieces placed side-by-side was measured in the 13 GHz resonator. In this scheme, the c-axis contri- bution to the surface resistance measurement is doubled in the second measurement. 204 8.5. Microwave Spectroscopy of Ba(Fe0:95Co0:05)2As2 There was no observed change in the measured RS(!; T ) indicating that the data pre- sented in this chapter are primarily due to contributions from planar surface currents (and not from currents along the sample edges). A nal factor to consider is enhanced bulk impurity scattering in sample A. This explanation does not seem likely, as both Ba0:72K0:28Fe2As2 samples A and B were grown under the same conditions. Furthermore, the dominant scattering source is al- most certainly the K-doping and the magnetization measurements of the thick sample A suggest homogeneous doping of the crystal. 8.5 Microwave Spectroscopy of Ba(Fe0:95Co0:05)2As2 Section 8.3 was ended by presenting an anomalous feature in the (T ) measure- ments that remains unexplained. This section presents anomalous behaviour in the microwave spectroscopy of the Co-doped sample which also presently remains unex- plained. The measured surface resistance spectra of the Ba(Fe0:95Co0:05)2As2 crystal (sample C) are shown in Fig. 8.15 both on linear and logarithmic scales. It is immediately clear that the data are qualitatively dierent than those of the Ba0:72K0:28Fe2As2 samples. Very surprisingly, there is a dip in the RS(!; T ) spectra that is centred at approximately 10 GHz and is approximately 5 GHz wide. At higher frequencies, the surface resistance increases rapidly. In the previous section several possible sources of extrinsic microwave loss were discussed. All of these sources lead to excess loss, and it is very dicult to imagine an extrinsic mechanism that would pro- duce a dip in the measured microwave power absorption. Quite remarkably, Fig. 8.16 shows that the surface resistance spectra at frequencies below the dip frequency quan- titatively match the RS(!; T ) spectra of Ba0:72K0:28Fe2As2 sample B. The top plot directly compares the surface resistance spectra of the two samples. These are un- corrected data sets without any scaling or oset adjustments. For clarity, only the lowest and highest temperature spectra are shown. In the bottom plot, the same RS(!; T ) = RS(!; T )  RS(!; T0) analysis that was done for the Ba0:72K0:28Fe2As2 samples has been repeated for the Ba(Fe0:95Co0:05)2As2 sample. This procedure re- moves the dip feature entirely and RS(!; T ) / !2 (dashed lines) up to 18 GHz above which data rapidly rise. The dashed lines in the gure are identical to those used in Fig. 8.11 allowing for direct quantitative comparisons of the K- and Co-doped 205 8.5. Microwave Spectroscopy of Ba(Fe0:95Co0:05)2As2 0 5 10 15 20 0.0 0.1 0.2 0.3 0.4 0.5    14 K  7.8  5.4  2.7  1.2 R S  (m ) Frequency (GHz) 1 10 1E-3 0.01 0.1 1 10   R S  (m ) Frequency (GHz) Figure 8.15: Measured RS(!; T ) spectra of the Ba(Fe0:95Co0:05)2As2 crystal (sample C). Top: There is a dip centred at 10 GHz and then a rapid increase in the surface resistance at higher frequencies. Bottom: The same data plotted on a logarithmic scale. 206 8.5. Microwave Spectroscopy of Ba(Fe0:95Co0:05)2As2 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 Ba(Fe 0.95 Co 0.05 ) 2 As 2  Sample C  15 K  1.8 Ba 0.72 K 0.28 Fe 2 As 2  Sample B  14 K  1.2  R S  (m ) Frequency (GHz) 1 10 1E-4 1E-3 0.01 0.1 1 10  14 K  7.8  5.4  2.7   R S (T )= R S (T )- R S (T 0)   ( m ) Frequency (GHz) Figure 8.16: The surface resistance spectra of Ba(Fe0:95Co0:05)2As2 sample C is com- pared directly to that of Ba0:72K0:28Fe2As2 sample B. At low frequencies, the data from the two samples are quantitatively similar. Bottom: Below 18 GHz, RS(!; T ) of the Ba(Fe0:95Co0:05)2As2 sample is proportional to ! 2 (dashed lines). For ease of comparison, the dashed lines are identical to those in Fig. 8.11. 207 8.5. Microwave Spectroscopy of Ba(Fe0:95Co0:05)2As2 samples. In temperature units, 18 GHz corresponds to: T18 GHz Tc = ~! kBTc  0:04; (8.8) which is exactly the location of the step in the (T ) data in Fig. 8.9. This analysis suggests that perhaps the anomalous behaviour in penetration depth and surface resistance are related. Each of the experimental observations (RS goes through a dip rather than an enhancement, at low frequency RS(!; T ) of the Ba(Fe0:95Co0:05)2As2 and Ba0:72K0:28Fe2As2 samples are in quantitative agreement, and RS(!; T ) / !2 below 18 GHz) tend to suggest that the observed anomalous behaviour in the microwave spectroscopy of sample C may be intrinsic rather than extrinsic. However, while all of the observed features in sample C are reproducible, they have not been seen in any other Ba(Fe0:95Co0:05)2As2 or Ba0:72K0:28Fe2As2 samples. It is natural to ask what makes sample C dierent from all of the other samples measured? One possible clue might come from the normal state resistivity. Figure 8.17 compares normal state surface resistance spectra of samples B and C. These two samples are similar in that they are both very thin (thickness t . 10 m) as determined using an optical micro- scope and are therefore not expected to suer from delamination eects at the sample edges. The thin-limit normal state surface resistance has been previously discussed in x3.1.1 and x6.2 and the data are t to ReS (!) given by Eq. 3.16 in which the sample thickness t and normal state resistivity  are the only parameters. Because the samples are of similar thickness, the dierence in the normal state surface resis- tance spectra can only be due to dierences in the resistivities of the two samples. Table 8.4 gives t and  for samples A, B, and C as determined by ts to normal state surface resistance spectra. Comparing the two thin samples (B and C) one sees that the resistivity of the Co-doped sample is an order of magnitude greater than that of the K-doped sample. It is possible that the anomalous features in the spectroscopy of the Co-doped sample are due to an eect that is activated by increased defects and/or impurities. Sample C is a very thin foil-like sample that was peeled from the surface of a much larger Co-doped single crystal. So, like sample A, just prior to measurements it had one freshly cleaved surface and one surface that was cleaved weeks earlier. A second Co-doped sample was cut and cleaved from the same single crystal from which sample C was taken. This second Co-doped sample had a thickness of t  19 m and a resistivity of   0:70  m. Spectroscopic measurements of this sample yielded 208 8.5. Microwave Spectroscopy of Ba(Fe0:95Co0:05)2As2 Sample T (K) Tc (K) t (m)  ( m) A 32 29.4 50 0.56 B 31 27.3 7.8 0.28 C 23 20.0 6.8 2.8 Table 8.4: The thickness and resistivity of all three iron-arsenide samples determined from ts to normal state surface resistance spectra. The Co-doped sample resistivity is signicantly greater than that of either of the K-doped samples. results very similar to those of the K-doped samples and did not show any of the anomalous features exhibited by sample C. The absolute measured RS(!; T ) spectra at 5 K lies between those measured for samples A and B. The magnitude of RS(!; T ) is consistent with that seen in the K-doped samples, but it follows a power law that 0 5 10 15 20 0.00 0.05 0.10 0.15 0.20 Ba 0.72 K 0.28 Fe 2 As 2  Sample B - 27 K  data  fit Ba(Fe 0.95 Co 0.05 ) 2 As 2  Sample C - 23 K  data  fit  R S  ( ) Frequency (GHz) Figure 8.17: Normal state RS(!; T ) spectra of samples B (Ba0:72K0:28Fe2As2 ) and C (Ba(Fe0:95Co0:05)2As2). The data from sample B is the same as shown in Fig. 6.2. These two samples have similar thicknesses and the extreme dierence in the shape of the thin-limit surface resistances is due to large dierences in the normal state resistivities. 209 8.6. Iron-Arsenide Summary is closer to !1:8 than !2. 8.6 Iron-Arsenide Summary The microwave electrodynamics of three 122-type iron-arsenide single crystals were studied (two K-doped and one Co-doped). SQUID magnetometery measurements in a dc magnetic eld imply that these are homogeneously doped samples. Remarkable consistency of (T ) as a function of T=Tc among the three samples also points to homogeneous samples and suggests that something near to the intrinsic temperature dependence is being measured. For reasonable values of (0), the super
uid density 2(0)=2(T ) approaches Tc linearly, indicative of mean eld type behaviour. The data deviate from this linear approach to Tc only within 1 K of Tc, another sign of homogenously doped samples. Measurements on these samples compare very favourably to published super
uid density measurements made on similar samples which show wider regions of curvature as T ! Tc. For T  Tc, the (T ) data do not follow the temperature dependence expected for a single s-wave energy gap. The data can be t to a model which includes two independent s-wave gaps with a common Tc. However, the t parameters are sensitive to the t range which may be an indication that the model is failing. Moreover, the extracted gap magnitudes 20;S=kBTc < 20;L=kBTc < 3:5 are small and there is a very large mismatch in the contributions to the measured super
uid density from the small and large gaps, with 0;L accounting for all but 2% of the total super
uid density. The low-T super
uid density can also be t to a power law model and the t parameters are less sensitive to the t range. The power ranges between the values 2:1 < n < 2:7 with the Co-doped sample at the higher end of the range and the K-doped samples at the lower end. A T 2 temperature dependence of the super
uid density could be explained by a d-wave gap (cylindrical fermi surface with line nodes) in the dirty limit as in the HiTc cuprates or by a gap with point nodes in the clean limit as in PrOs4Sb12 [179]. In the K-doped samples there is a 28% substitution of the Ba sites and, in the Co-doped sample, the Co replaces Fe in the FeAs plane. In both of these cases, a high density of scatterers is expected and it is very unlikely that one is in the clean limit. Understanding the origin of non-integer powers greater than two requires detailed theoretical models of various gap scenarios, probably in the presence of a dense population of scatterers. The observed power law behaviour is consistent 210 8.6. Iron-Arsenide Summary with similar measurements on the same compounds by other researchers [162, 168]. Finally, the (T ) data of all three samples show a step-like feature at T=Tc  0:04 that is approximately 1 A in size. The origin of this feature is unknown. Microwave surface resistance spectra of the two K-doped samples were measured from 0.1 to 20 GHz. The absolute surface resistance at 1 K of the thicker of the two samples is larger by a factor of four. However, the temperature evolution of the two data sets are identical, suggesting that there is an extrinsic loss that is more prominent in the thick sample. Sources of the extrinsic loss might be associated with degradation of the sample surfaces or with delamination of the sample edges. When the extrinsic loss is removed by subtracting the low-temperature spectrum from the higher temperature spectra the resulting RS(!; T ) all follow ! 2. This frequency dependence is expected for a quasiparticle conductivity that is 
at over the measurement bandwidth. The quasiparticle scattering rate is estimated to be  100 GHz which is at least an order of magnitude larger than the scattering rates observed in ortho-II ordered YBa2Cu3O6:5. The microwave spectroscopy of the Co-doped sample shows an anomalous dip feature followed by a rapid increase. The dip in RS(!; T ) is dicult to explain by any extrinsic source which would be expected to be additive. When the RS(!; T ) analysis is applied, the dip feature is removed and below 18 GHz the !2 dependence is recovered. Above 18 GHz the rapid rise of the microwave loss persists. In units of T=Tc, 18 GHz corresponds to 0.04, exactly where the step in (T ) occurs. The dip-rise behaviour has not been observed in any other samples. The resistivity of the anomalous sample is an order of magnitude larger than other the samples studied suggesting that the observed behaviour might be defect/impurity activated. 211 Chapter 9 Ultra-Low-Temperature Microwave Spectroscopy To this point, all of the broadband spectroscopic measurements presented have been limited to base temperatures of 1.2 K. These experiments are performed by immersing the entire apparatus into a temperature-regulated pumped bath of super
uid 4He. The super
uid bath is ideal since, because of its very high thermal conductivity, it will not support any temperature gradients [180]. Moreover, the large super
uid bath has a high heat capacity so that its temperature is insensitive to small changes to the heat 
ux delivered to the bath. A stable base temperature is essential to the success of the broadband spectrometer because the sensitivity of the measurement (i.e. the calibration factor) is a strong function of temperature. For example, the sensitivity of the sample side thermal stage (hot nger) is determined by the thermal conductance of the quartz tube which follows a T 1:8 power law [107]. In addition to a stable average temperature of the bath, the 
uctuations about the mean temperature must also remain very small. The experimental signal is an oscillating sample temperature due to power absorption of incident microwave magnetic elds modulated at low frequency (typically, 1 Hz). Any temperature oscillation of the base temperature at this same frequency will lead to spurious background signals. One could imagine, for example, that the semirigid coaxial cable that runs through the bath and delivers microwave power to the spectrometer could cause the bath temperature to 
uctuate. This chapter discusses broadband spectroscopy at sub-kelvin temperatures using a dilution refrigerator. The operation of the apparatus, the signal conditioning, and signal acquisition all remain exactly as described in chapter 5. The only change is the cryostat and the focus here will be on implementing a mounting scheme that provides sample temperatures that are stable and suitably free of low-frequency 
uctuations. 212 9.1. Dilution Fridge Mounting 9.1 Dilution Fridge Mounting The scheme used to mount the broadband apparatus onto a S.H.E. Corporation dilution refrigerator is shown in Fig. 9.1. The main body of the apparatus is bolted (a) 1 K Pot Vacuum Flange Still (700 mK) Heat Sheild (cutaway) Mixing Chamber (50 mK) Thermal Filters (cross-section) Microwave Spectrometer Vacuum Can (cutaway) (b) Figure 9.1: (a) Scale CAD drawing of the microwave spectrometer installed on a S.H.E. Corporation dilution refrigerator. Sections of the vacuum can and heat shield have been cutaway. A cross-section of the thermal lters is shown. The blue lines rep- resent copper braids that link the thermal stages to the mixing chamber through the thermal lters. (b) Digital photograph of the fully installed microwave spectrometer. The vacuum can is not in place. The microwave coaxial cable is thermally anchored to the heat shield. 213 9.1. Dilution Fridge Mounting to the base of a heat (or radiation) shield which is in turn bolted to the still of the fridge which typically operates at approximately 700 mK. The thermal stages (sample and reference sides) are isolated from the main body of the experiment via the Vespel tubes allowing their temperatures to be independently set. The thermal stages are linked to the mixing chamber of the dilution refrigerator through a set of massive low-pass thermal lters. The mixing chamber typically operates at about 50 mK. The ultimate base temperature of the sample is therefore determined by both the thermal resistance between the sample stage and the experiment body (700 mK) and the thermal resistance between the sample stage and the mixing chamber (50 mK). Figure 9.2 shows a CAD drawing and a digital photograph of the low-pass thermal lters. These lters can be analyzed in exactly the same way as the Vespel lters of x5.5. The signal from the normal metal alloy of the reference stage is large and only a single low-pass lter is used. Signals from superconducting single crystals of the sample stage, on the other hand, can be very small and two cascaded low-pass lters are used. Here we focus on the two-stage sample-side lter. In the high frequency limit !  11 ; 12 , a temperature oscillation of amplitude eTmc and frequency ! at the mixing chamber will be suppressed by a factor of 1=!212, where  = RssC. Rss is the thermal resistance of the thin-walled stainless steel tubes and C is the heat capacity of the copper blocks. The subscript \1" refers to the upper lter closest to the mixing chamber and the subscript \2" refers to the bottom lter. Before evaluating the 1 and 2 time constants, we rst nd the typical operating temperature of the lter components. Figure 9.3 gives an easy way to estimate the relevant temperatures. Thermal connections between the lters and the sample stage of the spectrometer are made using short sections of copper braid. For simplicity, it is assumed that the copper braids are thermal shorts. In this approximation, the equivalent thermal circuit is given in the gure. Here one has P1 = P3 and P2 = 0, such that: P1 = TB  Tmc Rvp2 +Rss1 ; (9.1) and: T2  T1 = Tmc +Rss1P1: (9.2) 214 9.1. Dilution Fridge Mounting (a) A A Thin-walled stainless steel tubes Copper blocks Mounting flanges (b) Figure 9.2: (a) Scale drawing of the low-pass thermal lters used with the dilution refrigerator setup. The cross-sectional view shows that the single-stage lter used for the reference side ts inside the dual-stage lter used for the sample side. (b) Digital photograph of the reference side lter (left) and sample side lter (right). To determine P1 requires that one rst estimate Rss1 and Rvp2: 1 Rss1 = Gss1 = 1 T1  Tmc Ass1 `ss1 Z T1 Tmc  0:141 W K2m  TdT (9.3a) 1 Rvp2 = Gvp2 = 1 TB  T1 Avp2 `vp2 Z TB T1  0:0017 W K3m  T 2dT (9.3b) 215 9.1. Dilution Fridge Mounting Figure 9.3: Left: The cascaded low-pass lters are thermally connected to the mi- crowave spectrometer using copper braid (represented by blue lines). For clarity the sample loading guide pins have been suppressed on the left side of the microwave spectrometer. Right: The equivalent thermal circuit assuming that the copper braids are thermal shorts. where ss = (0:141 WK 2m1)T and vp = (0:0017 WK3m1)T 2 are the low- temperature thermal conductivities of stainless steel and Vespel-SP22 respectively. Note that both Rss1 and Rvp2 depend on the value of T1 which is unknown. The apparatus has been designed to ensure that Rvp2  Rss1 such that T1 ' Tmc is ex- pected. As an initial estimate, P1 is calculated using T1 = Tmc which is, in turn, then used in Eq. 9.2 to calculate a new value of T1 from which updated estimates of Rss1 and Rvp2 are made. This process can be repeated until self consistency is reached. Stable values of P1 and T1 are reached after only two iterations and the results are summarized in Table 9.1 along with the relevant lter dimensions and thermodynamic quantities. A 1 Hz temperature oscillation of the mixing chamber is therefore expected to be suppressed by a factor of 1=!2ss1ss2  2:6  104. A full numerical analysis of the complete thermal circuit was done in Ref. [97] and will not 216 9.1. Dilution Fridge Mounting P1 T1 T2 (W) (mK) (mK) 0.33 57 57 Stainless Steel/copper lters Capacitors Resistors RC V C A ` R  (106 m3) (J/K) (106 m2) (mm) (103 K/W) (s) C1 110 450 Rss1 46 6.9 20. ss1 9.0 C2 12 49 Rss2 5.0 8.9 220 ss2 11 Vespel-SP22/copper lters Capacitors Resistors RC V C A ` R  (106 m3) (J/K) (106 m2) (mm) (106 K/W) (s) C1 0.98 4.0 Rvp1 11 3.3 54 vp1 220 C2 0.54 2.2 Rvp2 11 6.7 2.0 vp2 4.4 Table 9.1: Top: Self consistent values of the power P1 and temperature T1 from Fig. 9.3. Treating the copper braids as thermal shorts gives T2  T1. Middle: Heat capacitance, thermal resistance, and time constants for the cascaded low-pass stain- less steel/copper lters. Bottom: Heat capacitance, thermal resistance, and time constants for the cascaded low-pass Vespel/copper lters. 217 9.1. Dilution Fridge Mounting be repeated here. However, all of the approximate results given above are accurate to within 15% or better. Figure 9.4 shows the stability of the mixing chamber temperature controlled us- ing a Cryogenic Control Systems model CryoCon 62 temperature controller. The 0 1 2 3 59.96 59.98 60.00 60.02 60.04   T m c ( m K ) Time (hours) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 59.96 59.98 60.00 60.02 60.04   Time (minutes) Figure 9.4: Right: The dilution refrigerator mixing chamber temperature over a period of 3.5 hours. The set-temperature was 60 mK. Left: The rst three minutes of the data shown on the right. A data point was taken once every second. The 
uctuations in the data occur on a time scale that is longer than one second suggesting that the source of the noise is variations in the mixing chamber temperature and not the electrical noise in the detection electronics. temperature set-point was 60 mK and the temperature was stable to within 20 K of the set point. For a low-loss superconductor like YBCO, the power absorbed by the sample due to the modulated microwave magnetic eld is of order 1 nW at 1 K and results in an oscillation of the sample temperature with an amplitude less than 10 K45. These considerations highlight the need for thermal ltering of the mixing chamber temperature. Since the sample temperature is determined by both the mix- ing chamber and the still temperatures, they should both be controlled for maximum stability of the sample temperature. A Fourier transform of the data of Fig. 9.4 shows 1=f noise but is otherwise featureless. The automated data acquisition software has the capability to update the temperature-dependent sample and reference stage cali- bration factors as often as deemed necessary. Therefore, a slow drift of the thermal 45At sub-kelvin temperatures the power absorption of the sample is expected to be even smaller, however, the reduced loss will be compensated for by an increased measurement sensitivity due to a reduction the in the quartz tube thermal conductance G (T = P=G). 218 9.2. Preliminary Measurements: Sr2RuO4 stage temperatures can be managed somewhat with frequent re-calibrations. 9.2 Preliminary Measurements: Sr2RuO4 This section presents preliminary broadband surface resistance measurements of a Sr2RuO4 (strontium ruthenate) single crystal at dilution refrigerator temperatures. The samples were grown at Kyoto University in Japan using the 
oating zone method in an image furnace as described in appendix F. The platelet Sr2RuO4 crystal studied is shown in Fig. 9.5. With a maximum transition temperature Tc = 1:5 K for the Figure 9.5: The top four digital photographs are of the cleaved ab-plane surfaces of a Sr2RuO4 single crystal. The cleaved surfaces are highly re
ecting as shown by the photographs on the righthand side taken with glancing incident light. Step edges are visible on the cleaved surfaces. The bottom photograph shows the sample edge when mounted on a sapphire plate. The broad face of the sample has an area of 1.38 mm2 and the sample thickness is 28 m. purest samples, Sr2RuO4 has attracted a lot of interest as a possible p-wave supercon- ductor in which the electrons form symmetric spin-triplet pairs [34]. The transition 219 9.2. Preliminary Measurements: Sr2RuO4 temperature of the sample studied was estimated to be Tc  1:42 K from ac sus- ceptibility measurements of a dierent sample cut and cleaved from the same single crystal (see appendix F for further details). The measured surface resistance spectra of the Sr2RuO4 sample, in both the nor- mal and superconducting states, are shown in Fig. 9.6. The superconducting tran- sition is signaled by a qualitative change in the shape of the RS(!; T ) spectra at temperatures of 1.2 K and below. The normal state spectra were t to Eq. 6.6 from x6.3: RS(!) = r 0! 20 hp 1 + (!)2  ! i1=2 ; (9.4) which assumes a Drude conductivity. Table 9.2 summarizes the t parameters. The T (K) 0 (10 6 1cm1)  = 1 0 ( cm)  2 = 1 2  1   (GHz) 1.9 1:253 0:011 0:798 0:007 21:3 0:3 3.8 0:916 0:007 1:092 0:008 25:0 0:4 6.3 0:588 0:008 1:70 0:02 32:0 1:0 Table 9.2: Parameters extracted from Drude model ts to the measured normal state Sr2RuO4 RS(!; T ) spectra. eective measured surface resistance spectra will contain both ab-plane and c-axis contributions: RS;e = (1 x)RS;ab + xRS;c; (9.5) where, for the platelet sample studied and the measurement geometry used, x  0:029. Since RS / p, the eective resistivity extracted from the t is also a mixture of ab and c such that: r e ab = 1 + x r c ab  1  : (9.6) The normal state resistivity of Sr2RuO4 is very anisotropic with c;0=ab;0 reported to be between 400 and 4000, where 0 is the T ! 0 residual resistivity [34]. Despite the large anisotropy, the normal state of Sr2RuO4 obeys Fermi-liquid theory and both the ab-plane and c-axis resistivities follow a T 2 temperature dependence. In the bottom panel of Fig. 9.6 the extracted e obey the expected T 2 power law with an eective 220 9.2. Preliminary Measurements: Sr2RuO4 0 5 10 15 20 25 0 5 10 15 20 25 30    6.3 K  3.8 K  1.9 K  1.2 K  1.04 K  810 mK  600 mK  400 mK  310 mK R S  (m ) Frequency (GHz) 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0    ( cm ) Temperature (K) Figure 9.6: Top: RS(!; T ) spectra of Sr2RuO4 for temperatures above and below Tc. The measured surface resistance is a mixture of ab-plane and c-axis contributions. Bottom: The resistivity extracted from the three normal state spectra. The line is a t to a T 2 temperature dependence. residual resistivity of e;0 = 0:72  cm. Equation 9.6 can be used to make order of magnitude estimates of the expected value of ab;0 as in Table 9.3. The key result is that, assuming 400 < c;0=ab;0 < 4000, one expects 0:31 > ab;0 > 0:096  cm. The Tc of Sr2RuO4 is a very sensitive function of ab;0. There is a complete suppression 221 9.3. Quartz Tube (T ) c;0 ab;0 e;0 ab;0 ab;0 c;0 ( cm) m cm 4000 7.5 0.096 0.38 400 2.3 0.31 0.12 Table 9.3: Estimates of the ab-plane residual resistivity in Sr2RuO4. of Tc when ab;0 > 1  cm and Tc > 1:4 only when ab;0 < 0:2  cm [181, 182]. The estimates of ab;0, therefore, bracket the expected range of values considering that the Tc of the measured sample is approximately 1.42 K. The only way to reliably separate the ab-plane and c-axis contributions to the resistivity would be to cut the crystal to systematically alter the sample aspect ratio as, for example, was done to separate a(T ) and c(T ) in the superconducting state of YBa2Cu3O6+y [86]. 9.3 Quartz Tube (T ) In x5.10 the expected temperature dependence of the thermal diusivity  = =cV of vitreous quartz was plotted in Fig. 5.38. As temperature is decreased,  ini- tially increases and then peaks at 600 mK before rapidly decreasing. An incident ac microwave magnetic eld causes an oscillation of the sample temperature with an amplitude that has the approximate frequency response of a single pole low-pass lter. The characteristic corner frequency of that response, as given in x5.10, is: !C  p 6B(T ) `2 ; (9.7) where B(T ) is the temperature dependent thermal diusivity of the quartz tube of length `. Figure 9.7 shows the raw signals obtained from the broadband spectrometer at four dierent sample temperatures. These signals are due to a 0.5 Hz square-wave modulation of the incident microwave power. The shape of the response is deter- mined by low- and high-pass ltering in the detection electronics and by the thermal diusivity of the quartz tube. The change in the shape of the signals is due only to changes in the quartz thermal diusivity with temperature. The time constant 222 9.3. Quartz Tube (T ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -1.0 -0.5 0.0 0.5 1.0    350 mK  1.2 K  2 K  4 K N or m al iz ed  S ig na l Time (s) 0.00 0.25 0.50 0.75 1.00 -0.5 0.0 0.5 1.0   N or m al iz ed  S ig na l Time (s) Figure 9.7: Signal response due to a 0.5 Hz square-wave modulation of the microwave power. Top: Two full periods of the signal at four dierent sample temperatures. The signals have been scaled for ease of comparison. Bottom: The same data as above, but showing only the rst half-period. The lowest and highest temperature data sets have the slowest rise times indicating that quartz tube thermal diusivity peaks between these two temperatures. 223 9.4. Future Directions  = 1=!C of the rise and decay of the signal is inversely proportional to B(T ). Note that the signal rises most rapidly at the two intermediate temperatures (small  , large B) indicating that as temperature goes from 4 K to below 0.5 K, B passes through a peak as expected. The two low-temperature data sets have lower signal-to-noise ratios because they are obtained in the superconducting state where the sample loss is much lower than it is in the normal state. 9.4 Future Directions Sub-kelvin microwave spectroscopy of low-loss materials will be continued by James Day and Jordan Baglo. This section brie
y introduces some outstanding issues that can be addressed using this apparatus at sub-kelvin temperatures. 9.4.1 Ortho-I YBa2Cu3O6:99 Chapter 7 discussed the microwave spectroscopy of YBa2Cu3O6:5. Patrick Turner et al. also studied the microwave conductivity of fully doped YBa2Cu3O6:99 and the a-axis data are shown in Fig. 9.8 [94]. These spectra are noticeably distinct from the YBa2Cu3O6:52 spectra; the zero-frequency intercepts do not tend to a common value and the spectra do not scale in the same manner as the ortho-II spectra. Even more striking is that the data seem to evolve from cusp-like line shapes expected for weak limit scattering at low temperatures to Drude-like Lorentzian line shapes at the higher temperatures. This point is made clear by the Drude ts (solid lines in Fig. 9.8) that progressively improve with increasing temperature. It has been speculated that the ortho-I sample may be in an intermediate scattering regime. This view is advocated by Hill et al. as a result of their thermal conductivity measurements on a UBC-grown fully doped YBCO single crystal [122]. The conductivity model given in x7.2 and repeated here: (!; T ) = ne2 m?  1 i! + 1(")  " ; (9.8) takes the form of a sum of Drude line shapes and was derived in the context of point scatters [120, 121]. Recalling that for a d-wave superconductor, 1(") has a nontrivial dependence on the energy " one would not expect Lorentzian line shapes from this model. The observation of these line shapes in Fig. 9.8 suggests that the scattering mechanism in ultra-pure YBCO crystals may not be point-like. Recently, Nunner and Hirschfeld have convincingly modeled the temperature dependence of 224 9.4. Future Directions 0 5 10 15 20 0 20 40 60 80 100 0 5 10 0 10 20 30  1.3 K  3.0 K  5.0 K  7.0 K  9.0 K σ 1 (10 6 Ω - 1 m - 1 ) Frequency (GHz)  1 d from 1/λ2(T) from n n e2 /m  (10 17 Ω - 1 m - 1 s - 1 ) T (K) Figure 9.8: The measured a-axis conductivity spectra of ortho-I YBa2Cu3O6:99. The solid lines are Drude ts to the data. The inset compares the normal 
uid den- sity found by integrating the conductivity ts to that found from penetration depth measurements. Figure provided courtesy of P.J. Turner [94, 99]. conductivity data for both YBa2Cu3O6:99 and Bi2Sr2CaCu2O8+ by including both point scatters and extended scatters due to out of plane disorder [127]. Broadband measurements at temperatures down to  100 mK would be extremely valuable to clarify our understanding of the electrodynamics of YBa2Cu3O6:99. If a crossover from Born-limit scattering to some intermediate scattering strength is being observed, then the low-temperature data should become increasingly cusp-like and may move towards a temperature independent zero frequency intercept. Furthermore, the low temperature data may also recover some form of a !=T scaling behaviour. Whatever the result, these measurements are needed to further our understanding of the low-energy quasiparticle excitations in fully doped YBCO. 9.4.2 Universal Conduction Another issue that has yet to be satisfactorily examined, despite being in the liter- ature for well over a decade now, is the universal electrical conductivity of a d-wave superconductor rst predicted by Patrick Lee in 1993 [183]. He showed that the resid- 225 9.4. Future Directions ual conductivity (! ! 0; T ! 0) of a d-wave superconductor tends to a non-zero result, that is independent of impurity concentration, and is given by: 00 = ne2 m0(0) ; (9.9) where 0(0) is the maximum gap at T = 0 K [183]. Physically, this eect arises as a result of the competition between the growth of the normal 
uid density with increasing impurity concentration and a corresponding reduction in the scattering mean free path. Shortly after Lee's original result Hirschfeld et al. showed that in the nite temperature limit the conductivity varies quadratically with temperature: xx(! ! 0; T )  00 " 1 + 2 12  kBT  2# ; (9.10) where 
 is a constant set by 0(0) and the normal state impurity scattering rate  [121]. These authors also showed that universal conduction is only observable in the so-called gapless regime dened by kBT < 
  kBTc. In the limit of unitary scat- terers 
  p0 and the universal behaviour should be observable at experimentally accessible temperatures. In contrast, in the Born limit 
  0 exp(0=N), where N  =c2 and c is the cotangent of the scattering phase shift (c  1 in the Born limit and c ! 0 in the unitary limit). Thus in the Born limit, even for large values of N, universal conductivity is observable only at exponentially small temperatures. In 2000 Durst and Lee improved upon the results of Ref. [183] by including so- called vertex and Fermi-liquid corrections [184]. In this work they examined the universal limits of electrical, thermal, and spin conductivity. The vertex corrections account for anisotropic scattering potentials, that could for example, have dier- ent amplitudes for forward and backward scattering, while Fermi-liquid corrections account for Fermi-liquid interactions between electrons in the superconductor and depend on the strength of the interactions. These authors found that while thermal conductivity is not aected by these corrections, the spin conductivity was modied by Fermi-liquid corrections and the electrical conductivity is modied by both vertex and Fermi-liquid corrections. If all three universal limits were measured using a sin- gle sample, then the magnitudes of the vertex and Fermi-liquid corrections could be independently determined. It should be noted that the Fermi-liquid correction factor can also be obtained from a measurement of the super
uid density (/ 2(T )), how- ever this method depends sensitively on the absolute value of (0) which is dicult 226 9.4. Future Directions to measure accurately [184]. Equation 9.9 can also be cast in the following form: 00 = e2 2~d vF v ; (9.11) where d is the average spacing of the CuO2 planes, vF = @"k=@k is the Fermi ve- locity, and v = @k=@k is the gap velocity [184]. The universal limit for thermal conductivity takes on a similar form and since there are no vertex or Fermi-liquid corrections a measurement of its value gives the ratio vF=v. The universal limit in thermal conductivity measurements has been observed in both of the d-wave super- conductors Bi2Sr2CaCu2O8 and YBa2Cu3O6:9 [122, 185]. In the YBCO sample the ratio of the Fermi velocity to gap velocity was measured to be vF=v = 14. With a CuO2 plane spacing of 5.85 A the expected electrical conductivity limit (without vertex or Fermi-liquid corrections) is 00  0:6  106 1m1. From Fig. 9.8, UBC broadband measurements of YBa2Cu3O6:99 at 1.3 K give (! ! 0; T = 1:3 K)  40  106 1m1  00. As was argued above, it is believed that the overdoped YBa2Cu3O6:99 samples are best described by an intermediate scattering picture (nei- ther Born nor unitary) and that universal electrical conductivity should be observable. We therefore speculate that, at 1.3 K our samples are not yet in the gapless regime required to observe universal behaviour. This scenario is consistent with the thermal conductivity measurements that observed the universal limit over a temperature range spanning 40 to 700 mK and also suggest an intermediate scattering regime [122, 185]. A thorough investigation of the universal behaviour of the electrical conductivity of YBCO is critical to further test and explore the current intermediate scattering picture in YBCO. If the picture is shown to be correct, one may be able to determine the temperature 
=kB at which the samples crossover into the gapless regime, which in turn could help determine to what extent the samples are in the Born or unitary limits for both YBa2Cu3O6:99 and YBa2Cu3O6:52. If universal behaviour is observed, then the deviation from 00 will quantify the combined magnitude of the vertex and Fermi-liquid corrections. If on the other hand, universal behaviour is not observed, we are certain to encounter new physics. The new broadband spectrometer is the ideal tool to carry out this investigation. 227 9.4. Future Directions 9.4.3 Heavy Fermions There are a variety of novel experiments that can be performed on heavy fermion su- perconductors using the low-temperature microwave spectrometer. The conductivity of common metals typically follows the Drude model: (!) = 1  i2 =  ne2 m  1  i! ; (9.12) where  is a frequency independent scattering rate. At high temperatures heavy fermion materials also adhere to this description, however, at low temperatures their behaviour is drastically altered due to the onset of strong electron correlations. As temperature is reduced below a coherence temperature T ? these materials show a large increase in magnetic susceptibility and the electronic contribution to the spe- cic heat. These eects are normally explained as an enhancement of the eective mass m? which can be of the order of 100me where me is the free electron mass. The mass enhancement renormalizes the plasma frequency (!?p) 2 = 4ne2=m? shifting the electronic excitations, characterized by an eective scattering rate ?, to low ener- gies. As the temperature drops further below T ?, m? continues to increase while ? decreases. To model the energy dependence of the renormalization eects one uses a frequency dependent scattering rate ?(!) and eective mass m?(!) in Eq. 9.12 to in- terpret the observed conductivity spectrum. Many of the interesting features, such as the renormalized Drude peak, are expected to reside within the microwave frequency regime making the low-temperature microwave spectrometer the ideal apparatus to investigate the conductivity of many heavy fermion materials [186]. Once cooled below the superconducting transition temperature Tc (typically  1 K), heavy fermion materials may exhibit absorption peaks in surface resistance measurements due to order-parameter collective modes. These collective modes are due to macroscopic quantum oscillations of the pair condensate and can be excited by ultrasound or electromagnetic waves. It is thought that the heavy fermion materials may be characterized by unconventional 3He-like order parameters and theoretical predictions of mode frequencies for various order parameter symmetries have been made [187]. Characteristic frequency scales in these materials are determined by the gap function =~  kBTc=~  20 GHz for Tc = 1 K. Feller et al. have observed collective modes in surface resistance measurements of UBe13 by cavity perturbation techniques [188]. The broadband apparatus would resolve the shape of these modes in much greater detail and allow for a systematic investigation of their temperature 228 9.4. Future Directions dependence. While evidence for collective modes in UBe13 does seem to exist, it is lacking for other heavy fermion materials such as UPt3. 9.4.4 Cooling by Adiabatic Demagnetization Finally, without presenting the details, it is remarked that cooling by adiabatic de- magnetization is an ideal way to make spectroscopic measurements between 100 mK and 1 K using the broadband apparatus. In this scheme one would use a paramag- netic salt to cool only the sample-side thermal stage (10 g copper block, quartz tube, sapphire plate, sample, bolometers, and chip heater), just as the mixing chamber only cools the thermal stages in the dilution refrigerator setup. Isolated from the cold thermal stage by the Vespel tube, the rest of the apparatus would be cooled to 1.2 K in a bath of pumped super
uid 4He. FAA (ferric ammonium alum, Fe3+2 (SO4)3(NH4)2SO424H2O) is an example of a paramagnetic salt that would be suitable for this application. The spontaneous magnetic ordering temperature for this salt is approximately 30 mK. Run in single shot mode and starting from an initial temperature of 1.2 K and a modest initial magnetic eld of 2 T, a base temperature of 100 mK can be reached and maintained for several hours [189]. A number of technical challenges are encountered when designing an adiabatic de- magnetization stage for the microwave spectrometer. A challenge encountered in all adiabatic demagnetization refrigerator (ADR) systems are the low thermal conductiv- ities and long thermal time constants of the paramagnetic salts. When adiabatically decreasing the applied magnetic eld to cool the paramagnetic salt, one can have large thermal gradients within the salt pill. In addition, it can be dicult to make good thermal contact to the salt pill to transfer the cooling to the experimental ap- paratus. To overcome these diculties the salt can be grown directly onto an array of long, thin gold (or copper) wires. These wires extend out past one end of the salt pill and serve two purposes: rst, they reduce thermal gradients within the salt pill, and second, the free ends of the wires are used to make good thermal contact to the apparatus being cooled. Once grown, the salt pill is typically sealed to prevent chemical changes (i.e. dehydration) that can cause premature deterioration of the pill. To cool the salt pill to the initial launch temperature of 1.2 K it will need to be in good thermal contact with the 4He bath. When cooling the salt pill to its operating temperature it will need to be isolated from the 4He bath. Therefore, a reliable heat switch, either mechanical or electro-mechanical needs to be designed. The University 229 9.4. Future Directions of California group of Richards et al. have published a number of excellent articles de- scribing their design and construction methods for making small-scale ADR systems for space and laboratory applications [190{192]. Two technical challenges that are particular to the microwave spectrometer are now raised. First, the paramagnetic salt pill will be in close proximity to the spectrom- eter and therefore the sample being investigated. The salt pill and the electromagnet used to apply the dc eld will need to be enclosed in a tight ferromagnetic shield. Second, exquisite control of the sample temperature will be required. In practice, the salt pill will be cooled by reducing the applied eld to some nite value. The temperature of the paramagnetic salt (and therefore the sample) will then slowly drift upwards due to the heat leak from the spectrometer body through the Vespel tube. 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uid and normal 
uid contributions 1(!) = 1s(!) + 1n(!). The super
uid delta-function is easily integrated (Eq. 3.27) such that Eq. A.3 becomes: 2 ( )  1   nse 2 m? (T ) + 2   Z 1 0 1n (!) d!: (A.4) At extremely high frequency, the electrons in the material can be considered as free 242 Appendix A. Oscillator Strength Sum Rule and 2 is the same in normal and superconducting states and has the Drude form: 2( ) = 2s( ) + 2n( )  ne 2 m?  ; (A.5) where n is the total electron number density. Combining Eqns. A.4 and A.5 produces desired sum rule: nne 2 m? (T )  ne 2 m?  nse 2 m? (T )  2  Z 1 0 1n (!; T ) d!: (A.6) 243 Appendix B Properties of CuGa Alloy at 1 K The reference-side thermal weak link of the microwave spectrometer was made using an alloy of uncertain composition. The alloy rod used for the reference stage was cut from a cylinder of material found inside a box labeled \CuGa (14.4%)". B.1 Resistivity and Thermal Conductivity The resistance/resistivity of the alloy was determined by measuring the voltage across two contacts with a current of 3 amps passing through a cylinder of the material. This measurement was done at room temperature and at 77 K in liquid nitrogen. The Wiedemann-Franz law was then used to extract the thermal conductivity  at these two temperatures:  = LT  ; (B.1) where L is the Lorentz number: L = 2k2B 3e2 = 2:45 108 W  K2 : (B.2) Table B.1 summarizes the results of these measurements. Making the assump- tion that  is linear from 1 K to room temperature yields a slope of (T = 1 K)=1 K = 0:16 WK2m1. This value is comparable to values known for other common alloys (stainless steel: 0.1 WK2m1, phosphor bronze: 0.2 WK2m1, brass: 0.8 WK2m1). Temperature (K)  ( m)  (WK1m1) 295 0.14 51 77 0.11 17 Table B.1: Resistivity and thermal conductivity of the CuGa alloy measured at 295 K and 77 K. 244 B.2. Specic Heat B.2 Specic Heat To make a measurement of the specic heat, the broadband spectrometer apparatus was used along with the results of Eqns 5.39 and 5.40. The frequency response of the quartz tube thermal stage was measured as described in x5.10. The data and t are repeated in Fig. B.1. Next a sample of the CuGa alloy was cut and mounted onto the sapphire plate using silicone grease. The sample was purposely cut relatively large so that its heat capacity would dominate that of the sapphire plate, bolometers, heater, . . . and cause a signicant change in the frequency response. The dimensions of the cut sample are L  W  H = 3:15  1:04  0:25 mm3 resulting in a total volume of 0:82 mm3. The frequency response with the CuGa sample loaded is also shown in Fig. B.1. As in Table 5.7, the t to the blue data with ` = 13:0 mm gives B = 2:62  103 m2/s and !0 = GB=CA = 2(18:8) rad/s. The addition of the CuGa sample can only modify !0 through a change to CA. The magenta line is a 0.1 1 10 1  No CuGa  Fit  CuGa  CuGa  Fit  N or m al iz ed  R es po ns e Frequency (Hz) Figure B.1: The frequency response of the quartz thermal stage at 1.2 K t to Eq. 5.39 is shown in blue and is a repeat of the data shown in Figs. 5.36 and 5.37. The response at 1.2 K after installing the large CuGa sample is shown in magenta. The open and solid symbols represent two dierent measurements. The magenta line is a t to the data (open and solid points) using the value of B obtained from the t to the blue data. 245 B.2. Specic Heat t where both ` = 13:0 mm and B = 2:62  103 m2/s are xed. The data points were obtained by applying ac power to heater and the measured response clearly deviates from the expected shape above 4 Hz. Moreover, at high frequency, the data tends to merge with the blue data. These observations indicate that above 4 Hz the CuGa sample is losing contact with the sapphire plate. Specically, above 4 Hz, the modulation frequency exceeds the time constant associated with the grease contact between CuGa sample and the sapphire plate. The !0 extracted from the magenta t is 2(2:37 0:09) rad/s. Assuming a quartz (T ) given by Eq. 5.41(a), CA=V of the CuGa sample is 171 JK1m3. 246 Appendix C Cascaded Low-Pass Filters In x5.5 the suppression of a temperature oscillation through a pair of cascaded low- pass thermal lters was considered (see Fig. 5.13). That section considered the high- frequency limit where !RC  1. This appendix will present the general solution for eT1.eTB. Equations 5.7 and 5.8 established the general relations: eT1eTB  =  eT2eTB   eT1eT2  ; (C.1) and:  eT1eT2  = 1p1 + !2R21C21 : (C.2) To evaluate  eT2.eTB, it is necessary to consider the parallel combination Ze = ZC2 jj(R1 + ZC1). After a page or two of complex algebra Ze can be sepa- rated into its real and imaginary components as follows: Ze = [!R1C1(C1 + C2) + !R1C1C2] j [(C1 + C2) + !2R21C21C2] !(!R1C1C2)2 + !(C1 + C2)2 : (C.3) Then, the desired quantity is given by: eT2eTB  =  ZeR2 + Ze  : (C.4) This quantity was evaluated with the aid of the software package Mathematica 6.0.0 and found to be: 247 Appendix C. Cascaded Low-Pass Filters  eT2eTB  = [(C1 + C2)2 + !2R21C21(C21 + 6C1C2 + 6C22) + !4R41C41C22 ]1=2  [C22 + !2R22C42 + !2C41(1 + !2R21C22)((R1 +R2)2 + !2R21R22C22) + 2C1(C2 + 2! 2(R1 +R2)R2C 3 2) + 2!2C31C2(3R 2 1 + 4R1R2 + 2R 2 2 + 2! 2R21R2(R1 +R2)C 2 2) + C21(1 + 2! 2(5R1R2 + 3R 2 2 +R 2 1(3 + ! 2R22C 2 2))C 2 2)] 1=2: (C.5) This is a remarkably complicated result for what one might consider a relatively simple circuit. It is not dicult to conrm that this expression has the expected high-frequency limit  eT2.eTB  !112 for !2  1. Figure C.1 compares  eT2.eTB obtained from Eq. C.5 to the approximate expres- sion 1= p 1 + !2 22 using the 1.2 K values of R1, R2, C1, and C2 from Table 5.3. Finally, Fig. C.2 compares  eT1.eTB obtained from Eqs. C.1, C.2, and C.5 to the  !"#  !"$ %&% %& %&% %&    ∣∣∣∣∣ T̃2 T̃B ∣∣∣∣∣ ω/2pi (s−1) Figure C.1:  eT2.eTB as a function of frequency for the spectrometer low-pass thermal lter. The solid green line is the full result given by Eq. C.5 and the dotted magenta line is the approximate solution given by Eq. 5.9. 248 Appendix C. Cascaded Low-Pass Filters approximate expression given in Eq. 5.10a using the same R1, R2, C1, and C2 values used in Fig. C.1.  !"#  !"$ %&% %& %&% %&    ∣∣∣∣∣ T̃1 T̃B ∣∣∣∣∣ ω/2pi (s−1) Figure C.2:  eT1.eTB as a function of frequency for the spectrometer low-pass thermal lter. The solid red line is the full result the dotted blue line is the approximate solution given by Eq. 5.10a. It is clear that, for modulation frequencies above 0.1 Hz, the full and approximate solutions are in good agreement. 249 Appendix D Ortho-I YBa2Cu3O6:50 Doping In x7.1 the doping of the ortho-I phase of YBa2Cu3O6+y (randomly distributed chain oxygen atoms) was said to be given by: p = p0  y2; (D.1) where 0  y  1 and p0 = 0:194 is the doping for fully oxygenated YBCO with y = 1. When all Cu-O chain chain sites have equal probability of being occupied, y gives the average occupation of each chain. The rst section of this appendix will numerically conrm Eq. D.1 for y = 0:5. It also goes through the useful exercise of presenting all of the possible oxygen congurations with six and eight half-lled Cu-O chain sites. Sections D.2 and D.3 then give two separate proofs of Eq. D.1 that are valid for any value of y. D.1 Numerical Conrmation Consider rst the relatively simple case of Cu-O chains with six available oxygen sites. Assume that every Cu-O chain contains three oxygen ions. There are: 6 3  = 6! 3!3! = 20; (D.2) distinct ways to ll six sites with three oxygen ions. All of the possible arrangements are shown in Fig. D.1. For the random ortho-I ordering all of the twenty distinct arrangements are equally likely. There are: 4 1  = 4; (D.3) 250 D.1. Numerical Conrmation Figure D.1: There are 20 ways to ll six Cu-O chain sites with three oxygen ions. Of the 20, four contribute two fractional holes, twelve contribute one fraction hole, and the remaining four do not contribute any holes to the CuO2 plane. arrangements with chainlets of length three that contribute two fractional holes to the CuO2 plane. There are: 2  5 2   4  = 12; (D.4) arrangements with chainlets of length two that contribute a single fractional hole to the plane. The remaining four arrangements do not contain any hole-contributing chainlets. Therefore, on average, there are a total of: ny = (2)(4) + (1)(12) + (0)(4) = 20; (D.5) fractional holes contributed to the CuO2 planes when 20 half-led Cu-O chains with six available sites are considered. Twenty fully occupied Cu-O (y = 1) chains of length six would result in: n1 = (5)(20) = 100; (D.6) fractional holes. If Eq. D.1 is correct, then the ratio ny=n1 is expected to be ap- proximately equal to y2. For the case considered, ny=n1 = 0:200 and y 2 = 0:250. 251 D.1. Numerical Conrmation The simple counting argument presented above underestimates the doping because of end eects. Chainlets are necessarily interrupted at the leftmost and rightmost Cu-O chain sites. A value of ny=n1 that is much closer to y 2 can be obtained by using circular boundary conditions. That is, arrange the available Cu-O chain sites into loops so that there are no ends. Alternatively one expects the end eects to be downplayed in the limit of very long Cu-O chains. Figure D.2 shows all of the: 8 4  = 70; (D.7) possible arrangements of half-lled Cu-O chains with eight oxygen sites. Table D.1 summarizes the counting of the fractional holes contributed to the CuO2 planes. The 70 distinct congurations contribute a total of ny = 105 fractional holes to the CuO2 planes. When the 70 Cu-O chains are fully oxygenated they contribute n1 = (7)(70) = 490 fractional holes, such that ny=n1 = 0:214 which, as expected, chainlet fractional holes frequency total fractional length per chainlet holes contributed 4 3  5 1  = 5 15 3 2 2  6 2   5  = 20 40 2 2 1+1  6 2   5  = 10 20 2 1 3  7 3   5 10 10  = 30 30 1 0  8 4   5 20 10 30 = 5 0 70 105 Table D.1: Summary of chainlet lengths and fractional hole contributions of half-lled Cu-O chains with eight available oxygen ion sites. There are 105 fractional holes due to the 70 equally likely Cu-O chain congurations. 252 D.1. Numerical Conrmation Figure D.2: There are 70 ways to ll eight Cu-O chain sites with four oxygen ions. Of the 70, 5 contribute three fractional holes, 20 contribute two fraction holes, 10 contribute (1+1) fractional holes, 30 contribute one fractional hole, and the remaining 5 do not contribute any holes to the CuO2 planes. is closer to y2 = 0:250 than the same ratio for Cu-O chains with six available sites. Figure D.3 conrms that ny=n1 approaches y 2 as the Cu-O chain lengths are increased. 253 D.2. Combinatorics Analysis of Ortho-I YBa2Cu3O6+y Doping 0 10 20 30 40 50 0.00 0.05 0.10 0.15 0.20 0.25 0.30   pr op or tio na l t o C uO 2 p la ne  d op in g # Cu-O Chain Sites y2 = 0.25 Figure D.3: The ratio ny=n1 is proportional to the doping of the CuO2 plane. For completely random ortho-I ordering, ny=n1 approaches y 2 as the Cu-O chain lengths are increased. The data shown are for y = 0:5 which corresponds to half-lled Cu-O chains. D.2 Combinatorics Analysis of Ortho-I YBa2Cu3O6+y Doping We now show that the doping of ortho-I YBa2Cu3O6+y is in general proportional to y2. The clever proof is due to Mona Berciu [195] and is presented with her permission. Consider a Cu-O chain with a total of N +m sites, of which m are lled, so that: y = m N +m : (D.8) The total number of possible distinct congurations is: NT = C m N+m =  N +m m  = (N +m)! m!N ! ; (D.9) where the more compact notation CmN+m has been introduced for the binomial co- ecient. These congurations can be subdivided into congurations with a single chainlet, with two distinct chainlets, with three distinct chainlets, and so on up to 254 D.2. Combinatorics Analysis of Ortho-I YBa2Cu3O6+y Doping congurations with m distinct chainlets. Note that, in this context we consider a sin- gle isolated oxygen atom as a chainlet. For example, congurations with m distinct chainlets correspond to m isolated oxygen atoms. Consider the congurations with a single chainlet. These are possible if all m oxygen atoms come one after another. The total number of such congurations is: n1 = C 1 N+1: (D.10) The counting is illustrated in Fig. D.4. If the m consecutively oxygen atoms are One chainlet Two distinct chainlets Figure D.4: The black circles are occupied sites and the white circles are unoccupied. Left: Combinatorics for one chainlet. The large black circle is a single composite object that corresponds to all of them consecutively lled sites. Right: Combinatorics for two distinct chainlets. An empty site is attached to the right end of the left chainlet. grouped into a single composite object, one has N + 1 objects in total, any one of which can be selected to be the chainlet. Each of these congurations will contribute m 1 holes to the Cu-O chain layer or m 1 fractional holes to the CuO2 plane. Next, consider congurations with two distinct chainlets. First, assume that the left chainlet has m1 sites and therefore the right one will have m2 = m  m1. Of course, m1 can be anything from 1 to m1. The rst question is: How many distinct congurations exist for these two chainlets arranged in this order (m1 on the left and m2 on the right)? The answer is: n2 = C 2 N+1; (D.11) and again Fig. D.4 demonstrates how the counting works. To ensure that there is at least one empty site between the two chainlets (so that they remain distinct), a single unoccupied site is attached to the right end of m1 and is included as part of 255 D.2. Combinatorics Analysis of Ortho-I YBa2Cu3O6+y Doping the composite object of m1 lled sites and a single empty site. So there are still a total of N +1 objects and one needs the number of arrangements of the two chainlets with m1 always to the left of m2. Following the same logic, the number of ways to have q chainlets (in some well dened order, the rst always with m1 sites, the second with m2 sites, . . . ) is: nq = C q N+1: (D.12) By considering q composite objects representing the chainlets and attaching an empty site to the leftmost q  1 of them, there is no possible contact between the chainlets and there are a total of N + 1 objects. The number of ordered arrangements nq is then given by selecting the q chainlets from the N + 1 objects. The next question is: In how many distinct ways can the m lled sites be divided into q  m chainlets? If q = 1, of course, there is only one way. If 2  q  m, there are: fq = C q1 m1; (D.13) distinct ways. Figure D.5 illustrates the counting. One must insert q  1 boundaries Figure D.5: The crosses represent breaks in the chainlets. These crosses can represent either a single empty Cu-O chain site or a collection of continuous empty sites. One must insert q  1 crosses into m 1 available sites to create q chainlets. to split the m lled Cu-O chain sites into q chainlets. There are m 1 positions for these boundaries (there is always one chainlet at the beginning and one at the end of the sequence of chainlets and boundaries, see the gure). Before proceeding to calculate the doping, we rst conrm that all of the NT distinct congurations as given by Eq. D.9 are accounted for. If the analysis above is correct, one expects that NT should be equivalent to the sum over q of nq (the number of ways to have q chainlets in some well dened order) multiplied by fq (the number of ways m lled sites can be divided into q chainlets): NT = mX q=1 fqnq = mX q=1 Cq1m1C q N+1: (D.14) 256 D.2. Combinatorics Analysis of Ortho-I YBa2Cu3O6+y Doping This sum can be evaluated by making use of the following two identities from the binomial expansion: (1 + x)N+1 = N+1X k=0 CkN+1x k; (D.15a)  1 + 1 x m1 = m1X q=0 Cqm1 1 xq : (D.15b) Multiplying these two identities together gives: (1 + x)N+1  1 + 1 x m1 = N+1X k=0 CkN+1 m1X q=0 Cqm1x kq: (D.16) Now consider only the terms that are proportional to x and equate the coecients from the left hand side (lhs) and right hand side (rhs) of the equation. From the rhs, we need only consider terms with k = q + 1 such that the rhs coecient is: m1X q=0 Cqm1C q+1 N+1 = mX q=1 Cq1m1C q N+1; (D.17) where the index q has been shifted by one q ! q1. Note that this is exactly the sum that we are trying to evaluate from Eq. D.14 and it must be equal to the coecient of x on the lhs of Eq. D.16. Rearranging the lhs of this equation gives: (1 + x)N+1  1 + 1 x m1 = (1 + x)N+m xm1 = 1 xm1 N+mX `=0 C`N+mx `; (D.18) such that the coecient of the term proportional to x is CmN+m and: NT = mX q=1 fqnq = mX q=1 Cq1m1C q N+1 = (N +m)! m!N ! ; (D.19) has been conrmed. Cu-O Chain Holes To nd the average number of holes in the Cu-O chain layer, it is essential to notice that any conguration with one chainlet contributes m  1 holes; any conguration with two chainlets contributes m  2 holes; any conguration with three chainlets 257 D.2. Combinatorics Analysis of Ortho-I YBa2Cu3O6+y Doping contributes m  3 holes. . . It does not matter how the lengths of the chainlets are chosen, all fq ways to dividem lled sites into q chainlets contribute the same number (m q) of holes. Therefore: hNhi = 1 NT mX q=1 (m q)fqnq: (D.20) This equation may be expressed in the following form: hNhi = m 1 1 NT mX q=1 (q  1)fqnq; (D.21) since: m 1 1 NT mX q=1 (q  1)fqnq =  1 NT " NT(1m) + mX q=1 (q  1)fqnq # =  1 NT mX q=1 [(1m)fqnq + (q  1)fqnq] = 1 NT mX q=1 [(m q)fqnq] : (D.22) Now the quantity N0 is dened such that: hNhi = m 1 N0 NT : (D.23) Once the sum: N0  mX q=1 (q  1)fqnq = mX q=1 (q  1)Cq1m1CqN+1 = m1X q=0 qCqm1C q+1 N+1; (D.24) is evaluated one will have the average number of holes in the Cu-O chain layer. We employ the same strategy that was used to the evaluate the sum in Eq. D.14, except now the derivative of Eq. D.15a is used which, after a relabeling of the indices, gives: (m 1)(1 + x)m2 = m1X q=0 Cqm1qx q1: (D.25) 258 D.2. Combinatorics Analysis of Ortho-I YBa2Cu3O6+y Doping Multiplying this expression by Eq. D.15b, again after relabeling the indices, results in: (m 1)(1 + x)m2  1 + 1 x N+1 = m1X q=0 Cqm1q N+1X k=0 CkN+1x pk1: (D.26) This time, we keep only terms that are proportional to 1=x2 which requires that k = q + 1. The rhs coecient of this term is: m1X q=0 qCqm1C q+1 N+1 = N0; (D.27) which is exactly the sum we want to evaluate and must equal the 1=x2 coecient of: (m 1)(1 + x)m2  1 + 1 x N+1 = (m 1)(1 + x) N+m1 xN+1 ; (D.28) which is: N0 = (m 1)CN1N+m1 = (m 1)(N +m 1)! (N  1)!m! = (m 1)CmN+m1: (D.29) Finally, substituting this result into Eq. D.23 one nds the average number of holes to be: hNhi = m 1 (m 1)(N +m 1)! (N  1)!m! m!N ! (N +m)! = (m 1)  1 N N +m  = m(m 1) N +m : (D.30) There are a total of N +m oxygen sites in the Cu-O chain, so the concentration of holes is: p = hNhi N +m = m(m 1) (N +m)2 : (D.31) For large enough m, the -1 is negligible and p = y2 which is desired result. Of course, since the Cu-O chain holes are partially lled by electrons from the CuO2 planes, the hole doping of the copper-oxide planes also goes as y2. 259 D.3. Ortho-I YBa2Cu3O6+y Doping from Expectation Values D.3 Ortho-I YBa2Cu3O6+y Doping from Expectation Values This elegant (and more compact) proof is due to Chenggang Zhou [196] and makes use of some simple expectation values. Each oxygen site in a Cu-O chain has two possible states, empty (i = 0) and occupied (i = 1). If two neighboring sites are both occupied, hole number Nh increases by one. Assuming that the state of site i is completely independent of the state of site i + 1, the probability that any single site is occupied is given by y. It follows that the expected number of oxygen atoms in the chain is E[ P i i] = yN . The notation E[: : : ] stands for the expectation value of whatever appears inside the square brackets. The number of holes can be written as: Nh = N1X i=1 ii+1: (D.32) If at least one of i or i+1 is zero the number of holes is unchanged. Nh increases by one only when both the sites are occupied. The remaining step is to calculate the expectation value of Nh: E[Nh] = N1X i=1 E[ii+1] = (N  1)E[12] = (N  1)y2: (D.33) The second equality arises because all of the sites are independent and identical so that E[12] = E[ii+1] where i can be any integer between 1 and N  1. The probabilities P of the four possible combinations of ii+1 and the desired expectation value E[ii+1] are summarized in Table D.2. Once again, the result is that in ortho-I YBa2Cu3O6+y the hole concentration in the Cu-O chain layer, and therefore in the CuO2 plane, is proportional to y 2. 260 D.3. Ortho-I YBa2Cu3O6+y Doping from Expectation Values i i+1 ii+1 Probability, P Pii+1 0 0 0 (1 y)2 0 0 1 0 (1 y)y 0 1 0 0 y(1 y) 0 1 1 1 y2 y2 E[ii+1] = y2 Table D.2: Probabilities of the four distinct pairs of states of two neighbouring oxygen sites in a Cu-O chain. The expectation value E[ii+1] is obtained by summing the contributions along the last column. A nonzero contribution occurs only when both sites are occupied. 261 Appendix E Extracting 1(!) from Measurements of RS(!) This appendix outlines procedures for extracting quasiparticle conductivity 1(!; T ) spectra from surface resistance RS(!; T ) measurements. Some of the key results obtained from chapter 3 will be repeated for convenience. The surface impedance of a conductor is dened by: ZS(!; T ) = RS(!; T ) + iXS(!; T ) = s i0! (!; T ) ; (E.1) where (!; T ) = 1(!; T )i2(!; T ) is the complex conductivity. After some algebra, the real and imaginary parts can be separated such that: RS = s 0!( p 21 +  2 2  2) 2(21 +  2 2) ; (E.2a) XS = r 0! 2(21 +  2 2) 1qp 21 +  2 2  2 : (E.2b) Up to this point, the expressions are completely general and are equally valid for all conductors within the limit of local electrodynamics. For a superconductor with T  Tc, it is common to simplify these expressions by making use of the fact that 2  1 at microwave frequencies, in which case: RS  1 22 r 0! 2 ; (E.3a) XS  r 0! 2 : (E.3b) The reactive part of the conductivity 2(!; T ) is responsible for screening the magnetic 262 E.1. Two-Fluid Model elds. For a superconductor, the magnetic penetration depth (T ) is dened as: 1 2(T )  0!2(!; T ): (E.4) Making this substitution the simplied forms for the surface resistance and reactance are: RS (!; T )  1 2 20! 23 (T )1 (!; T ) ; (E.5a) XS (!; T )  0! (T ) : (E.5b) To extract the real part of the conductivity, independent measurements of both RS(!; T ) and (T ) are needed. E.1 Two-Fluid Model For a superconductor, one generally adopts a two-
uid model for the conductivity consisting both of normal 
uid and super
uid channels: (!; T ) = f1n(!; T ) + 1s(!; T )g  i f2n(!; T ) + 2s(!; T )g : (E.6) The perfect dc conductivity of a superconductor is captured by writing 1s as a delta- function: 1s(!) =  2 nse 2 m? (!); (E.7) where the coecient is set by the oscillator strength sum rule: 2  Z 1 0 1(!)d! = ne2 m? : (E.8) All of the surface resistance measurements made are at nite frequency and Eq. E.5a can therefore be rewritten as: RS(!; T )  1 2 20! 23 (T )1n (!; T ) : (E.9) Ultimately, the quantity we wish to extract from the measured surface resistance is 1n(!; T ) which is often referred to as the quasiparticle conductivity. At low temper- ature, a further approximation (T )! (0) is regularly made. There is a subtle point to consider, as dened in Eq. E.4, the penetration depth 263 E.2. Extracting 1n(!; T ) from RS(!; T ) includes contributions from both the super
uid and the normal 
uid. Normal 
uid screening is more eective and becomes nonnegligible at high frequency (skin depth decreases). At low frequency and when T  Tc, 2  2s and: 1 2L(T )  0!2s; (E.10) where L(T ) is the London penetration depth. All of the precision penetration depth measurements done by our group are made using the 1 GHz resonator46. At this frequency, the super
uid screening dominates the normal 
uid screening and (T )  L(T ) to a very good approximation except near Tc. E.2 Extracting 1n(!; T ) from RS(!; T ) The simplest way to obtain the quasiparticle conductivity from the measured surface resistance is to use Eq. E.9: 1n(!; T )  2RS(!; T ) 20! 23(0) ; (E.11) which is a very good approximation at microwave frequencies when T  Tc. This approximation neglects the normal 
uid contribution to the screening. To do better requires substantially more work because the full expression for RS(!; T ) = < [ZS(!; T )] given by Eq. E.2a must be used. As already discussed, at nite frequency: 1(!; T ) = 1n(!; T ); (E.12a) 2(!; T ) = 2n(!; T ) + 1 0!2L(T ) : (E.12b) The approach used is to phenomenologically model the quasiparticle conductivity as: 1n(!; T ) = 0 1 + (!=)y ; (E.13) where y = 2 is the usual Drude conductivity. The imaginary part of the normal 
uid 46The cavity measurements provide only the change in the penetration depth (T ) = (T )  (T0) where T0 is typically 1.2 K. The absolute zero-temperature penetration depth (0) must be obtained from another source. 264 E.2. Extracting 1n(!; T ) from RS(!; T ) conductivity is then obtained using the Kramers-Kronig transformation: 2n (!) = 2  P Z 1 0 !  2  !21n ( ) d ; (E.14) where the P denotes the principal value of the integral. The low-temperature mea- sured penetration depth (T ) at 1 GHz is assumed to be equivalent to L(T ). The sequence of steps used to generate 1n(!; T ) spectra from the RS(!; T ) mea- surements is now outlined. 1. Generate an approximate quasiparticle conductivity spectrum from the RS(!; T ) data using Eq. E.11. 2. Fit the approximate quasiparticle conductivity to the phenomenological expres- sion given by Eq. E.13 to extract approximate values for the parameters 0, , and y. 3. Fit the RS(!; T ) data to Eq. E.2a. As a rst cut, the contribution of 2n to 2 is neglected. However, the correct value of (T ) is used. The new estimates of 0, , and y should be very close to the nal parameters that are ultimately extracted. 4. Fit the RS(!; T ) data to Eq. E.2a and include 2n which is obtained from Eq. E.14. The value of 2n(!) must be calculated separately for each microwave frequency in the RS(!) data set. A specic model for 1n(!) is also needed. Equation E.13 is used with the parameters extracted from step 3. This form of 1n allows the integration over all frequencies be carried out. The input param- eters are varied to achieve a least squares minimization between the model and data set. The nal parameters 0, , and y are generated by this minimization. 5. Finally, Eq. E.2a is solved for 1n(!) and experimental data points are generated for the quasiparticle conductivity starting from the measured surface resistance: data1n (!) = s 8R4S(!)22(!) 40!R2S(!)2(!) + 20!2  p830!3R2S(!)2(!) + 40!4 8R4S(!) ; (E.15) where: 2(!) = 2  P Z 1 0  !  2  !2   0 1 + ( =)y  d  + 1 0!2L(T ) : (E.16) 265 E.2. Extracting 1n(!; T ) from RS(!; T ) Figure E.1 shows the extracted 8.6 K a-axis quasiparticle conductivity spectrum of an ortho-II YBa2Cu3O6:5 single crystal. There are two curves shown. The open circles represent the approximate analysis as given by Eq. E.11. The lled circles were obtained using the full analysis (Eqs. E.15 and E.16). Most of the dierence between the two curves is due to the dierence between (0) and (8:6 K). However, including the normal 
uid screening does result in a noticeable dierence. The approximate and full spectra become even more alike at lower temperatures. 0 5 10 15 20 0 5 10 15 20 25 30  Full Analysis  Approximate Analysis   1n  (1 06  -1 m -1 ) Frequency (GHz) Figure E.1: Extracted quasiparticle conductivity spectra of YBa2Cu3O6:5 at 8.6 K. The full analysis includes the temperature dependence of the penetration depth and screening by the normal 
uid. 266 Appendix F Sr2RuO4 Growth Unconventional superconductivity is extremely sensitive to even tiny amounts of im- purities and/or disorder making the preparation of these complex materials very challenging. This appendix describes just one of the many dierent crystal growth techniques used to make high-quality single crystals of unconventional superconduc- tors. The goal here is not to give a detailed description of the growth procedure start-to-nish, but rather to give an account of some of the most interesting aspects of Sr2RuO4 crystal growth. Shonichiro Kittaka and Taketomo Nakamura were both enormously helpful with regards to the crystal growth which was done in the labora- tory of the Quantum Materials group at Kyoto University. Structurally, Sr2RuO4 is nearly identical to the high-Tc cuprate (La,Sr)2CuO4. In addition, transport in both materials is highly anisotropic with conduction occurring in two-dimensional oxide planes. Yet, despite these similarities, there are remarkable dierences between the cuprate superconductors and Sr2RuO4. For example, the superconducting transition temperatures dier by nearly two orders of magnitude and whereas the \normal" (or non-superconducting) state of the cuprates is highly anomalous, above Tc, Sr2RuO4 is a well behaved Fermi liquid [197]. Most signicantly, spin susceptibility measure- ments through the superconducting transition show drastically dierent behavior in the cuprates and Sr2RuO4 [198]. In the normal state, a weak external magnetic eld polarizes the Fermi surface into a spin up and spin down components which gives rise to a spin susceptibility. In the superconducting state of a spin singlet superconductor, Cooper pairs of electrons with antiparallel spins do not contribute to the polarization of the Fermi surface and the spin susceptibility is suppressed. This suppression has been observed in the spin-singlet d-wave superconductor YBa2Cu3O7. In contrast, there is no change in the spin susceptibility of Sr2RuO4 when one crosses over to the superconducting state. The simplest interpretation of this result is that Sr2RuO4 is a spin triplet superconductor with electron pairs of parallel spins. The application of a weak magnetic eld changes the relative number of pairs with spins parallel and an- tiparallel to the magnetic eld, but leaves the spin susceptibility unchanged [197, 198]. 267 F.1. Floating Zone Crystal Growth F.1 Floating Zone Crystal Growth Single crystals of Sr2RuO4 were grown in an image furnace by the 
oating zone method [199]. The image furnace focuses light rays from an infrared lamp onto a small region within the furnace where temperatures up to 2500C are obtained. The focusing is obtained by placing the infrared lamp at one of the two foci of an ellipsoid with highly re
ecting walls. Any light ray that leaves the lamp will pass through the other focus after a single re
ection47. The image furnace is equipped with a camera allowing one to watch and record the entire crystal growth. As shown in Fig. F.1, during the growth (a) the end of a polycrystalline feed rod is heated until Figure F.1: Digital images from a Sr2RuO4 growth run. See the text for descriptions of photographs (a) through (d). molten and (b) brought into contact with a seed crystal. (c) As the molten material is moved out of the focus of the light it solidies as single crystal Sr2RuO4. Finally, after approximately one hour, (d) the feed rod is disconnected from the single crystal. Typically Sr2RuO4 crystals grown by this method are 5 to 8 cm long and 23 mm2 in cross-section. Figure F.2 shows two single crystals of Sr2RuO4 grown by the 
oating zone method. Since RuO2 actively evaporates during the single crystal growth, the starting composition needs to be carefully selected. The polycrystalline feed rod is prepared by mixing SrCO3 and RuO2 powders in the ratio 2SrCO3+nRuO2. The powder is then placed into a mold and compressed under high pressure. The resulting rod is baked at 1420C and the reaction 2SrCO3 + nRuO2 !Sr2RunO2n+2 + 2CO2 proceeds. This polycrystal feed rod is then suspended in the furnace such that its 47To see how an image furnace works, consider the classic way to draw an ellipse. Put two nails in a board and place a loop of string on the board such that both nails pass through the loop. Pull the string taunt with a pencil and trace a line by encircling the two nails while always keeping the string taunt. The resulting line is an ellipse and the two nails mark the foci. Ignoring the straight line path from one focus to the other, note that at all points on the ellipse the string connects to two foci. The string represents the path of a light ray from one focus to the other after a single re
ection. 268 F.2. O2 Annealing Figure F.2: Digital photograph of two as-grown Sr2RuO4 single crystals. The scale on the ruler is in centimeters with millimeter increments. tip is at the focus of the heating. Motorized stages allow for control of the vertical positions of feed rod and seed crystal. In this way, the entire feed rod can be passed through the focus of the heating. Crystals are grown in a 10% O2 in Ar atmosphere at a pressure of 2.5 bar. During the 
oating zone growth RuO2 evaporates from the feed rod via Sr2RunO2n+2 !Sr2Run0O2n0+2 + (n  n0)RuO2 and n must be chosen such that n0  1 is achieved. A typical value of n is 1.15. Values of n0 6= 1 lead to the formation of a non-stoichiometric compound. Since RuO2 more readily evaporates from the surface of the feed rod, the centre of the nal crystal can be Ru rich. Under some growth conditions, Sr2RuO4 crystals cut and polished to expose the core region reveal dilute patches of Ru metal in a stripe pattern. Associated with the Sr2RuO4-Ru composite structure is an enhancement of Tc up to 3 K. This phenomenon is not well understood, but thought to be related to an increased ruthenium oxide interlayer coupling mediated by the Ru metal [199]. F.2 O2 Annealing Good measures of Sr2RuO4 sample quality are the superconducting transition tem- perature and its width. The superconducting transition of samples cut and cleaved from the as-grown single crystals were measured using the ac susceptometer described in x4.2. The in-phase component of the signal 0 represents the diamagnetic response of the superconducting state and the out-of-phase component 00 is due to dissipative processes. The red data in Fig. F.3 shows the temperature dependence of 0 and 00 for a sample cut from an as-grown Sr2RuO4 single crystal. The feature above 1.4 K is a 269 F.2. O2 Annealing 0.0 0.2 0.4 0.6 0.8 1.0  as grown  O2 annealed   ' ( ar bi tra ry  u ni ts ) 1.00 1.25 1.50 1.75 2.00 0.0 0.2 0.4 0.6 0.8 1.0    " (a rb itr ar y un its ) Temperature (K) Figure F.3: The temperature dependence of 0 and 00 of the same Sr2RuO4 sample before (red) and after (blue) a 12 hour O2 anneal at 900 C. The O2 anneal suppresses the anomalous Tc enhancement mediated by excess Ru metal. result of the Tc enhancement associated with excess ruthenium in the sample. The blue data is from the same sample after an anneal in one atmosphere of 
owing oxygen at 900C for 12 hours. The O2 anneal completely suppresses the anomalous feature leaving just the narrow superconducting transition. The peak of 00 is at a temperature of Tc = 1:41 K and the full width at half maximum (FWHM) of the peak 270 F.2. O2 Annealing is Tc = 25 mK. This width compares favorably with the results of Mao et al. who report Tc = 32 mK for a Tc = 1:49 K sample [199]. The improvement is a result of the high-temperature O2 anneal causing the excess Ru to migrate to the surface where it oxidizes to become RuO2. Rinat Ofer and Pinder Dosanjh are continuing the Sr2RuO4 oxygen annealing studies. 271


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