UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Shubnikov-de Haas measurements and the spin magnetic moment of YBa₂Cu₃O₆.₅₉ Ramshaw, Brad 2012

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2012_fall_ramshaw_brad.pdf [ 31.81MB ]
Metadata
JSON: 24-1.0072925.json
JSON-LD: 24-1.0072925-ld.json
RDF/XML (Pretty): 24-1.0072925-rdf.xml
RDF/JSON: 24-1.0072925-rdf.json
Turtle: 24-1.0072925-turtle.txt
N-Triples: 24-1.0072925-rdf-ntriples.txt
Original Record: 24-1.0072925-source.json
Full Text
24-1.0072925-fulltext.txt
Citation
24-1.0072925.ris

Full Text

Shubnikov-de Haas Measurements and the Spin Magnetic Moment of YBa2Cu3O6.59 by Brad Ramshaw  Bachelor of Science (Hons.) - The University of British Columbia, 2007  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) July 2012 c Brad Ramshaw 2012  Abstract High-temperature superconductivity (high-Tc ) was discovered in 1986 in copper-oxide materials, and since that time the goal of understanding high-Tc has driven the advancement of theoretical and experimental condensed matter physics. Despite the concerted efforts of some of the brightest minds in physics over the past 26 years, there is still no microscopic understanding of these materials. One of the main problems is an uncertainty as to whether Fermi liquid theory, which has been the foundation of our understanding of conventional metals for over 50 years, can be used to describe the strange pseudo-metallic properties of the cuprates. This thesis studies the resistivity of the high-Tc superconductor YBa2 Cu3 O6+x (YBCO) in magnetic fields up to 70 Tesla. These resistivity measurements show oscillatory behaviour as a function of magnetic field, which is a clear signature of a Fermi surface. The development of an advanced technique (based on a genetic algorithm) for analyzing the oscillatory resistance is presented, and the Fermi surface of YBa2 Cu3 O6.59 is determined with great precision by analyzing the field, angle, and temperature dependences of the oscillations. Analysis of the data shows that the electronic g factor, related to the strength of quasiparticle spin magnetic moment, does not experience strong renormalization in YBCO, in contrast with previous experimental studies. This lack of renormalization has important implications for theoretical descriptions of YBCO. A full description of the shape of the Fermi surface of YBCO is presented, and measurements of YBCO with different oxygen concentrations give the evolution of the Fermi surface with hole doping. A novel technique for fine-tuning the hole doping in YBCO is presented in the context of a Hall coefficient experiment. The result is a detailed doping dependence of the Hall coefficient, indicating that the Fermi surface seen in quantum oscillation experiments is influenced by some type of electronic order—such as charge and spin stripe order—competing with superconductivity near 1/8th hole doping. The behaviour of the superconducting vortex lattice in a magnetic field is analyzed as a function of temperature, and this behaviour also indicates that something is competing with superconductivity near 1/8th doping.  ii  Preface The study of the spin-zeros in YBa2 Cu3 O6.59 , presented in chapter 7, has been published in Nature Physics, 7(3):234-238, Mar 2011 [79]. The samples were prepared by D. A. Bonn, W. N. Hardy, Ruixing Liang, and B. J. Ramshaw (crystal growth, annealing, detwinning, and contacts); the measurements were performed at the Laboratoire National des Champs Magn´etiques Intenses in Toulouse, France by James Day, Cyril Proust, B. J. Ramshaw, and Baptiste Vignolle; the data analysis was carried out by B. J. Ramshaw; the manuscript was written by D. A. Bonn and B. J. Ramshaw, and the project was supervised by D. A. Bonn. Everything presented in chapter 9—the study of the Hall effect—was my work, including the sample preparation, the measurement, and the analysis. Part of this work on the Hall effect was used in a larger project which has been published in Nature, 450(7169):533536, Nov 22 2007 [53]. The samples were prepared by S. Adachi, D. A. Bonn, W. N. Hardy, Ruixing Liang, and B. J. Ramshaw; measurements were performed by D. A. Bonn, L. Balicas, J.-B. Bonnemaison, R. Daou, N. E. Hussey, Nicolas Doiron-Leyraud, David LeBoeuf, Cyril Proust, and B. J. Ramshaw; the manuscript was written by Nicolas DoironLeyraud, David LeBoeuf, and Louis Taillefer, and the project was supervised by Louis Taillefer. Results from the study of the Hall effect, presented in chapter 9, were also published in Physical Review B, 83:054506, Feb 2011 [54]. The samples were prepared by D. A. Bonn, W. N. Hardy, Ruixing Liang, and B. J. Ramshaw; the measurements were performed by D. A. Bonn, L. Balicas, Johan Chang, Olivier Cyr-Choini`ere, R. Daou, Nicolas Doiron-Leyraud, Y. J. Jo, Francis Lalibert´e, David LeBoeuf, J. Levallois, Cyril Proust, Mike Sutherland, B. J. Ramshaw, and B. Vignolle; the manuscript was written by David LeBoeuf and Louis Taillefer, and the project was supervised by Louis Taillefer.  iii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Preface  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  Glossary  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Acknowledgments  xxi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xxii  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  The Free-Electron Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1.1  3  The Lattice Potential . . . . . . . . . . . . . . . . . . . . . . . . . .  1.2  Fermi Liquids  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  1.3  The Cuprates  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  1.3.1  History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7  1.3.2  The Cuprate Phase Diagram . . . . . . . . . . . . . . . . . . . . . .  9  1.3.3  Fermi Surfaces in the Cuprates . . . . . . . . . . . . . . . . . . . . .  12  2 Quantum Oscillations  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19  2.1  Landau Levels  2.2  Cyclotron Orbits in a Magnetic Field  . . . . . . . . . . . . . . . . . . . . .  22  2.3  Thermodynamic Potential and Density of States . . . . . . . . . . . . . . .  26  2.3.1  . . . . . . . . . . . . . . . . . . . . . . . .  28  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  29  2.4  2.5  Quasi-2D Fermi Surfaces  Phase Smearing 2.4.1  Spin Splitting  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  30  2.4.2  Finite Quasiparticle Lifetime . . . . . . . . . . . . . . . . . . . . . .  31  2.4.3  Finite Temperature  . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  2.4.4  Full Expression for the Oscillatory Thermodynamic Potential . . . .  34  Angle Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  34  iv  Table of Contents 2.6  Shubnikov-de Haas  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3 Experimental Technique  38  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  41  3.1  DC Electrical Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  42  3.2  Pulsed Magnetic Fields  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  44  3.3  Temperature Control  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  3.4  Field Angle Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  4 Sample Preparation  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.1  As-Grown YBa2 Cu3 O6.x  4.2  Selecting the Appropriate Sample 4.2.1  4.3 4.4  48  . . . . . . . . . . . . . . . . . . . . . . .  49  . . . . . . . . . . . . . . . . . . . . . . . . . .  52  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  Sample Modification  Electrical Contacts 4.3.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  48  Masking and Gold Evaporation  . . . . . . . . . . . . . . . . . . . .  55  Oxygen Homogenization and Contact Annealing . . . . . . . . . . . . . . .  58  4.4.1  58  Citric Acid Etch . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.5  Oxygen Ordering  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  59  4.6  Attaching Wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  61  5 Data Acquisition and Analysis 5.1  5.3  63  . . . . . . . . . . . . . . . . . . . . . . . . .  64  Calculating the Angle . . . . . . . . . . . . . . . . . . . . . . . . . .  68  Angle and Field Measurement 5.1.1  5.2  . . . . . . . . . . . . . . . . . . . . . . . . .  Background Removal  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  73  5.2.1  User-Controlled Background Removal . . . . . . . . . . . . . . . . .  74  5.2.2  Fitting the Background . . . . . . . . . . . . . . . . . . . . . . . . .  74  Building a Model  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  78  5.3.1  Field Dependence  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  78  5.3.2  Preliminary Temperature Dependence . . . . . . . . . . . . . . . . .  79  5.3.3  Preliminary Angle Dependence . . . . . . . . . . . . . . . . . . . . .  81  5.3.4  Preliminary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .  83  Limitations of the Fourier Transform Technique in YBa2 Cu3 O6.x . . . . . .  87  6 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  92  5.4  6.1  C  ++  Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  93  6.1.1  Optimization Goals  . . . . . . . . . . . . . . . . . . . . . . . . . . .  94  6.1.2  Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  95  6.1.3  Algorithm Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . .  96  7 Angle Dependence of YBa2 Cu3 O6.59  . . . . . . . . . . . . . . . . . . . . . .  100  7.1  Angle Dependence Up to 35 Degrees . . . . . . . . . . . . . . . . . . . . . .  100  7.2  Angle Dependence from 40 to 57 Degrees . . . . . . . . . . . . . . . . . . .  109 v  Table of Contents 7.3  Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . .  111  7.4  Summary of YBa2 Cu3 O6.59 . . . . . . . . . . . . . . . . . . . . . . . . . . .  112  7.4.1  Fermi Pocket Locations . . . . . . . . . . . . . . . . . . . . . . . . .  113  7.4.2  Quasiparticle Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . .  114  7.4.3  Cyclotron Mass  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  116  7.4.4  Quasiparticle Spin Magnetic Moment . . . . . . . . . . . . . . . . .  117  7.4.5  Parameter Uncertainties  . . . . . . . . . . . . . . . . . . . . . . . .  118  8 Other Dopings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  120  8.1  Oxygen 6.47  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  120  8.2  Oxygen 6.51  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  124  8.3  Oxygen 6.59  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  128  8.4  Oxygen 6.67  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  131  8.5  Oxygen 6.75  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  135  8.6  Conclusions from the Doping Dependence . . . . . . . . . . . . . . . . . . .  138  8.6.1  Doping Dependence of the Fermi Surface . . . . . . . . . . . . . . .  138  8.6.2  Doping Dependence of the Vortex-Lattice Melting Behaviour . . . .  139  9 In-Plane Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  145  9.1  Order-Disorder Doping  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  145  9.2  Conclusions from the Hall Experiment . . . . . . . . . . . . . . . . . . . . .  155  10 Conclusion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  157  10.1 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  158  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  161  Appendices A Genetic Algorithm Code  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  174  A.1 DataExtractor.cpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  174  A.2 Parameters.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  176  A.3 GeneticAlgorithm.cpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  177  A.3.1 initializeParameters . . . . . . . . . . . . . . . . . . . . . . . . . . .  177  A.3.2 calculateMinimum . . . . . . . . . . . . . . . . . . . . . . . . . . . .  179  A.3.3 calculateNewGenerations . . . . . . . . . . . . . . . . . . . . . . . .  179  A.3.4 startResidualThread . . . . . . . . . . . . . . . . . . . . . . . . . . .  182  A.3.5 residualCalculatingThread  . . . . . . . . . . . . . . . . . . . . . . .  182  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  183  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  183  A.3.6 calculateResidual A.3.7 getYValue  vi  Table of Contents A.3.8 integrateLegendre . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 FittingProgram.cpp  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  B Background Subtraction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  184 185 187  vii  List of Tables 5.1  The rotator angle: nominally, and as calculated using Equation 5.6 and fits to the data in Figure 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5.2  The sample angle: nominally, and as shown in Table 5.1 plus an offset of 1.06 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7.1  70 73  The ranges for the fit parameters for the two pieces of Fermi surface. These estimates are based on the qualitative data analysis performed in section 5.3. 101  7.2  The two sets of fit parameters corresponding to the minima reached by the purple and red curves in Figure 7.1. The warping on surface 2 is essentially zero. The only qualitative difference between the two minima is the value of ms on sheet 1, as well as the phase. The phase, however, is different by about π, which is equivalent to multiplying the amplitude by a negative sign, thus the only real difference lies in the ms value. This difference is plotted in Figure 7.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7.3  103  The Fermi surface parameters, and their uncertainties, for YBa2 Cu3 O6.59 . The parameter gmin is the minimum value for g assuming zero electronphonon coupling, as described in subsection 7.4.4. The uncertainties labelled ∆θ , ∆T , and ∆B come from the uncertainty in angle, temperature, and field, respectively. The column ∆T otal is the quadrature sum of all of the uncertainties.  8.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  119  Fit parameters for the temperature and field dependence of the oscillatory magnetoresistance of YBa2 Cu3 O6.51 . ms is not reported because extraction of that parameter requires the angular dependence of the oscillations. . . .  8.2  126  Fit parameters for the temperature dependence of YBa2 Cu3 O6.67 . ms is not reported because extraction of that parameter requires the angle dependence of the oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8.3  133  The Fermi surface parameters for YBa2 Cu3 O6.51 , YBa2 Cu3 O6.59 , and YBa2 Cu3 O6.67 . The parameter gmin is the minimum value for g assuming zero electronphonon coupling, as described in subsection 7.4.4.  . . . . . . . . . . . . . .  139  viii  List of Tables 8.4  The superconducting correlation length as obtained by fitting the vortex lattice melting curves to Equation 8.2, with cL = 0.3. There is an absolute uncertainty of ±6 ˚ A for the ξ values, coming from the choice of cL between 0.2 and 0.4, but for a fixed cL the uncertainty is relatively small. . . . . . .  9.1  142  The various hole doping values obtained during the order-disorder experiment. Note that “re-ordered” is not supposed to imply that the sample is as ordered as it was to begin with; this would require a much longer annealing period than was used to obtain doping levels in the third column.  . . . . .  147  ix  List of Figures 1.1  Left: dispersion relation for free electrons, drawn in the reduced zone scheme. The red bands can be thought of as free electron bands centred around (k ± 2π), or as the original black band folded from k = π → −π, k = −π → π. Right: a lattice potential with period 1 is introduced, opening up gaps wherever the bands in the previous panel crossed. . . . . . . . . . . . . . . .  1.2  4  The crystal structure of YBa2 Cu3 O7 . The copper-oxide chain layer, responsible for doping, can be seen at the top and bottom of this figure, and are completely full in O7 . The copper oxide planes, responsible for superconductivity, are in the middle of the diagram on either side of the yttrium ion. Image used with permission from Peets [73]. . . . . . . . . . . . . . . . . . .  1.3  8  The phase diagram of the high-Tc cuprates, with data for YBa2 Cu3 O6.x . Region I is the superconducting dome: the blue line is the function 1 − Tc /Tc,max = 82.6(p − 0.16)2 , with Tc,max = 94.6 K for YBCO [59]. This function is phenomenological, but seems to describe Tc as a function of p for many cuprates. The blue data points are actual Tc values for YBCO, taken from Liang et al. [59]. The suppression of Tc around p = 0.125 is also characteristic of many cuprates. In region II, YBCO is an antiferromagnetic insulator. The red data points are from unpublished cˆ-axis resistivity measurements of the N`eel ordering temperature, TN [80]. The orange line is a parabolic fit to this data, including TN = 450 at p = 0 (not shown on this scale) from Tranquada [102]. Region III is the pseudogap region, bounded above by T . The green data points mark the deviation from linear resistivity [20], which is characteristic of the “strange-metal” found in region IV. Region V displays properties consistent with a Fermi liquid description of the quasiparticles, including a coherent three-dimensional Fermi surface in Tl2 Ba2 CuO6+x , and T 2 resistivity in several cuprates [42, 44]. The boundary between regions IV and V is purely schematic, as the crossover from T -linear to T 2 resistivity is gradual. The boundary between regions III and IV is a linear fit to the green T data, and should also be considered phenomenological.  . . . . . .  10  x  List of Figures 1.4  The Fermi surface of Tl2 Ba2 CuO6+x with a Tc of 30 K, as determined by angle-resolved photoemission spectroscopy (ARPES) [78]. The black line is a fit to a tight binding model. The area of the hole pocket agrees with the area measured by angular magnetoresistance oscillations (AMRO) and quantum oscillations. Figure reproduced with permission from Plat´e et al. [78]. . . .  1.5  16  Left:the Fermi surface of as-cleaved YBa2 Cu3 O6.50 , with an effective hole doping of p = 0.28. The Fermi surface is qualitatively similar to that seen in Tl2 Ba2 CuO6+x , shown in Figure 1.4. Right: the Fermi surface of YBCO after potasium deposition on the surface. The large hole pocket has shrunk to “arcs” along the nodal directions of the Brillouin zone. The line-like objects running left to right near the centre of the Brillouin zone are the chain bands. Figure reproduced with permission from Hossain et al. [40]. . . . . . . . . .  2.1  17  Schematic of Landau levels, zoomed in near the Fermi energy of a 2D-Fermi surface (a cylinder). Landau levels are Lorentzians in energy, and the FermiDirac distribution at finite temperature is shown as a dashed red line. Occupied states are filled in black. In the left panel the value of the magnetic field is such that the highest occupied state falls between the peaks of two Landau levels. As the magnetic field is increased (or decreased), the panel on the right shows that there is a large occupation of states near the peak of a Landau level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.2  22  A quasi-2D Fermi surface, with an extremal quasiparticle orbit shown as the thick black line. The orbit is perpendicular to the magnetic field, which is applied at an angle θ to the cˆ-axis. Equation 2.69 gives the area of this orbit in terms of an integral around the angle φ. Reproduced with permission from Bergemann et al. [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.3  36  The Rs spin factor term, Equation 2.73, plotted as a function of angle. The first spin zero is around 28◦ . Here, the oscillation amplitude is zero, and, beyond this angle, the oscillation phase shifts by π with respect to its value before the spin zero. This can also be thought of as the oscillations picking up an overall minus sign at each spin zero. . . . . . . . . . . . . . . . . . . .  3.1  38  Circuit diagram for the measurement, with an in-plane-contact-geometry (as opposed to cˆ-axis) shown for clarity. The voltage across the 1 kΩ resistor is measured with a separate voltmeter, giving the exact current through the sample, which will be near 5 milliamp (mA) since the lead, sample, and contact resistance are all much less than 1 kΩ. As long as the contact resistances at V1 and V2 are not too large, then the potential drop measured by the voltmeter will be equal to the potential drop across the sample between the voltage contacts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  43 xi  List of Figures 3.2  Magnetic field as a function of time during a pulse. The capacitor bank is disconnected from the circuit and the coil shorted with a shunt near the peak field at around 80 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.3  45  The resistance of a YBa2 Cu3 O6.56 sample during the rising and falling of the magnetic field, nominally at 8 kelvin. Note that the resistive transition happens at a lower field value on the down sweep. This is an indication that the sample has heated above the base temperature during the pulse. . . . .  4.1  46  BaZrO3 crucible after a crystal growth. Most of the crystals attached to the walls have been removed. The crucible is a pale cream colour before the growth, but the Y2 O3 -BaO-CuO melt leaches into the walls during the growth, turning it black and eventually destroying it.  4.2  . . . . . . . . . . . .  49  a ˆ-ˆ c face of a YBa2 Cu3 O6.x crystal showing twin boundaries. In the bottom half of the picture the twins run continuously from left to right; in the top half, there are several places where the twins run half way across the sample, and then abruptly change pattern on the right hand side. This is either a cˆ-axis stacking fault or an internal crack. The upper right of the picture also shows flux attached to the surface of the crystal. This crystal is approximately 70 µm thick in the cˆ direction, and approximately 1 mm by 1 mm in-plane. . .  4.3  51  An a-b face, showing the twin domains. Applying force to the two parallel edges at high temperature (200-300◦ C) will force the sample into one domain, provided that internal flaws are not so great as to pin the domains. This crystal’s dimensions are approximately 600 by 750 by 50 microns. . . . . . .  4.4  52  Detwinning stage showing the YBCO ceramic sample platform (the grey triangular sliver in between the gold pads) and the gold pads which apply pressure to parallel edges of the sample. The entire apparatus is housed in a vacuum chamber. The sample is placed across the end of the sample platform, as shown in the inset (rotated 90◦ from the main figure). When the gold pads are moved together, they make contact with the sample and detwin it (the top and bottom edges of the sample in the inset will have pressure applied to them when placed between the gold pads, orienting the crystallographic a ˆ-axis with the long-dimension of the sample). . . . . . . .  4.5  54  Copper sheet with 10 µm aluminum foil attached with Crystalbond. This provides a hard and flat surface for preparing masks. Several masks have been cut from this piece of foil already: the bow-tie shaped masks are for a ˆˆb-plane contacts, and the square sections are where cˆ-axis masks were made. A typical YBCO sample for cˆ-axis measurements is shown in the middle, and in the zoomed-in inset. This sample is 650 µm long, 350 µm wide, and about 30 µm thick. This is the same sample that is shown in Figure 4.6. . . . . .  56  xii  List of Figures 4.6  Top left: The foil mask is shown taped down to a larger sheet of backing foil. The sample— 650 µm x 350 x 30 µm—is shown above the mask, and will be slid underneath (the same sample was shown in Figure 4.5.) Top right: The sample has been slid between the mask and the backing foil, and the edges of the mask have been crimped around the sample, using tweezers, to prevent the sample from moving during the evaporation. Mid left: The backing foil, the mask, and the sample are taped to a large copper finger which is held during the evaporation. Mid right: The sample after the gold has been evaporated onto one side. This image is taken with cross-polarized light which emphasizes the edge of the gold, and also reveals twin boundaries (this sample was prepared only for illustrative purposes, and was not detwinned). The crystal will be flipped over, re-positioned under the mask, and gold will be evaporated on the other side. Bottom left: In-plane contact geometry used in chapter 9. The contacts are prepared in a similar way to the cˆ-axis contacts, but they wrap around the sides of the crystal to prevent cˆ-axis contamination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.7  57  Clockwise from top left: No chain oxygen, ortho-II, ortho-V,ortho-I, orthoVIII, and ortho-III. The a ˆ −ˆb crystallographic directions are shown in the first panel. Copper atoms are shown in red, and oxygen atoms, which have a larger ionic radius than copper, are shown in blue. The black arrows encompass one period of order, with the label of that state printed above the arrow.  4.8  .  60  The probe tip with two samples mounted on sapphire blocks. The wires are attached to the samples with silver paint in this image. The samples have been G.E.-varnished down to sapphire blocks to minimize vibration and movement due to torque during the pulse. The loose wire ends at the top and bottom of the picture will be trimmed and connected to the probe wiring. 62  5.1  Resistance of a YBCO sample as a function of time during the magnetic field pulse. The data is asymmetric about the maximum because the field rises faster than it falls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  64  5.2  Voltage in the pickup coil used to measured the field, as a function of time.  65  5.3  Magnetic field as a function of time during a pulse, obtained by integrating the data in Figure 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5.4  66  Raw resistance versus magnetic field for both the rising and falling field. Note the difference between the two: the sample was torqued to a slightly different angle during the rise of the field. . . . . . . . . . . . . . . . . . . . . . . . .  5.5  67  Voltage across the rotator pickup coil during the pulse plotted against the voltage across the field pickup coil. The slope is related to the angle of the rotator via Equation 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  68  xiii  List of Figures 5.6  The voltage-voltage curves for each of the angles of rotation during the angular dependence of YBa2 Cu3 O6.59 experiment. The nominal angles—the angles which were “aimed” for—are given in the legend.  5.7  Top: The resistance curves at the nominal angles of 1 B cos θ .  . . . . . . . . . .  ±30◦ ,  69  plotted against  The angles used in the scaling are those from Table 5.1, and the  temperature is 4.2 K. Bottom: The same data as above, but now plotted using an offset angle of 1.06◦ , which accounts for the fact that the sample is not perfectly aligned when mounted on the rotator. . . . . . . . . . . . . . . 5.8  71  Top: All of the raw resistance data as a function of angle and field. Middle: The same data shown in the top panel, but with the x-axis scaled by B → 1 B cos θ ,  with the angles taken from Table 5.1. Bottom: The same data as the  upper two panels but with the added offset of 1.06◦ . Note that the transition from the vortex solid to the vortex liquid now occurs at the same field value, as predicted in Blatter et al. [11]. 5.9  . . . . . . . . . . . . . . . . . . . . . . .  72  The Mathematica graphical interface for removing a background from the raw resistivity data. The grey circles are the control points for the red Bspline curve, which can be moved with the mouse. The inset shows the data with the background removed, which is updated in real time. The slider bar labelled ‘d’ sets the order of the B-spline: the default is a cubic spline.  5.10 Top panel : The data at  40◦  . .  75  after the background has been removed by hand.  Middle panel : The fit to the data in the upper panel is combined with an 8th order polynomial to fit both the background and the data at the same time. The total fit is shown in red, and the background by itself is shown in green. Bottom panel : The oscillatory component after a second iteration. Note the improvement at lower field values compared to the first iteration in the top panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 All of the data from 0◦ to 57◦ with the background subtracted.  . . . . . .  77 78  5.12 The absolute value of the oscillatory part of the resistance. The features of central importance are the beats near 32 and 47 tesla, and the fact that the beat does not drive the amplitude fully to zero. 5.13 The temperature dependence at an angle of of m for a quasi-2D Fermi surface is just  . . . . . . . . . . . . . . .  −30◦ .  m cos θ ,  79  The angular dependence  and so we can measure the  effective mass at any angle and get a consistent result. . . . . . . . . . . . .  80  5.14 The amplitude of four of the peaks seen in Figure 5.13 as a function of temperature. These amplitudes have been fit to the expression for RT to obtain an estimate for m . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  81  5.15 Top panel : The beat structure, easily visible in Figure 5.12, has almost completely disappeared near 35◦ . Bottom panel : The beat structure has begun to return by 47◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  82 xiv  List of Figures 5.16 The first six data sets, up to 35◦ . Note that the phase remains fairly constant, except perhaps near the node around 0.03 tesla−1 where the Bessel function changes sign. The constant phase recovers on either side of the Bessel function node.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5.17 The five data sets from  30◦  45◦  and is constant between angles. The data at  40◦  to  47◦ .  The phase between  and  47◦ ,  30◦  and  35◦  is constant,  with an offset from the previous two  seems to bridge the gap between them. . . . . . . .  5.18 The four data sets from  45◦  55◦ .  to  83  The phase between  45◦  and  47◦  84  is  constant, as is the phase between 50◦ and 55◦ , with an offset from the previous two angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  85  5.19 Three representative data sets showing the phase offset in each of the 3 regimes shown in Figure 5.16, Figure 5.17, and Figure 5.18. The phase shifts a bit when the spin zero is passed through for one Fermi surface just above 35◦ , and then the full 180◦ phase shift is seen near 50◦ once both surfaces have gone through the spin zero. Note that the data at 45◦ and 50◦ have had their amplitudes scaled in order to be visible when plotted with the data at 35◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  86  5.20 The relatively large Fermi surface of Sr2 RuO4 (shown in the upper inset) yields several thousand periods of oscillation between about 3 and 33 tesla. This large number of oscillations results in sharp peaks when the data is Fourier transformed (the lower inset). Figure reproduced with permission from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  88  5.21 Top panel : Individual Fourier transforms for Fermi surface pockets with areas of 450, 510, and 530 tesla. The data is taken between 20 and 70 tesla. Bottom panel : The magnitude of the Fourier transform of the sum of the three frequency components that were shown in the upper panel. The relative heights of the peaks depends strongly on the phase of the oscillations, which is allowed to vary between 0 and a phase of  π 4,  π 2  for the 510 tesla signal. Also note that for  the upper peak in the green curve is at 550 tesla: a frequency  that is not present in the actual data. . . . . . . . . . . . . . . . . . . . . .  90  5.22 Fourier transforms of the oscillatory magnetoresistance for O6.51 , O6.59 , and O6.67 . The field ranges for the transforms are 34.5→59.8 T, 27.5→69.2 T, and 36.5→60.0 T, respectively. The data has been apodized with a Kaiser window with α = 2.5 [33], and the amplitudes have all been normalized to 1. 6.1  A plot of cos  e  Sk (κz ,θ) B  91  , which is the argument of integral Equation 6.2.  Typical values needed to fit the YBa2 Cu3 O6.59 data were used. Judging by the number of times the derivative changes sign, one might expect to need a polynomial of at least order 18 to roughly approximate this curve. . . . . .  98  xv  List of Figures 7.1  Plot of the residuals from the fit to Equation 7.2 for various parameter values, with a Log-Log scale. There are 3 trials, in shades of purple, with s = .3, p = .9, and N = 200, which reach a minimum after 300 iterations. Two trials, in shades of blue, with s = .03, p = .9, and N = 200, reach their minima after 30 iterations, but the residual is much larger than when s = .3. Two trials, in shades of green and yellow, with s = .3, p = .09, and N = 200, indicate that they may eventually reach the same minimum as the p = .9 trials, but the convergence will take more than 5000 generations. Two trials, in shades of red, with s = .3, p = .09, and N = 1000, reach the same minima as with N = 200, but take more generations to minimize. . . . . . . . . . .  7.2  102  The absolute value of the spin factors (Rs ) for surface 1 and 2 from minimum 1 in Table 7.2. The blue curve is surface 1, and the red is surface 2. Note that there is no angle where the total amplitude is zero, thus searching for a “spin zero” is a hopeless task. The equivalent plot for minimum two would look very similar, except with slightly different behaviour for surface 1 beyond 35◦ (see Figure 7.3.)  7.3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  103  The two spin factors (Rs ) for surface 1 for the two different minima tabulated in Table 7.2. They have both been multiplied by the amplitude, A, to show that they are almost identical up to 35◦ , which is the the limit of the data used in the fits. The negative sign is accounted for by the π phase shift between the two minima for surface 1. . . . . . . . . . . . . . . . . . . . . .  7.4  The fit from minimum 2 in Table 7.2, plotted with the data, from 35◦ .  0◦  104  to  The second harmonic of the oscillations, which can make up roughly  10% of the signal at high field, was not included in the fits. This accounts for the deviations near the node at low angle. The background subtraction is inaccurate at the boundaries when fitting with a polynomial, leading to deviations in the fit from the data around 57 tesla, especially at 10◦ and −30◦ . 106 7.5  Plots of the two pieces of Fermi surface, using parameters from Table 7.2, minimum 2. The relative dimensions are to scale. The colour scale on the warped surface has been used to highlight the warping. . . . . . . . . . . . .  7.6  107  The warped section of Fermi surface, with F = 478 and ∆F = 40, in the Brillouin zone of YBCO. The positioning is for illustrative purposes only, to show the relative size of the pocket with respect to the zone. This image should should not imply that this is where the Fermi pocket is located, nor should it imply that there is only one copy of this pocket. The kx and ky axes have been scaled to run from −π to π, and all dimensions are to scale.  7.7  The fit to the data (using parameters from minimum 2) at θ =  40◦ .  108  The fit  would line up if the phase was shifted slightly. . . . . . . . . . . . . . . . . .  109  xvi  List of Figures 7.8  The oscillation amplitude as a function of angle, fitted individually without the Rs term, holding all other parameters except phase constant. The fit through the data is the Rs term, fitted to the first six data points (up to 35◦ .) The fit is quite good up to 40◦ , as might be expected since the model parameters (when fitting at each angle) came from a global fit to the data up to 35◦ . While the amplitude is overestimated at higher angles, the shape appears to be at least somewhat correct. The masses extracted from the fit (shown in the key) are very close to those in Table 7.2. Two values for ms for surface 1 are shown in blue: the smaller value fits the data better at low angles, and does not have the extra node around 55◦ as the higher ms value does. It appears that ms = 1.174 provides a better fit to the data. . . . . .  7.9  Fits to the temperature dependence at θ =  −30◦ ,  111  using m for each surface  as the free parameter, and fixing the other parameters to the values obtained in minimum 2 of Table 7.2. The fits are worst at the highest field values, as expected, since the background subtraction is most inaccurate near the boundaries. These fits yield m1 = 1.52me and m2 = 1.65me . . . . . . . . . .  112  7.10 Left: The blue lines are a schematic of the Fermi surface for a high Tc cuprate. With an ordering wave-vector Q = (π, π), shown in black, there is a second copy of the Fermi surface in the centre of the zone, shown in green. Right: An energy gap, proportional to the energy cost U of having two electrons doubly-occupy a copper site, breaks the degeneracy of the two bands and gives electron (red) and hole (black) pockets [16]. . . . . . . . . . . . . . . . 8.1  115  The magnetoresistance of YBa2 Cu3 O6.47 , at temperatures ranging from 1.5 to 120 K. No oscillatory behaviour is seen; the small bumps in the data ( near 57 tesla at 1.5 K, for example) are due to movement of the whole and cryostat during the field pulse. . . . . . . . . . . . . . . . . . . . . . . . . .  8.2  121  The resistance as a function of temperature for YBa2 Cu3 O6.47 in the cˆ-axis direction. The red dots are taken at 50 tesla, and the black curve is in zero field. It should be noted that there is strong magnetoresistance below about 30 K (see Figure 8.1 for the magnetoresistance).  8.3  . . . . . . . . . . . . . . .  122  The magnetic field where the vortex lattice melts and a resistive signal is first seen, plotted as a function of temperature. Models of this transition depend on the London penetration depth, the superconducting coherence length, the dimensionality of the superconductivity, and assumptions about how melting takes place [26].  8.4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  123  Magnetoresistance of YBa2 Cu3 O6.51 at temperatures ranging from 1.5 to 60 K. Oscillations can be seen above about 40 tesla, and persist up to around 10 kelvin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  124  xvii  List of Figures 8.5  The resistance as a function of temperature for YBa2 Cu3 O6.51 in the cˆ-axis direction. The red dots are taken at 50 tesla, and the black curve is in zero field. The abrupt dip below about 10 kelvin is right where quantum oscillations become visible (the resistance value is taken from the background average between the oscillation peaks.) . . . . . . . . . . . . . . . . . . . . .  125  8.6  The vortex lattice melting field Bmelt as a function of temperature up to Tc . 126  8.7  Quantum oscillations for YBa2 Cu3 O6.51 with the background subtracted. The fits are made to the same model used in section 7.1. The fit is worse at high fields and low temperatures because only the first harmonic is fit, and background subtraction is difficult near the boundaries. . . . . . . . . . . .  8.8  127  Magnetoresistance of YBa2 Cu3 O6.59 up to 60 tesla, from 1.5 to 200 K. At the highest fields, the peak-to-peak oscillation size is 50% of the size of the background.  8.9  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  128  Resistance of YBa2 Cu3 O6.59 in zero field from Tc = 60 K to 270 K, and at 50 tesla below Tc . As was seen in O6.51 , the resistance begins decreasing with temperatures below about 8 kelvin.  . . . . . . . . . . . . . . . . . . . . . .  129  8.10 The magnetic field value Bmelt where the vortex lattice melts. This shape is similar to what was seen in O6.47 and O6.51 . . . . . . . . . . . . . . . . . . .  130  8.11 Magnetoesistance from 1.5 to 60K. Quantum oscillations are visible at 1.5 and 4.2 K above ˜ 45 tesla. . . . . . . . . . . . . . . . . . . . . . . . . . . . .  131  8.12 cˆ-axis resistance at 0 and 50 tesla up to 270 K for O6.67 . The rollover at low temperature seems to be associated with the appearance of quantum oscillations, in this material as well as in O6.51 and O6.59 . . . . . . . . . . .  132  8.13 The vortex lattice melting transition in YBa2 Cu3 O6.67 , as determined by the onset of a finite resistance, up to Tc . The maximum Bmelt at low temperature is much lower for O6.67 than it is for both O6.59 and O6.75 . . . . . . . . . . .  133  8.14 The oscillatory component of the magnetoresistance at 1.5 and 4.2 K. The data is shown in shades of purple, and the fits are shown in shades of orange. The higher frequency component visible near the oscillation peaks at 1.5 K may come from a Fermi pocket, but the signal is near the noise level and so higher field measurements are needed to confirm this.  . . . . . . . . . . . .  134  8.15 Magnetoresistance of YBa2 Cu3 O6.75 to 70 tesla, from 1.5 to 30 K. No quantum oscillations are visible. The small bumps near 65 tesla are due to vibrations of the cryostat during the magnetic field pulse. . . . . . . . . . . . . .  135  8.16 The zero field resistance from 4.2 to 270 K, and the 60 tesla resistance value at 1.5, 4.2, 20, and 30 kelvin. The resistance at low temperature does not turn over—a feature this sample shares with the other doping with no oscillations: O6.47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  136  xviii  List of Figures 8.17 The vortex-lattice melting transition field as a function of temperature up to Tc . The maximum field value is greater than it was for O6.67 , reversing the trend of decreasing Bmelt with increasing doping from O6.47 to O6.67 . . . . .  137  8.18 The vortex lattice melting field for YBa2 Cu3 O6.59 , with fits to Equation 8.2. The red and black curves correspond to different bounds on the data, as indicated in the legend. These fits are made with cL = 0.3 and a single fit ˚ for the upper curve and ξ = parameter, ξ. The fit values are ξ = 37 A ˚ for the lower curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.5 A  141  8.19 Top: The cˆ-axis magnetic penetration depth for YBCO as a function of hole doping, shown in red, as measured by infrared reflectance [39]. The dashed red line is a guide to the eye. The deviation in Tc from the parabolic form (see Figure 1.3) is also shown, in order to illustrate that the penetration depth shows no anomaly centred around 1/8th doping. Bottom: The a ˆ-ˆbplane penetration depth, as measured by electron-spin resonance (ESR) and muon-spin rotation (µSR) [75, 96]. The dashed red line is a guide to the eye. Although the data is sparse near 1/8th , it is clear that there is no peak in λab , as there is in ∆Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  143  8.20 The superconducting coherence length as a function of doping, along with the deviation in transition temperature from the parabolic form of Tc vs p (the parabola can be seen in the phase diagram in Figure 1.3). The peak in coherence length roughly corresponds to the maximum in the suppression of Tc , which is thought to be caused by the competition of superconductivity with some other form of electronic order [54]. . . . . . . . . . . . . . . . . . 9.1  144  Top panel : YBa2 Cu3 O6.49 in-plane resistance as a function of temperature for the ordered sample, the disordered sample, and the same sample again after reordering. Bottom panel : The same as above but for YBa2 Cu3 O6.51 . .  9.2  148  Top panel : YBa2 Cu3 O6.49 in-plane resistivity as a function of magnetic field at 1.5, 20, 30, and 100K. Note the strong magnetoresistance at low temperature. Bottom panel : The hall coefficient, Rxy /B, as a function of magnetic field at the same temperatures shown in the top panel. Quantum oscillations are barely visible at the highest field and lowest temperature. The Hall data has been truncated around 20 tesla because otherwise noise (from dividing by small field values to obtain RH ) obscures the plot.  9.3  . . . . . . . . . . .  149  The same YBa2 Cu3 O6.49 sample shown in Figure 9.2, except that now the sample has been disordered and the doping is lower. Note that the Hall coefficient is now positive at high fields, at all temperatures.  . . . . . . . .  150  xix  List of Figures 9.4  The same YBa2 Cu3 O6.49 sample as the previous two figures, but with the chain oxygen partially reordered. The negative Hall coefficient at high fields and low temperatures has returned.  9.5  . . . . . . . . . . . . . . . . . . . . . .  151  Top panel : YBa2 Cu3 O6.51 in-plane resistance as a function of magnetic field at 1.5, 20, 30, and 100 K. Bottom panel : Hall coefficient. Quantum oscillations are clearly visible at the lowest temperature and highest field.  9.6  . . . .  152  Top panel : The same YBa2 Cu3 O6.51 sample as Figure 9.5, but with the oxygen disordered. Bottom panel : The Hall coefficient is now positive at high field, at all temperatures. The noise has also increased due to sample contact degradation while in the furnace. . . . . . . . . . . . . . . . . . . .  9.7  153  Top panel : The in-plane magnetoresistance of the same YBa2 Cu3 O6.51 sample as above, but with the oxygen partially reordered. Bottom panel : The Hall coefficient, which is negative again at high fields and low temperatures. 154  9.8  The temperature T0 where the Hall coefficient changes sign at high field as a function of hole doping. Tc is also included in this plot for reference. The green data points are from this work, and the blue data is taken from [54]. .  156  xx  Glossary AMRO angular magnetoresistance oscillations ARPES angle-resolved photoemission spectroscopy BCS Bardeen, Cooper, and Schrieffer CDW charge-density wave DFT density functional theory dHvA de Haas-van Alphen DOS density of states high-Tc high temperature superconductivity LDA local density approximation LNCMI Laboratoire National des Champs Magn´etiques Intenses mA milliamp meV milli-electronvolt mΩ milliohm ms millisecond NMR nuclear magnetic resonance SdH Shubnikov-de Haas SDW spin-density wave STM scanning-tunnelling microscopy Tc critical temperature Vpp volts peak-to-peak YBCO YBa2 Cu3 O6+x  xxi  Acknowledgments Dr. Doug Bonn, my supervisor, has been extremely helpful not only in guiding the research contained in this thesis, but also in the development of my career as a scientist. Throughout my doctoral studies I have taken part in many collaborations around the world, and presented my research at several international conferences. These opportunities taught me what it really means to be a scientist, beyond researching in a laboratory. Doug gave me free rein to pursue the projects that I found interesting, and he put a lot of trust in my abilities. This also extends to the physics 109 undergraduate lab at UBC, which Doug ran, and in which I was a teaching assistant. This course has undergone a lot of changes (for the better) over the past five years, and thanks to Doug my teaching has improved along with it. Beyond being a “good boss”, Doug has been a friend and mentor to me, and I hope to one day achieve his exceptional balance of “work” and “life”. Dr. Walter Hardy and I have spent countless hours discussing all aspects of physics, both related to my research and not, and he helped satisfy my curiosity for all things science. He is always willing to help out on any project, and his invaluable experimental expertise, and ability to see the underlying physics in any problem, have helped me throughout my studies at UBC. Outside of physics, Walter has also become a friend and mentor to me, and inspired me to start putting some serious effort into my piano playing again. I am indebted to Dr. Ruixing Liang not only for his dedication to the improvement of single-crystal YBa2 Cu3 O6+x , but also for training me in the use of the SQUID magnetometer, the detwinning apparatus, and for generally teaching me about sample preparation. I am also thankful for his patience every time I came to him asking for yet another sample; without his careful work none of this thesis would have been possible, and quantum oscillations would probably still not have been seen in the underdoped cuprates. All of the experiments in this thesis were performed in the lab of Dr. Cyril Proust, and I am extremely grateful for all of the help he has given me and for making me feel welcome in the lab. He builds beautiful experimental setups, and is always enthusiastically involved in the measurements of the users who come to the pulsed field facility. I would like to thank the entire lab at the LNCMI in Toulouse, and especially Dr. Baptiste Vignolle for working closely with me on the experiment (and for always promptly sending me data sets that I forgot to save!) Dr. James Day came with me to Toulouse for a number of the experiments, and the work was always much more enjoyable (and productive) with him there. As a new graduate  xxii  Acknowledgments student I was awfully suspicious of what exactly a postdoctoral researcher did, and I am glad he showed me that they can be invaluable members of the lab (and good friends too!). I would like to thank the rest of the lab at UBC for their help, especially Jordan Baglo for putting up with my barging into his office with physics or computer-related questions, and Pinder Dosanjh for his help on sample preparation and advice on life in general. I have received a lot of help from outside the lab as well during my doctoral work, and I am grateful for the extensive discussions I have had with Drs. Sudip Chakravarty, Andrea Damascelli, Neil Harrison, Stephen Julian, David LeBoeuf, Mike Norman, George Sawatzky, Suchitra Sebastian, Arkady Shekter, Philip Stamp, Louis Taillefer (and the entire Sherbrooke lab), and David Vignolles. If I have forgotten someone from this list it is not because they didn’t make a valuable contribution, but because I am tired and ready to hand this thing in. I want to thank my parents Bob and Tracy, my brother Cole, Lysandra (you made it!), and Natasha for their constant support during the research and the writing of this work. They have always been there, whether to help with moving, to listen to me vent, or to help keep my life organized: I could not have done it without you.  xxiii  Chapter 1  Introduction This thesis is based on electrical resistance measurements of the high temperature superconductor YBa2 Cu3 O6+x (YBCO), in pulsed magnetic fields of up to 70 tesla. Analysis of these measurements reveals considerable detail about how charge carriers behave in YBCO. Much of this thesis will be devoted to the analysis of this resistance data, and to the extraction of the relevant parameters that describe YBCO. Of central importance to this data analysis, and to the conclusions I draw from it, is the concept of a Fermi surface. In the simplest terms, the Fermi surface is a boundary at zero temperature that divides quantum mechanical states occupied by charge carriers from those that are unoccupied. This is a deceptively simple statement, crafted with carefully chosen language, that raises the following questions: • How is something defined at zero temperature of any use, when experiments are always carried out at finite temperature? • What are “charge carriers”: are they not simply electrons? • How can a useful set of “quantum mechanical states” be constructed for a solid that has of order 1023 electrons? The answers to these questions lay the foundation for much of condensed matter physics. In this introduction I will outline the basic arguments that give support to the concept of a Fermi surface in metals, and discuss its applicability to YBCO. I will start with a discussion of what a Fermi surface is, illustrated with the simple example of a free electron gas. I will then discuss Lev Landau’s Fermi liquid theory, originally developed for helium-3, and end with a discussion of Fermi surface measurements in cuprate superconductors, of which YBCO is an important member.  1.1  The Free-Electron Fermi Surface  The simplest starting point for calculating the properties of a collection of electrons is to ignore all interactions and to treat them as a free electron gas. The Schr¨odinger equation can then be solved for each free electron, giving plane wave solutions of the form Ψ(r) ∝ eik·r ,  (1.1) 1  1.1. The Free-Electron Fermi Surface where p = k is the electron’s momentum [95]. The energy eigenvalues of these states are (k) =  2 k2  2me  ,  (1.2)  where me is the electron’s mass. There is an uncountably infinite number of possible wave functions, since all k values provide legitimate solutions to the Schr¨odinger equation. If the electrons are constrained to move in a box with sides of length L (which could be, for example, a crystal), Equation 1.1 can be normalized, and k is restricted as follows: 2 3 sin (kx ·x) sin (ky ·y) sin (kz ·z) L (nx x ˆ + ny yˆ + nz zˆ)π k= , L  Ψ(r) =  (1.3) (1.4)  where nx , ny , and nz are positive integers from zero to infinity. There is now a countably infinite number of quantum mechanical states, and each of these states is labelled by its unique k value. To give this free electron model at least some legitimacy, the Pauli exclusion principle should be imposed on the electrons. Since they are spin  1 2  particles, a maximum of two  electrons can occupy each k state in Equation 1.3—one for each spin orientation. This simple requirement has drastic consequences: instead of the lowest energy state for N electrons being the |k| =  3π 2 N L3  1 3  = 0 state, where all particles have zero momentum, all states up to  ≡ kF are occupied in the ground state. kF is known as the Fermi wave  vector, and the associated quantity  F  2 k2 F 2me  =  is the Fermi energy. At zero kelvin, when  the electrons are in this ground state, the set of all wave vectors k with magnitude kF make up the “Fermi surface”, defining the boundary between occupied and empty states. For the free electron model, the Fermi surface is a sphere of radius kF , since there is no preferred direction in space. The importance of the Fermi surface becomes apparent when a small amount of energy ∆ is added to the collection of electrons. This energy could come from a low energy photon, or from the application of a small electric field. Any electron occupying a state with energy <  F  −∆ cannot be excited by this perturbation, since all of the states with energy +∆  are occupied. Only electrons lying in the region ∆ below the Fermi surface can be excited to empty states above. When the system is at finite temperature, then the probability of occupation for a state of energy  is given by the Fermi-Dirac distribution, P( )=  1 1+  where µ is the chemical potential—equal to  F  e( −µ)/kB T  ,  at zero kelvin. For most metals,  (1.5) F  is of order 2  1.1. The Free-Electron Fermi Surface a few electron volts, or a few tens of thousands of kelvin [6], and therefore Equation 1.5 resembles a step function, even at room temperature. At finite temperature, it is the electrons within an energy ∼kB T of  F  that are in close proximity to empty states, and can  therefore be excited by low energy excitations. Taking a generously long mean free path of 1 µm for a charge carrier in a metal [6], and applying 5 millivolts across a 1 millimetre long sample of this metal (typical numbers for a resistivity experiment), only 0.005 meV of energy, or about 0.06 millikelvin, is imparted to this charge carrier. This energy transfer is clearly much smaller than  F  for a typical metal.  If the charge carrier scatters elastically—off of an impurity, for example—its momentum direction changes but its energy, by assumption, stays the same. The restriction on the momentum change is that the carrier must “scatter into” an unoccupied state. In this case clearly only carriers within kB T of  F  can scatter at all; carriers far below the Fermi surface  have no nearby empty states, and therefore cannot scatter. Similar arguments hold when the scattering is inelastic, because the amount of energy lost during the scattering process must be small compared to  F  if the carrier is to scatter to an empty state.  This exercise shows that, at least for free electrons, only those electrons lying very close to the Fermi surface take part in scattering events, or more generally, have their energy and momentum altered by small perturbations. In order to calculate quantities such as the resistivity, which depends on the scattering rate, or the specific heat, which depends on the density of states, then it is clear that we are interested in scattering rates and densities of states near the Fermi surface. Knowledge of the Fermi surface geometry in a metal, which will in general not be spherical, is essential for the calculation of many different material properties.  1.1.1  The Lattice Potential  The addition of a crystal lattice to the free electron gas is surprisingly straightforward. Bloch’s theorem states that solutions to the Schr¨odinger equation in a periodic potential are plane waves modulated by a function that has the same periodicity as the potential [6]. The quantity k is no longer the electron’s momentum, since the Hamiltonian no longer has complete translational invariance. k remains a useful quantity, however, and is referred to as the “crystal momentum” for these “Bloch electrons.” If the periodic potential of the lattice is known, then the single-particle Bloch electron dispersion relations can be calculated for any arrangement of atoms, at least in principle. Rather than dealing with the bare Coulomb potential for every electron and ion in the material, it is often sufficient to treat the potential from the ions and the inner core electrons as providing a net effective potential for the outer valence electrons. The calculated energy as a function of crystal momentum, (k), often looks similar to the free electron energy-momentum dispersion relation with a couple of modifications:  3  Εk  1.1. The Free-Electron Fermi Surface  Π  0  Π  Π  0  Π  k  k  Figure 1.1: Left: dispersion relation for free electrons, drawn in the reduced zone scheme. The red bands can be thought of as free electron bands centred around (k ± 2π), or as the original black band folded from k = π → −π, k = −π → π. Right: a lattice potential with period 1 is introduced, opening up gaps wherever the bands in the previous panel crossed. • k is restricted to lie inside of the “Brillouin zone”: a small subsection of momentum space. For a 1D lattice with spacing a, k lies between  −π a  and  π a.  Higher values of  k are folded back into this Brillouin zone by subtracting the reciprocal lattice vector ± πa , and a second index n is used to label the higher energies. See the first panel of Figure 1.1. • Near the Brillouin zone boundaries, at ± πa , the electron wavefunction experiences Bragg reflection and gaps open up in the dispersion relation. See the second panel of Figure 1.1. The concept of filling up k states to a maximum Fermi energy applies in the same way to the Bloch electrons as it did to free electrons. The resulting Fermi surface may no longer be spherical since rotational symmetry is broken by the lattice. If the highest occupied band is completely filled, then there will be no Fermi surface at all, since there are no unoccupied states above the highest occupied level (there is an energy gap to the next state). A material with completely full and empty bands at zero temperature is a band-gap insulator. If the highest occupied energy level is not at the top of a band, then there will be a Fermi surface between the occupied and unoccupied states. Such a material is a metal, a semi-metal, or a semi-conductor, depending on the number of filled states in the highest occupied band. The Fermi surface in this case may no longer be a continuous object in k space, and may have disconnected pockets separated by gaps. This has been a whirlwind overview of the subject known as band structure, which is discussed in greater detail in Ashcroft and Mermin [6] and many other solid state textbooks. The upshot of band structure calculations is that Fermi surfaces, from where all low energy excitations occur, still exist when a lattice potential is introduced. It is important to remember that up to this point the electrons have only been allowed to interact with a 4  1.2. Fermi Liquids static, classical lattice potential. In a real metal there are dynamic interactions between all of the electrons, and between the electrons and the lattice, and so the important question is: is the Fermi surface concept still valid when electron-electron interactions are included? The answer to this question is the domain of Fermi liquid theory, and is the topic of the next section.  1.2  Fermi Liquids  I will not pretend to give a full account of Fermi liquid theory here, but I will try to put forth a plausible argument as to why we can continue to talk about charge carriers with spin ± 12 , charge e, and momentum p. A more complete introduction to this topic can be found in Ashcroft and Mermin [6] or Mahan [63]. The excellent review article by Shankar [91] gives a very different perspective on Fermi liquid theory, from the renormalization group, and in many ways is much more satisfying. I will follow the notation of Coleman [17] for second quantization. Fermi liquid theory hinges on two key concepts: the Pauli exclusion principle, and the adiabatic theorem. The Pauli exclusion principle forbids two fermions from occupying the same state. It should be clear from the previous discussion, on the free electron and Bloch-electron Fermi surfaces, that this restricts weak perturbations of the system to a narrow range in energy around the Fermi energy. The adiabatic theorem says that if a noninteracting system in the ground state slowly has the interaction potential between electrons turned on, then the system remains in the ground state throughout this process. In fact, there will be a one-to-one correspondence between each state in the non-interacting system and a state in the interacting system. The energy levels of the states remain separated while the potential is turned on because of level repulsion: two states approaching each other in energy have matrix elements between them, and these matrix elements cause the states to “repel” one another and remain separated in energy [17]. The resulting ground state will look qualitatively similar to the non-interacting ground state, and will have the same symmetries. One possible caveat to this scenario is if two states have different symmetries and no matrix elements exist between them. This allows the energy levels to cross as the interaction potential is turned on. Then the system will evolve into a new ground state with new symmetries at some critical interaction strength—for example, a superconducting ground state, or a ferromagnet. The following example assumes that the system remains in the same ground state as interactions are turned on. Consider a single electron placed at some small energy ∆ above the Fermi surface 1,  with momentum p. Using the language of second quantization, this can be written as 1  The Fermi surface need not be the spherical free electron Fermi surface: it could be a Fermi surface of Bloch-electrons from band structure calculations. What is important is that the electrons initially do not interact with each other.  5  1.2. Fermi Liquids † cpσ |Ψ0 , where σ is the electron’s spin, and |Ψ0 is the non-interacting ground state of  the (Bloch-)electrons. Using the Coulomb potential between electrons as an example, the interaction Hamiltonian is [63] Hint =  1 2V  k,k ,σ,σ q=0  4πe2 † c c† c c , q 2 k+q,σ k −q,σ k,σ k,σ  (1.6)  where V is the system volume, and σ is a spin index. The important thing to note here is that each electron creation operator in the Hamiltonian has a corresponding hole creation operator: further on I will show that these pairs of operators are responsible for creating the cloud of excitations that are associated with a quasiparticle. Turning on the interactions slowly amounts to applying a time-evolution operator conˆ, taining the interaction part of the Hamiltonian, U ˆ = Tˆe− i U  0 −∞  Hint (t)dt  ,  (1.7)  where Tˆ is the time-ordering operator, and Hint (t) is Equation 1.6 written in the interaction representation (see chapter 2 of Mahan [63].) ˆ , the new time-evolved state U ˆ c† |Ψ0 can be written as Exploiting the unitarity of U pσ ˆ c† U ˆ †U ˆ |Ψ0 . This can be split into two parts: U pσ  † ˆ c† U ˆ† apσ ≡U pσ  (1.8)  is the “quasiparticle” creation operator, and ˆ |Ψ0 |Ψ ≡ U  (1.9)  is the ground state of the interacting system[17]. This quasiparticle is no longer a single electron, since its creation operator now contains particle-hole pairs from the interaction Hamiltonian, Equation 1.6. It does preserve its momentum p and spin σ, and still obeys the Pauli exclusion principle. The “cloud” of electron-hole excitations that accompany the quasiparticle give it its effective mass m , effective g factor, and other renormalized parameters. A potential problem is that this quasiparticle is not in the ground state, and can decay by giving up energy to create particle-hole pairs. This decay process will happen during the switching-on of the interactions, unless the lifetime of the quasiparticle is long in comparison to the switching-on time. Furthermore if there are a large number of quasiparticles, then they can interact with each other and decay. This is where Landau made his important observation: for quasiparticles ∆ above the Fermi surface, the fraction of quasiparticles below the Fermi surface that they can interact with goes as  ∆  of empty states the quasiparticles can scatter into also goes as  F  . Furthermore, the number  ∆ F  . Thus the scattering rate 6  1.3. The Cuprates of the quasiparticle, or inverse of its lifetime, goes as 1 ∝ τ  ∆  2  .  (1.10)  F  This demonstrates that as the excitation energy of the quasiparticle goes to zero, its lifetime goes to infinity; as long as we are interested in low energy excitations, the quasiparticle remains a long-lived object. This argument also holds at finite temperatures where there are many excited quasiparticles: the number of quasiparticles goes as to scatter into, giving a similar contribution of  kB T F  , as does the number of empty states  kB T F  2  to the scattering rate. Since the  resistivity of a metal is proportional to the scattering rate (at least in simple models), electron-electron scattering gives a T 2 contribution to the resistivity. This exercise demonstrates that the non-interacting ground state evolves into a state that can still be described by the occupation of states with momentum p and spin 12 . While p is no longer a constant of motion for an individual electron, it is a constant of motion for the electron plus a surrounding cloud of electron-hole excitations. This object is referred to as a quasiparticle, and because of the accompanying cloud of excitations, the quasiparticle behaves differently to applied stimulus than the free electron did. These differences are quantified via renormalized parameters, such as the effective mass m . Luttinger demonstrated that these quasiparticles still have a sharp discontinuity in their occupation number as a function of momentum at zero temperature, allowing for the definition of a Fermi surface at this discontinuity [61]. He also showed, in a result now known as Luttinger’s theorem, that the Fermi surface defined in this way occupies the same volume in momentum space as a free-electron Fermi surface for the same number of particles does. I will now switch gears and discuss the cuprates—a set of materials with a long and controversial history over whether the above Fermi liquid picture is applicable.  1.3  The Cuprates  This section will be divided into two subsections: a brief discussion of the history of high temperature superconductivity (high-Tc ) in the cuprates, followed by a summary of the experimental progress that has been made in measuring their Fermi surfaces.  1.3.1  History  Transition metal oxides have been of great interest to condensed matter physicists for almost a century, with insulators, magnets, semiconductors, metals, and even high-Tc superconductivity all appearing in this family of compounds. The d-orbital electrons and holes on the transition metal ions in these oxides are not described well by either a free electron model  7  1.3. The Cuprates  Figure 1.2: The crystal structure of YBa2 Cu3 O7 . The copper-oxide chain layer, responsible for doping, can be seen at the top and bottom of this figure, and are completely full in O7 . The copper oxide planes, responsible for superconductivity, are in the middle of the diagram on either side of the yttrium ion. Image used with permission from Peets [73]. or a tightly bound atomic model. The effects of both the periodicity of the lattice, as well as the propensity for these holes and electrons to remain localized around their donor atoms, must be taken into account[41, 66]. While this fact gives rise to very interesting physics, it is a very difficult problem to solve theoretically, since the usual method of starting with plane waves or atomic orbitals, and adding interactions as a perturbation, is not particularly accurate for the d-orbital electrons[2]. From a practical perspective, these materials take part in magnetic storage devices, transparent conductors for displays, and high-Tc . These widespread uses have driven researchers to understand these materials better, and to look for new ways to further tune their properties. The term “cuprate” can be applied to a large number of copper-bearing compounds, but in the context of this thesis I will use it in the colloquial sense to refer to quasi-2D copper-oxide materials. These cuprates are characterized by a perovskite crystal structure (sharing the structure of calcium titanate) with planes of CuO2 [77]. The copper atoms form a square lattice, with oxygen atoms on the face of each square. The CuO2 planes are separated by larger ions or collections of ions, such as yttrium, barium, strontium, calcium, etc., which also interact with the oxygen atoms. These large ions are often substituted for ions with other valences, doping the CuO2 planes with holes or electrons. There are more complicated arrangements of atoms in some of the cuprates, but they are all variations on this scheme. The crystal structure of YBCO is shown in Figure 1.2. See section 9.1 for further discussion of the chemistry of YBCO. 8  1.3. The Cuprates In 1986, Bednorz and M¨ uller discovered superconductivity near 30 kelvin in the cuprate material Bax La5−x Cu5 O5(3−y) [9]. By changing the relative concentrations of the dopant atoms, and by applying pressure, the onset of superconductivity—where the material shows a diamagnetic response—was pushed as high as 57 kelvin. This critical temperature (Tc ) was much higher than than the highest Tc superconductor at the time, Nb3 Ge with a Tc of 23 K, and was well beyond what was thought possible with the standard Bardeen, Cooper, and Schrieffer (BCS) theory of superconductivity. This discovery set off a search for even higher Tc s among the cuprates, and Bax La5−x Cu5 O5(3−y) would soon be surpassed. The year following the discovery of high-Tc by Bednorz and M¨ uller, the group of C.W. Chu was exploring the multiple phases of YBaCuO (of which YBa2 Cu3 O6.x is an example), with the idea that one of these phases might show superconductivity at an even higher temperature [108]. They were in luck: a sample cut from one of the sintered pellets showed an onset of superconductivity at 93 Kelvin. While HgBa2 Ca2 Cu3 Ox would eventually surpass YBCO with a Tc of 135 Kelvin at ambient pressure, YBCO has remained of great experimental and theoretical interest because it can be made extremely pure. Single crystals of the high temperature superconductor YBCO, grown at the University of British Columbia (UBC), have been at the centre of many important results in condensed matter physics research, including measurement of the London penetration depth [32], polar Kerr effect measurements of broken symmetry in the pseudogap state [109], and determination of the Fermi surface by quantum oscillation measurements in high magnetic fields [22]. This thesis attempts to add to UBC’s distinguished history in this field by exploring in detail the high field cˆ-axis resistivity of YBCO.  1.3.2  The Cuprate Phase Diagram  The hole-doping phase diagram for high-Tc cuprates is shown in Figure 1.3. Unless otherwise specified, “doping” will always refer to the introduction of holes into the copper-oxygen plane, either via cation substitution or, for YBCO, changing the oxygen content in the copper-oxide chain layer. I will often refer to “undoped”, “underdoped”, “optimally doped”, and “overdoped” cuprates, and I will use the letter p for the doping variable. A brief discussion of the physics in each regime follows. Undoped Cuprates Undoped cuprates have a single hole per planar copper, and a large Coulomb interaction (the “Hubbard U ”) between the holes prevents double occupation on the copper sites, resulting in insulating behaviour. These undoped cuprates are often called “Mott” insulators, to be contrasted with traditional “band-gap” insulators. The difference between the two is that the insulating behaviour in Mott insulators is driven by on-site repulsion, rather than a gap in the band structure [2, 19, 66, 72]. Instead of a half-filled conduction band at the  9  1.3. The Cuprates  250  200  TN  T Kelvin  150  T III  IV  100  50  II  Tc  I  V 0 0.00  0.05  0.10  0.15  0.20  0.25  0.30  Hole Doping, p Figure 1.3: The phase diagram of the high-Tc cuprates, with data for YBa2 Cu3 O6.x . Region I is the superconducting dome: the blue line is the function 1 − Tc /Tc,max = 82.6(p − 0.16)2 , with Tc,max = 94.6 K for YBCO [59]. This function is phenomenological, but seems to describe Tc as a function of p for many cuprates. The blue data points are actual Tc values for YBCO, taken from Liang et al. [59]. The suppression of Tc around p = 0.125 is also characteristic of many cuprates. In region II, YBCO is an antiferromagnetic insulator. The red data points are from unpublished cˆ-axis resistivity measurements of the N`eel ordering temperature, TN [80]. The orange line is a parabolic fit to this data, including TN = 450 at p = 0 (not shown on this scale) from Tranquada [102]. Region III is the pseudogap region, bounded above by T . The green data points mark the deviation from linear resistivity [20], which is characteristic of the “strange-metal” found in region IV. Region V displays properties consistent with a Fermi liquid description of the quasiparticles, including a coherent three-dimensional Fermi surface in Tl2 Ba2 CuO6+x , and T 2 resistivity in several cuprates [42, 44]. The boundary between regions IV and V is purely schematic, as the crossover from T -linear to T 2 resistivity is gradual. The boundary between regions III and IV is a linear fit to the green T data, and should also be considered phenomenological. 10  1.3. The Cuprates Fermi surface, as band structure predicts, the band in a Mott insulator is split into lower and upper Hubbard bands (LHB and UHB, respectively), with an energy gap of size U separating them. A small number of holes doped into the material in this region remain localized and the material remains insulating. This behaviour is characteristic of region II in Figure 1.3. In the case of the cuprates, this is not quite the full picture. While the Coulomb repulsion preventing double occupation is indeed large, there is another important energy scale in the system: the charge-transfer energy ∆ between the copper cations and their neighbouring oxygen anions. The Hubbard U is so large that it actually pushes the LHB down below the oxygen band, so that the lowest quasiparticle excitation energy is now between the oxygen band and the UHB. Where the Hubbard U was the energy difference between having one or two electrons on a copper, the charge transfer energy ∆ is the energy difference between the oxygen 2p band and the upper Hubbard band. In this case the insulating energy gap is ∆ rather than U , and the materials are properly known as charge-transfer insulators rather than Mott insulators. This is of course a simplified picture, as there is hybridization between copper and oxygen bands. A good discussion of this topic can be found in Zaanen et al. [112]. Underdoped Cuprates Underdoped cuprates have between 0 and 0.16 holes per copper oxygen plane, occupying the region of the phase diagram from the charge-transfer insulator at low doping to the maximum Tc at 0.16 holes. This doping range corresponds to regions III, II, and half of region I in Figure 1.3. As more holes are introduced into the insulating state, the antiferromagnetic charge-transfer insulator weakens, and eventually gives way to superconductivity. At temperatures above the superconducting critical temperature Tc , there is a so-called “pseudogap” state. This pseudogap is characterized by a suppression of the density of states and the onset of weak magnetic order below an onset temperature defined as T [25, 38]. The pseudogap’s maximum and minimum magnitudes mirror the gap structure of the d-wave superconducting gap, where the gap is a minimum along the (π, π) direction. This has led to speculation as to whether the order in the pseudogap might be a precursor to superconductivity, while other theories suggest it competes against superconductivity [84]. The pseudogap in the high-Tc cuprates is one of the most puzzling (and contentious) issues in condensed matter physics, and interested readers should consult Timusk and Statt [100] for an experimental overview of the subject. Another feature of underdoped cuprates is the suppression of Tc from the smooth parabolic form around 0.125 holes per copper oxygen plane, as seen in Figure 1.3. Charge and spin density wave order—stripe order—has been observed in the cuprate La1.8−x Eu0.2 Srx CuO4 [48], and the strongest order is seen near p = 0.125. The possibility of stripe order has been explored in other cuprates as well, with the hope that a unifying picture of stripe order 11  1.3. The Cuprates competing with superconductivity can be constructed [52]. I discuss this in the context of my own experiments in chapter 9 and in section 8.6. Optimally Doped Cuprates Cuprates with a hole doping of approximately 0.16 holes per copper oxygen plane exhibit a maximum Tc . It is around this region in doping where T is thought to go to zero, and at high temperatures the pseudogap region gives way to the “strange metal” phase which exhibits T -linear resistivity (region IV in Figure 1.3) [42]. Overdoped Cuprates Cuprates with more than 0.16 holes per copper oxygen plane are overdoped, and occupy regions IV, V, and half of I in Figure 1.3. In this regime angle-resolved photoemission spectroscopy (ARPES) sees the large hole Fermi surface predicted by band structure for Tl2 Ba2 CuO6+x [74]. As the hole doping is increased, the T -linear resistivity seen at high temperatures crosses over to a Fermi-liquid-like T 2 resistivity, and a fully three-dimensional Fermi surface is seen by angular magnetoresistance oscillations (AMRO) and the de Haasvan Alphen (dHvA) effect [44, 103].  1.3.3  Fermi Surfaces in the Cuprates  There are three main experimental techniques that have given Fermi surface information about the cuprates: angle-resolved photoemission spectroscopy (ARPES), angular magnetoresistance oscillations (AMRO), and quantum oscillations. Before I present a summary of Fermi surface measurements in the cuprates, I will provide a brief explanation of these three techniques. These techniques should be viewed as complementary rather than competing, and I will talk about the strengths and weakness of each. Angle-Resolved Photoemission Spectroscopy Prior to 2005 almost all that was known about the Fermi surface of the cuprates came from angle-resolved photoemission spectroscopy (ARPES). An excellent introduction to the technique and an overview of ARPES measurements on the cuprates up to 2003 can be found in Damascelli et al. [19]. While there are many complications to the technique, at its heart ARPES is conceptually very simple. The principle ideas behind ARPES can be summarized succinctly: photons of a specified energy are directed at a crystal surface; electrons absorb the photons and are ejected from the sample; the kinetic energy of the ejected electrons is measured as a function of the emission angle. Since the incident photon imparts very little momentum to the electron (in comparison to typical conduction electron momentum near  F ),  and since translational  12  1.3. The Cuprates symmetry is not broken in the direction parallel to the surface of the crystal, the energymomentum dispersion relation can be reconstructed. The loss of translational symmetry in the direction perpendicular to the crystal surface poses a problem for accurate reconstruction, but for quasi-2D Fermi surfaces such as those in the cuprates, this is usually a minor issue. One major potential problem with ARPES is that the electrons excited by the incident ˚ for typical photon excitation photons have a very short mean-free-path—on the order of 5 A energies [19]. This means that energy-momentum data will be dominated by electrons coming from the first unit cell of the material, which is of order 10 ˚ A in the cuprates. This means that surfaces cleaved in ultra-high vacuum are almost always required to obtain good data, and that surface states and surface reconstructions can make interpretation of the data difficult. In Sr2 RuO4 , for example, surface problems led to ARPES results that conflicted with other Fermi surface measuring techniques—dHvA in particular [10]. This was resolved with higher resolution data, and a better understanding of the surface reconstruction that happens when Sr2 RuO4 is cleaved at low temperatures [18]. The great advantage that ARPES has over the next two techniques I will describe is that, in principle, it can map out entire bands in momentum space—not just where they cross the Fermi energy. It can also directly measure the quasiparticle’s spectral function, which can be related to its self energy—a quantity of great interest to theorists. It also provides Fermi surface information at temperature scales well above Tc in the cuprates, whereas AMRO is limited to a few tens of kelvin above absolute zero, and quantum oscillations are limited to near liquid helium temperatures. Angular Magnetoresistance Oscillations AMRO is a technique that makes use of the unique properties of a quasi-2D metal. The AMRO technique measures the cˆ-axis resistivity of a metal as a function of the angles θ and φ between the crystal axes and the magnetic field. In quasi-2D materials the quasiparticle velocity in the cˆ-axis direction, averaged around a cyclotron orbit, goes to zero at certain angles, and these angles depend on the Fermi surface geometry [110]. The quasiparticle velocity going to zero produces a maximum in the resistivity at these angles, and this effect is quasi-periodic in tan θ. Detailed analysis of the structure of the resistivity as a function of θ and φ allows reconstruction the Fermi surface geometry. Reviews of this technique and its application to quasi-2D superconductors can be found in Bergemann et al. [10] and Singleton [93]. AMRO provides much higher precision Fermi surface geometry information than ARPES does, and is a bulk probe that does not suffer from surface quality and cleaving issues. The downside is that it is limited to the range of temperature where the product of the cyclotron frequency and the scattering rate, ωc τ , is of order 1 or greater. This means that AMRO signals are usually lost around 30 kelvin, although this depends on the quality of 13  1.3. The Cuprates the material being measured. This technique also only provides Fermi surface information, whereas ARPES can provide full electron-momentum dispersion relations below the Fermi surface. Quantum Oscillations Quantum oscillations are oscillations in a material property—resistivity, magnetization, heat capacity, size, temperature, etc.—that arise from the quantization of quasiparticle orbits in a magnetic field. The Lorentz force causes quasiparticles to traverse orbits in a magnetic field, and quantum mechanics discretizes the area of the orbits in the plane perpendicular to the field. When the area of an orbit is equal to the extremal cross-sectional area of the Fermi surface there is a peak in the density of states at the Fermi surface. The orbital areas increase linearly with increasing magnetic field, causing periodic intersection of orbits with the Fermi surface. This gives oscillatory behaviour in the density of states as a function of magnetic field, and any property that depends on the density of states at periodically in  F  oscillates  1 B [92].  By measuring the oscillation frequency, F , the Fermi surface area, Sk , can be calculated from the Onsager relation [71]: Sk =  2πe  F,  (1.11)  where F is the measured frequency. Equation 1.11 depends only on fundamental constants, and has been shown to be robust across a wide variety of materials, from organic superconductors to heavy fermions to gallium arsenide heterostructures [10, 92, 93]. This relation even holds when electron-electron interactions are accounted for, at least under a certain set of assumptions detailed in Kohn [49]. The definitive text on quantum oscillations is Shoenberg [92], with Singleton [93] and Bergemann et al. [10] providing a nice update to Shoenberg’s text for quasi-2D systems. I will give a detailed description of the theory of quantum oscillations in chapter 2. The temperature dependence of quantum oscillations measures the cyclotron effective mass (m ) of the quasiparticles as renormalized by the band structure, electron-electron interactions, electron-phonon interactions, and all other interactions. This mass gives information about the curvature of the band at  F,  and can be compared to the mass obtained  from the electronic specific heat [92]. The amplitude of the oscillations as a function of magnetic field is determined by the product ωc τ , where τ is the quasiparticle lifetime, and ωc is the cyclotron frequency. This scattering rate can be extracted from quantum oscillation data provided that oscillations are seen over a large enough field range. In addition to providing detailed Fermi surface geometry information, the angle dependence of the oscillations also provides information about the spin magnetic moment of the quasiparticles, g µB , where µB is the Bohr magneton and g is the renormalized g factor [10, 24, 92]. The great disadvantage of quantum oscillation techniques is that samples must be of 14  1.3. The Cuprates extremely high purity, such that ωc τ  1, in order to see oscillations. Quantum oscillations  are inherently a quantum mechanical phenomenon, and get washed out when kB T ≈ ωc , which limits this technique to around 10 kelvin and below. The requirement of a magnetic field can also be problematic, as the system may undergo a quantum phase transition at a critical field value, changing the ground state of the system. See Mercure [64] for an example of this in Sr3 Ru2 O7 . Experimental Progress It is simplest to start with a discussion of the overdoped side of the phase diagram (above p = 0.16 in Figure 1.3), where experiments agree with band structure calculations, and there is evidence for Fermi liquid behaviour. Many cuprate materials have been studied via ARPES: I will focus on Tl2 Ba2 CuO6+x and YBCO for the overdoped side of the phase diagram, since YBCO is the focus of this thesis, and Tl2 Ba2 CuO6+x shows the cleanest ARPES results. Other cuprates, such as Bi2 Sr2 CuO6.x and La2−x Srx CuO4 , have complications due to surface reconstructions or stripe order [19]. The first fully three dimensional Fermi surface measurement in the cuprates was on Tl2 Ba2 CuO6+x , performed by Hussey et al. [44] using AMRO. At 45 tesla and 4.2 K they found a large hole Fermi surface occupying 62% of the Brillouin zone, agreeing with the doping as determined by a Tc of 20 K. The in-plane shape that was determined—a rounded square—agrees with band structure calculations. They were also able to measure the warping along the cˆ-axis direction, and found that the cˆ-axis hopping rate goes to zero along eight symmetric directions in the Brillouin zone [44]. In 2005 a very similar Fermi surface was seen by ARPES in a Tc = 30 K sample, along with the other quasiparticle excitation information that ARPES provides [78]. This Fermi surface is shown in Figure 1.4. This was the first time for quantitative agreement between a surface probe and a bulk probe in the cuprates. In 2008, dHvA and Shubnikovde Haas (SdH) oscillations were seen in Tl2 Ba2 CuO6+x , adding even stronger support for the existence of Landau quasiparticles at low temperature  2  [103]. An extremely detailed  account of the quantum oscillation measurements on Tl2 Ba2 CuO6+x can be found in Rourke et al. [82], and similarly for ARPES in Peets et al. [74]. YBCO cannot be doped in bulk beyond 0.19 holes per copper-oxygen plane, but when YBCO is cleaved for a surface-sensitive experiment (ARPES or scanning-tunnelling microscopy (STM)), there is a build-up of electric charge on the cleaved surface. This is because there is no neutral cleavage plane, and so charge moves to the surface to lower the potential energy of the system [40]. Despite the massive change in the surface doping (from 0.11 as-grown to 0.28 as-cleaved, for example), the resulting bands as measured by 2  Vignolle et al. [103] claim that their quantum oscillation measurements are first to confirm the existence of true long-lived quasiparticles, since AMRO is a semi-classical effect and because the ARPES excitation peaks seen in Tl2 Ba2 CuO6+x are too broad in energy to be attributed to long-lived quasiparticles.  15  1.3. The Cuprates  Figure 1.4: The Fermi surface of Tl2 Ba2 CuO6+x with a Tc of 30 K, as determined by ARPES [78]. The black line is a fit to a tight binding model. The area of the hole pocket agrees with the area measured by AMRO and quantum oscillations. Figure reproduced with permission from Plat´e et al. [78]. ARPES agree quite well with band structure calculations at the new effective doping [40]. Because there are two copper-oxygen planes, the large hole Fermi surface is doubled, as both the bonding and anti-bonding bands pass through the Fermi surface. Bands from the copper-oxygen chains also appear to cross  F.  Moving toward the underdoped side of the phase diagram, two main features become apparent in ARPES measurements: the opening of a gap-like structure well above Tc (the “pseudogap”), and the loss of Fermi surface crossings everywhere except near a line along the (0, 0) → (π, π) direction in the Brillouin zone [19]. The magnitude of the pseudogap has the same momentum dependence as the d-wave superconducting gap, which has led to speculation that the pseudogap is somehow associated with the formation of phase-incoherent Cooper-pairs well above Tc [84]. Unlike the overdoped regime, where band structure appears to work quite well, the underdoped regime is in complete disagreement with density functional theory (DFT) calculations. At zero hole doping, local density approximation (LDA) calculations predict a half-filled metallic band, which persists at all hole doping levels [23]. Experiments, on the other hand, reveal an antiferromagnetic insulator below about p = 0.05 (region II of Figure 1.3.) In 2008, Hossain et al. [40] found that by evaporating potassium ions onto the surface of cleaved YBa2 Cu3 O6.50 the surface was doped with electrons. Subsequent ARPES measurements revealed the same Fermi surface seen in other underdoped cuprates—disconnected “arcs” along the (0, 0)− > (π, π) direction, shown in Figure 1.5. As YBCO is underdoped further, the coherence of the quasiparticles decreases until the correlated charge-transfer16  1.3. The Cuprates  Figure 1.5: Left:the Fermi surface of as-cleaved YBa2 Cu3 O6.50 , with an effective hole doping of p = 0.28. The Fermi surface is qualitatively similar to that seen in Tl2 Ba2 CuO6+x , shown in Figure 1.4. Right: the Fermi surface of YBCO after potasium deposition on the surface. The large hole pocket has shrunk to “arcs” along the nodal directions of the Brillouin zone. The line-like objects running left to right near the centre of the Brillouin zone are the chain bands. Figure reproduced with permission from Hossain et al. [40]. insulator state is reached [27]. In a Fermi liquid, the coherence factor Z is a measure of how well the independent quasiparticle picture describes the electronic excitations [19, 61]. Clearly a cuprate with 0 hole doping has Z = 0, since the holes are completely correlated and form an antiferromagnet, and all single-particle excitations are gapped out. The ARPES study of Fournier et al. [27] found Z to go to zero near 0.15 holes per copper-oxygen plane. This loss of quasiparticle coherence implies that the “Fermi surface crossings” seen below this doping level are not single-quasiparticle states, but instead are the incoherent excitations of a strongly correlated system. In 2007 quantum oscillations were seen in the Hall coefficient of YBa2 Cu3 O6.50 [22]. This was the first time that long-lived quasiparticle excitations had been seen in the underdoped cuprates, in contrast with all the earlier information provided by ARPES. The Fermi surface area as measured by these oscillations occupies about 2% of the full Brillouin zone [22]. The overall sign of the Hall coefficient is negative at low temperatures, indicating that the large hole-like Fermi surface seen in the overdoped cuprates has reconstructed into small pockets, some of which are electron-like. Subsequent measurements revealed these oscillations to be robust across a large number of material properties, including the Hall, Nernst, and Seebeck coefficients, torque magnetization, in-plane resistivity, cˆ-axis resistivity, and heat capacity [7, 22, 52, 79, 81, 89]. With only one exception [94], all of these measurements were performed on crystals grown by Ruixing Liang at the University of British Columbia, and all of these measurements  17  1.3. The Cuprates were performed in the narrow oxygen region from 6.49 to 6.61 3 . In section 8.4 I will show unpublished data with oscillations on YBa2 Cu3 O6.67 , which has the Ortho-VIII oxygen superstructure. Only one other hole-doped cuprate has shown quantum oscillations, and that is the other YBCO material YBa2 Cu4 O8 [111]. This stoichiometric material has two copperoxygen chain layers, in comparison to the one in YBa2 Cu3 O6.x , and has a hole doping of 0.125 holes per copper-oxygen plane—the same as YBa2 Cu3 O6.70 . The size of the Fermi surface seen in YBa2 Cu4 O8 is very similar to that seen in YBa2 Cu3 O6.69 , even though the band structure of the chains is quite different between the two compounds [13, 94]. This gives support to the idea that the quantum oscillations are inherent to the copper oxygen planes, rather than a specific feature of the chains. The remaining questions from the quantum oscillation measurements are: what is the source of the Fermi surface reconstruction that yields the small observed pockets; is this phenomenon universal in the cuprates, or is it unique to YBCO compounds? In this thesis I will try to address the first question by looking at the behaviour of the quasiparticle spins. I will also give support to the argument that the oscillations are not due to the chain structure in YBCO, but the ultimate test for this is seeing oscillations in a different underdoped cuprate material 4 .  3  The exception to this is the measurement of YBa2 Cu3 O6.69 by Singleton et al. [94], although it should be noted that the doping is very uncertain on this sample, and only two periods of oscillation were seen. 4 Quantum oscillations have been seen in the electron-doped cuprate Nd2−x Cex CuO4 , with small pockets similar to those seen in YBCO that appear to be the result of a Fermi-surface reconstruction [36]. The chemistries of the hole and the electron doped cuprates are quite different, however, and until the order parameters that cause the reconstructions in these materials are seen directly by x-rays or neutrons, it is not clear whether direct comparisons should be made between the hole and electron doped materials. There are many different material classes—iron-arsenides, heavy fermions, organic superconductors—that have qualitatively similar phase diagrams to the cuprates, with magnetism in close proximity to superconductivity, but that does not mean that the physics is the same in these different materials.  18  Chapter 2  Quantum Oscillations Quantum oscillations are a broad class of observable phenomena that are characterized by the oscillation of a material property as the magnetic field B is increased. Specifically, quantum oscillations are periodic in  1 B,  and arise because quasiparticle cyclotron orbits are  quantized in a magnetic field. This chapter will begin with a motivation for the phenomenon of quantum oscillations, using the exactly-solvable model of free electrons in a magnetic field. With this motivation in hand, a more rigorous derivation for quantum oscillations in a metal is presented.  2.1  Landau Levels  The classical Lagrangian for an electron in a magnetic field is 1 L = mr˙ 2  2  + er˙ · A(r),  (2.1)  where m is the electron’s mass, e is its charge (a positive quantity), r and r˙ are respectively its position and velocity, and A(r) is the vector potential. The form of this Lagrangian is motivated by the fact that it reproduces the Lorenz force law, F = −ev × B, for a charged particle in a magnetic field [63]. From Equation 2.1 the electron’s canonical momentum is found to be p = mr˙ + eA(r). Through a Legendre transform the classical Hamiltonian can be written as p − eA(r) H=  2m  2  ,  (2.2)  ˙ Equawhere p is understood to be the electron’s canonical momentum, and not just mr. tion 2.2 can then be quantized (leaving A(r) as a classical field) by promoting p to an operator: p → −i ∇. B is chosen to lie along the z-axis, with no loss of generality, which means that the vector potential may be written as A(r) = Bxˆ y.  (2.3)  Noting that there is no x or z component to the vector potential, Equation 2.2 can be re-written as  2  H=−  2m  ∂2 ∂2 + ∂x2 ∂z 2  +  1 2m  −i  ∂ − eBx ∂y  2  .  (2.4)  19  2.1. Landau Levels A further step can be made by noting that the Hamiltonian is independent of the y coor∂ ∂y  dinate (and, thus, py = −i  commutes with H), allowing the momentum operator py to  be replaced by its eigenvalue ky : 2  H=− where ωc ≡  eB m  2m  ∂2 ∂2 + ∂x2 ∂z 2  ky 1 + mωc2 x − 2 mωc  2  ,  (2.5)  is the cyclotron frequency. In the z direction this is the free electron  Hamiltonian, while in the xy-plane it looks exactly like the quantum harmonic oscillator with the x coordinate shifted by an amount x0 =  ky mωc .  This positional shift produces no  shift in the energies, which means that energies are the same as for the quantum harmonic oscillator: (n, kz ) =  2 k2 z  2m  + ωc n +  1 2  .  (2.6)  The electron’s wavefunction can immediately be writen down as Ψ(n, ky , kz ) =  ei(ky y+kz z) ψn (x − x0 ), Ly Lz  (2.7)  where Ly and Lz are the dimensions of the sample, and ψn (x − x0 ) is the wavefunction of the nth mode of the quantum harmonic oscillator [83]. Equation 2.6 is independent of ky , and thus ky is a degenerate quantum number. Restrictions can be put on ky by noting that ky =  2πN Ly ,  for some integer N , and that  0 ≤ x0 ≤ Lx (i.e., x0 must be within the sample). Combining these two restrictions gives 0≤N ≤  mωc Lx Ly . 2π  Therefore the number of states per Landau level is N =2  Φ Φ = , 2Φ0 Φ0  5  (2.8)  where the factor of 2 in the numerator comes from the spin degeneracy, Φ = BLx Ly is the magnetic flux through the sample, and Φ0 = h/(2e) is the magnetic flux quantum. This can be re-written as the 2D density of states (DOS) per Landau level, D=  B , Φ0  (2.9)  This exercise shows that free electrons in a magnetic field occupy discrete states—Landau levels—labelled by the quantum number n. The DOS on each Landau level is proportional to the strength of the magnetic field. 5 The nth “Landau level” is the nth harmonic oscillator state for a free electron in a magnetic field. The name “Landau level” will also be used in this thesis to describe the quantized energy levels of quasiparticles in metals.  20  2.1. Landau Levels It is useful to examine the effect this quantization has on the free Fermi gas6 . For a collection of Ne free electrons in a box, the energy-momentum dispersion relation is k=  2 k2  2m  ,  (2.10)  and the maximum filled state at zero temperature has an energy of 2 F  =  2m  3π  2 Ne  V  2/3  ,  (2.11)  where V is the volume of the system. The conclusion from section 1.1 was that low-energy excitations must take place near  F,  and this still applies when a magnetic field is introduced.  The total number of electrons remains fixed as the field is increased, but the Landau level energies and their degeneracies are increasing linearly with magnetic field. The highest occupied states still make up a Fermi surface, but there are energy gaps between each Landau level. It is easiest to visualize what happens, as the magnetic field is increased, if the Landau levels are broadened from delta functions in energy to Lorentzians 7 . I will also assume a quasi-2D Fermi surface for this discussion, since this is the relevant geometry for a layered material such as YBa2 Cu3 O6+x (YBCO). At certain values of magnetic field the Fermi energy lies far away from any Landau level peak, and the DOS at empty states in a narrow range above  F.  F  When the magnetic field is increased, the energy  and DOS for each Landau level increases, and at certain field values Landau level peak. This results in a large DOS at of empty states just above  F.  is small, with very few  F  F  will line up with a  at this field value, and a large number  This is shown schematically in Figure 2.1. This alternating  high and low occupation of states at  F  is what gives rise to the oscillatory behaviour of  material properties as a function of magnetic field.  6 Up to this point we have been working with a single free electron in a magnetic field. A free Fermi gas still has no interactions, but the electrons must obey Fermi-Dirac statistics and occupy individual states. 7 It will be shown in subsection 2.4.2 that this is indeed what happens when impurity scattering is introduced.  21  Occupation  2.2. Cyclotron Orbits in a Magnetic Field  Energy  Energy Increasing magnetic field →  Figure 2.1: Schematic of Landau levels, zoomed in near the Fermi energy of a 2D-Fermi surface (a cylinder). Landau levels are Lorentzians in energy, and the Fermi-Dirac distribution at finite temperature is shown as a dashed red line. Occupied states are filled in black. In the left panel the value of the magnetic field is such that the highest occupied state falls between the peaks of two Landau levels. As the magnetic field is increased (or decreased), the panel on the right shows that there is a large occupation of states near the peak of a Landau level.  2.2  Cyclotron Orbits in a Magnetic Field  While section 2.1 dealt with a free electron gas, the same ideas apply when the more complicated band structure of a real metal is included. Finding an exact solution for even a single electron may no longer be possible, but a semi-classical argument put forth by Onsager [71] shows that the essential oscillatory behaviour at the Fermi energy remains. The clearest and most intuitive explanation of this argument is given by Slater [95], and it is his text that I will follow in the next section. I will work with a semi-classical wavepacket (see Ashcroft and Mermin [6] chapter 12 or Slater [95] chapter 4) with crystal momentum k, and relate its change in momentum to the Lorentz force,  e ˙ k = − r˙ × B . c  (2.12)  The velocity of the wavepacket is related to the gradient of the energy by 1 r˙ = ∇k (k).  (2.13)  The Lorentz force in Equation 2.12 is orthogonal to the electron’s velocity, and so the electron’s energy remains constant even as the direction of its momentum changes. Looking 22  2.2. Cyclotron Orbits in a Magnetic Field at Equation 2.13 it is clear that if  is held constant, then (k) is a level set (surface of  constant value) and ∇k (k) will be thus normal to the surface of constant energy. As ˙ What this means is Equation 2.12 shows, k will in turn be perpendicular to both B and r. that the electron will orbit around a path of constant energy, much like it did for the free electron case in section 2.1, but with the surface of constant energy now determined by the dispersion relation of the electron in the lattice (k). Integrating Equation 2.12 over time, the electron’s momentum vector can be related to its position vector: (k − k0 ) = −e(r − r0 ) × B.  (2.14)  Looking at the projection of the r orbit into a plane perpendicular to B (i.e., r =  r×B |B| ,  which is in the same plane as k), the magnitudes of the two vectors are related by r − r0 =  k − k0 .  eB  (2.15)  Equation 2.15 shows that the area of the position and momentum space orbits, which depend on the square of the position and momentum vectors, are related by Sk =  e2 B 2 2  Sr ,  (2.16)  where S denotes an area. Thus for a fixed orbit in momentum-space, such as one around the Fermi surface, the real space orbit will grow smaller as the field is increased. The next step is to quantize the motion of the quasiparticle in a manner analogous to what was done in section 2.1. Rather than attack the full Schr¨odinger equation directly, I will make use of the WKB approximation, which is essentially an expansion of the wavefunction in powers of  and a set of instructions for dealing with boundary conditions. To first  order and in 1-dimension, if the potential the quasiparticle is in varies slowly in space, the wavefunction looks like a plane wave with wavelength λ(x) =  h 2m ( − V (x))  ,  (2.17)  where V (x) is the potential energy of the quasiparticle. The phase of the wavefunction at position x is then x  φ(x) =  2m ( − V (x )) dx  ,  (2.18)  where the integral is along the quasiparticle’s trajectory. Assuming that the amplitude of the wave function varies slowly around an orbit (relative  23  2.2. Cyclotron Orbits in a Magnetic Field to its phase), then to first order in  the quasiparticle’s wave function is  Ψ(x) ≈  e  i  x  √  2m( −V (x ))dx +φ0  2m 2  (V (x) − )  .  1 4  (2.19)  If this wavefunction is to be single-valued around an orbit then the phase must be an 2m ( − V (r)) is the quasiparticle’s  integer multiple of 2π, up to some constant. Note that  k − eA in a magnetic field with vector potential A.  semiclassical momentum, equal to  With this we can write down what amounts to the Bohr-Sommerfeld quantization condition: k − eA (r) · dr = (n + γ)2π ,  (2.20)  where the integral is around a closed path perpendicular to B (the element of path length is dr ), and γ is a constant (equal to  1 2  for a parabolic band [92]).  Using Equation 2.14 to express k in terms of r, and discarding the constants which add nothing when integrated around a closed path, Stoke’s theorem is used to express Equation 2.20 as (n + γ)  2π =B· e  r × dr  B · dSr ,  −  (2.21)  Sr  where Sr is the surface bounded by the path of the integral in real space. Since r is by definition perpendicular to B, the first term on the right hand side of Equation 2.21 gives no contribution for the component of r parallel to B. Thus the integral in the first term just gives twice the area of the surface traced out by the path in real space 8 . The second term gives one factor of the area times the magnetic field, and so the total is BSr = (n + γ)  2π = (n + γ)Φ0 . e  This remarkable result, first published by Lars Onsager [71], states that the flux  (2.22) 9  through  the quasiparticle’s semi-classical orbit is quantized in units of the flux quantum Φ0 ≡  h . e  (2.23)  Equation 2.16 is used to re-state Equation 2.22 in terms of the area of the orbit in momentum space: Sk = (n + γ)B  2πe  .  (2.24)  8  Roughly speaking, the factor of two comes about because r × dr is the area of a quadrilateral, and to calculate the area of a circle the areas of infinitesimal triangular slices are added, which are half the size of the corresponding quadrilateral. 9 This flux quantum is twice the value of the flux quantum used in section 8.6. There, the charge carriers are Cooper pairs with charge 2e, whereas here they are single quasiparticles with charge e. This convention is somewhat confusing, but standard in the literature [92, 95].  24  2.2. Cyclotron Orbits in a Magnetic Field Note that this result does not depend on the energy-momentum dispersion relation or any other parameters. Rather, it relates the area of the electron’s orbit to fundamental constants and to the strength of the magnetic field. Walter Kohn showed that this remains true when electron-electron interactions are accounted for, a result known as “Kohn’s theorem” [49]. Looking at Equation 2.24, and referring back to section 2.1, note that these quantized orbits are very similar to the quantized energy levels found in the free electron case. Analogously, there will be an oscillatory effect the magnetic field is swept; henceforth I will refer to the quantized orbits given by Equation 2.24 as Landau levels. Regardless of the band structure of the material, a path of constant energy can always be traced along Fermi surface in momentum space, defining a surface of area Sk perpendicular to the magnetic field. If Bn is defined to be the field at which the area of the nth Landau level is equal to the area of the Fermi surface orbit perpendicular to B, then at the higher field value Bn−1 the (n − 1)th Landau level will have the same area. Using Equation 2.24, and by isolating n, the relation Sk Sk −γ = − γ + 1. 2πeBn 2πeBn−1  (2.25)  is obtained. By solving for the difference in the inverse fields, the definition F ≡  1 Bn  1 Sk = 1 2πe − Bn−1  (2.26)  is obtained, where F is the frequency with which Landau levels will pass through the Fermi surface, resulting in oscillations of frequency F . Before proceeding further, the definition of the cyclotron frequency for quasiparticles is needed. Equation 2.12 and Equation 2.13 are re-arranged to give the time increment dt needed to traverse an element dk of an orbit in momentum space: 2  dt =  dk  e ∇k × B  ,  (2.27)  where dt will be integrated over the whole period of the orbit, and dk will be integrated around the orbit in momentum space. Since the the gradient of  is perpendicular to the  Fermi surface and the cross product with B removes any component of the gradient parallel to B, then, in terms of magnitudes, 2  dk  e ∇k × B  2  =  eB  dk ∆ ∆k  2  =  ∆(dSk ) . eB ∆  (2.28)  ∆k is the change in the component of the wavevector perpendicular to the Fermi surface and to the magnetic field moving up the energy-momentum dispersion relation by an amount ∆ , and ∆ (dSk ) ≡ dk∆k is the corresponding change in the area of the quasiparticle’s 25  2.3. Thermodynamic Potential and Density of States orbit. Integrating around an entire orbit gives 2  τ=  ∂Sk , 2eB ∂  where it should be noted that the derivative  ∂Sk ∂  (2.29)  is taken with the component of k parallel  to B (which I denote κ, in accordance with Shoenberg’s notation) held constant. It is useful to define the quasiparticle effective mass at this point: 2  m ≡  ∂Sk ∂  2π  .  (2.30)  κ  For a free electron the orbital area is Sk = πk 2 and the energy =  2 k2 2m ,  and so Equation 2.30  yields the correct result. Combining Equation 2.29 and Equation 2.30 gives the cyclotron frequency for an arbitrary (k): eB . m  ωc =  2.3  (2.31)  Thermodynamic Potential and Density of States  In order to calculate the DOS as a function of magnetic field, one starts by calculating the thermodynamic potential (or Landau free energy) of the system: Ω = − T S − N ζ,  (2.32)  where T is the system’s temperature, S is its total entropy, N is the number of quasiparticles, and ζ is the chemical potential [92]. The calculation is easiest to perform at T = 0, with the effects of temperature added later. For a system of Fermions this potential is given by ζ−  Ω = −kB T  ln 1 + e kB T  ,  (2.33)  states  which, at zero temperature, reduces to Ω=  ( (n, κ) − ζ) .  (2.34)  states  It is important to emphasize that the quasiparticle energies depend on the index of the Landau level, n, and the component of the quasiparticle’s momentum parallel to the field, which was defined earlier to be κ. The sum is over all occupied Landau levels, which are all levels with energy less than ζ, and so it is helpful to have the DOS per Landau level. Using Equation 2.9, the DOS on a 2D slice of a Landau level with thickness dκ is Dκ =  2 eBLx Ly Lz eB = 2 , V 2π 2π 2π  (2.35) 26  2.3. Thermodynamic Potential and Density of States where V is the real-space volume of the sample. Although this result uses the DOS per Landau level derived for free electrons, Lifshitz and Kosevich pointed out that it must be true for an arbitrary dispersion relation since it is independent of (k) [50]. The free energy density (per unit volume) for a slice of momentum space perpendicular to the field can now be written as  ∞  δΩ = Dκ  (  n  − ζ) Θ(ζ −  n ),  (2.36)  n=0  where the sum is over all of the Landau levels, and the Heaviside step function Θ(ζ − (n)) ensures that the sum is only over levels that are filled (i.e., below the chemical potential at T = 0). The dependence on the wave vector κ has been suppressed here since this calculation is performed for a slice at fixed κ. The sum in Equation 2.36 includes contributions from all of the Landau levels, but it is clear that any oscillatory part of the thermodynamic potential will come from Landau levels passing through the chemical potential and becoming depleted, and not from fully occupied Landau levels well below  F.  Thus what is really wanted is just this oscillatory  contribution, and so the sum over Landau levels is transformed to a sum over harmonics via the Poisson summation formula (see appendix 3 of Shoenberg [92]): ∞  ∞  ∞  (  n − ζ)Θ(ζ −  n) =  ( 0  n=0  n  − ζ)Θ(ζ −  nm ∞  (  =  n  − ζ)dn + 2  =  ∂ ∂Sk  κ  n  − ζ) cos (2πpn) dn,  p=1  − γ is the highest occupied Landau level, which has an area equal to  the Fermi surface cross sectional area ∂ ∂n κ  ( 0  0  SkF 2πe  (2.37)  p=−∞ nm  where nm ≡  i2πpn dn n )e  ∂Sk ∂n  dynamic potential is  κ  10 .  Equation 2.37 is integrated by parts twice using  . This gives three terms and, using m =  nm  δΩ = Dκ  (  n − ζ) dn +  n=0  1 eB eB + 24 mn=0 2m  ∞ p=1  2  2π  ∂Sk ∂  cos (2πpnm ) . π 2 p2  κ  , the thermo-  (2.38)  As the first two terms are non-oscillatory in field, only in the third term is of interest, which is re-written as   δ Ω = Dκ   ωc 2  ∞ p=1   1 π 2 p2  cos 2πp  F −γ B  .  (2.39)  10  I am assuming that the chemical potential ζ is constant as a function of magnetic field. At very high fields this approximation breaks down, since the chemical potential is pinned to the highest occupied Landau level.  27  2.3. Thermodynamic Potential and Density of States  2.3.1  Quasi-2D Fermi Surfaces  Before proceeding further with an integration of Equation 2.39 along κ, the geometry of the Fermi surface needs to be specified. Since YBCO is a quasi-2D material, it is useful to consider quasiparticles that are free in the plane and are weakly coupled between planes. In a magnetic field the dispersion relation for this system is n, kz =  n+  1 2  ωc − 2t⊥ cos (kz d) ,  (2.40)  where t⊥ is the coupling strength between the layers, and d is the cˆ-axis lattice parameter (see Singleton [93] for a review.) Using Equation 2.24 the area of a momentum space orbit is then Sk =  2πm ζ 2  + 2t⊥  2πm  cos (kz d) ,  (2.41)  ζ 2t⊥ 1 + cos (kz d) − . ωc ωc 2  (2.42)  2  or, in terms of the maximum occupied Landau level, nm (kz ) =  This is an intuitive answer that can be guessed without any calculation: the highest occupied Landau level is just the ratio of the chemical potential energy to the cyclotron energy, and this is modulated along the kz direction by the ratio of the hopping energy 2t⊥ to the cyclotron energy. Combining Equation 2.42 with Equation 2.39 gives δ Ω = Dκ  ωc 2  ∞ p=1  1 π 2 p2  cos 2πp  ζ 2t⊥ 1 + cos (kz d) − ωc ωc 2  .  (2.43)  By defining ζm e 2t⊥ m ∆F ≡ , e F ≡  (2.44) (2.45)  the thermodynamic potential for a 2D slice is δ Ω = Dκ  ωc 2  ∞ p=1  1 π 2 p2  cos 2πp  F ∆F 1 + cos (kz d) − B B 2  Equation 2.46 can now be integrated along kz from − πd to  π d  .  (2.46)  if B is aligned with the cˆ-  axis and κ is identified as kz (the dependence on field angle will be dealt with in section 2.5.) Note that −2πp 12 inside the cosine gives an overall factor of (−1)p . To simplify notation  28  2.4. Phase Smearing the integral Ip is defined via δ Ωdkz = Dκ  ωc 2  ∞ p=1  1 Ip , π 2 p2  (2.47)  where π d  Ip =  −π d π d  =  2πpF 2πp∆F + cos (kz d) dkz B B  cos  2πpF B  sin  2πpF B  cos  −π d  sin  cos  2πp∆F cos (kz d) − B  2πp∆F cos (kz d) dkz . B  (2.48) (2.49)  The sin term in Equation 2.49 integrates to zero by symmetry. The first term can have its integration range shifted by Ip = =  π 2d 3π 2d  −π 2d  to give 2πpF B  cos  2π cos d  2πpF B  cos J0  2πp∆F sin (kz d) dkz B 2πp∆F B  ,  (2.50) (2.51)  where J0 (x) is the 0th order Bessel function of the first kind (see page 952 of Gradshteyn and Ryzhik [30] for the integral representation of J0 used here.) The full expression for the oscillatory part of the thermodynamic potential can now be written as Ω=  eB ωc π 2d  ∞ p=1  (−1)p cos π 2 p2  2πpF B  J0  2πp∆F B  .  (2.52)  Equation 2.52 is the final form for the free energy density of a quasi-2D Fermi surface for quasi-free electrons in a magnetic field at T = 0 K. The free energy oscillates with frequency F , corresponding to the average Fermi surface size, and is modulated by the Bessel function, which has a frequency corresponding to the scale of the interlayer tunnelling. In subsection 2.4.2 I will show that the sum in Equation 2.52 can often be truncated at the first or second term, as higher harmonics are dampened out exponentially with an increasing factor of p. I now turn to the problem of incorporating finite temperature, impurity scattering, and the spin of the quasiparticles into Equation 2.52.  2.4  Phase Smearing  The simplest way of dealing with amplitude damping effects—finite temperature, finite quasiparticle lifetime, and quasiparticle spin—is to consider each effect as introducing a  29  2.4. Phase Smearing change in the oscillation frequency with some characteristic distribution, and then integrating over the distribution of frequencies. The simplest example is quasiparticle spin: the magnetic field changes the energy of the spin up and spin down quasiparticles by an amount proportional to their spin magnetic moment and the field strength. This effectively gives two populations of quasiparticles—spin up and spin down—with chemical potentials in Equation 2.43 shifted by the Zeeman energy. Because the chemical potential comes into Equation 2.43 as  ζm eB  and the Zeeman energy is proportional to B, this adds a field-  independent phase factor for each population of quasiparticles. The end result is the whole oscillatory term gets multiplied by a constant factor that depends on the quasiparticle’s spin magnetic moment. I will show in the following section that all of the standard phasesmearing effects end up multiplying the oscillation amplitude by an overall non-oscillatoryin-field damping factor. These results can also be derived directly from the quasiparticle’s self-energy, but for standard assumptions about the form of the self-energy the same results are obtained (see section 2.6 of Shoenberg [92] for a discussion of this, and additional effects of many-body interactions.)  2.4.1  Spin Splitting  The Zeeman effect shifts a quasiparticle’s energy by ∆ =g  e e s · B = ±g B, 2m0 2m0 2  (2.53)  where g may be different from its free electron value of g ≈ 2.0023, and m0 is the free electron mass. This “spin splitting” adds a term to the chemical potential in Equation 2.43, and a sum needs to be performed over both spin orientations (additionally, pull out a factor of 2 from Dκ in Equation 2.43 since the sum over spin is now explicit.) The oscillatory term is now Ω∝  cos 2πp σ=↑↓  F e + g σB B 2m0 2  1 ωc  m F m F + πpg + cos 2πp − πpg B 2m0 B 2m0 m F = 2 cos 2πp cos πpg . B 2m0 = cos 2πp  (2.54)  Note that this sum returns the factor of two, and yields a field-independent factor, Rs ≡ cos πpg  m 2m0  , that modifies the amplitude of the oscillations.  For free electrons where g ≈ 2 and m = m0 , this Rs factor multiplies each harmonic by (−1)p . m was defined in Equation 2.30 as the rate of change of orbital area in momentum space as the orbit’s energy is increased, and I will show in section 2.5 that this causes m to increase as  m cos θ  when the field is tilted at an angle θ with respect to the cˆ-axis in a quasi-2D 30  2.4. Phase Smearing system. This behaviour of m gives this spin term interesting behaviour as a function of angle. This derivation is almost correct, except that I was a bit overzealous in using m — the fully renormalized cyclotron mass of Equation 2.30. A more detailed quantum mechanical calculation by Engelsberg and Simpson [24] shows that the mass appearing in Equation 2.54 should not include the electron-phonon interactions which renormalize m . Electron-electron interactions still play a role in renormalizing g and m, however, and so I have defined g and a new mass ms ≡  m 1+λ ,  where λ is the electron-phonon coupling  strength, as the correct parameters. The mass ms is now understood to include both the effects of the periodic lattice (band structure effects) and electron-electron interactions, but not electron-phonon interactions. The correct spin term can now be written as Rs ≡ cos πpg  2.4.2  ms 2m0  .  (2.55)  Finite Quasiparticle Lifetime  Following the procedure outlined in chapter 2 of Shoenberg [92] for incorporating phasesmearing effects into the oscillatory chemical potential, and dropping all non-oscillatory factors to simplify notation, the thermodynamic potential is Ω∝  ∞ −∞ cos  F 2πp B + φ D( φδ )dφ ∞ φ −∞ D( δ )dφ  ,  (2.56)  where D( φδ ) is a distribution of phases with characteristic width δ. For example, in subsection 2.4.1, D( φδ ) would have been two delta functions: one at each of the shifted Fermi surfaces from the Zeeman interaction. This distribution is assumed to be independent of kz . If it is not independent, then this phase-smearing procedure must be done before the integration in kz . This, in principle, presents no difficulty but makes for messy looking results. For a quasi-2D system, the scattering rate and the effective mass can be often be approximated as constants, which is what I will assume. Defining χ ≡ φδ , Equation 2.56 is re-written as Ω∝  F  ei2πp B  ∞ iδχ D(χ)dχ −∞ e ∞ −∞ D(χ)dχ  = cos 2πp  F B  ∞ iδχ D(χ)dχ −∞ e ∞ −∞ D(χ)dχ  .  (2.57)  The end result here is that the oscillations get multiplied by the real part of the normalized Fourier transform of the distribution of phases. Several scattering mechanisms are discussed in the original derivation of the Shubnikovde Haas effect by Adams and Holstein [1], including point defect scattering, low and high temperature acoustical scattering, ionized impurity scattering, and piezoelectric scattering. I will make use of the assumption that the dominant scattering mechanism is energy31  2.4. Phase Smearing independent (i.e., from point defects), which is crude, but works especially well in quasi-2D systems at low temperature. In this case, usually only one Landau level is intersecting the Fermi surface at any given field value, and so scattering between Landau levels of different energies can be ignored. Treating the impurities as randomly distributed and dilute (in the sense that scattering events do not interfere with each other i.e., no line-crossing diagrams in the self energy: see Mahan [63]), then a lifetime τ can be assigned to the quasiparticles such that a quasiparticle experiences a scattering event during a time dt with probability  dt 2τ .  In a hand-waving and  semi-classical way, it can be said that since the quasiparticle states have a finite lifetime their quantized energy levels are no longer delta functions, but rather are now broadened into Lorentzians with a full width at half maximum δ =  τ.  As long as  2τ  is smaller  than the Landau level spacing ωc , we can still think of n as a “good” quantum number describing the system. This affects the quantum oscillations in the following way: as long as τ is assumed to be constant as a function of , the Landau levels can be treated as delta functions, and the Lorentzian broadening now becomes a Lorentzian-weighted distribution of chemical potentials with width ∆ζ = phase shift of φ =  2πp (µ−ζ) ωc ,  τ.  Equation 2.57 can now be applied using the  where µ is the “effective” chemical potential being integrated  over, and ζ is the actual chemical potential of the material. The distribution of effective chemical potentials comes from the Fourier transform of t  the scattering probability P (t) = e− τ , which gives D( φδ ) = tion 2.57 now gives  1 2. (µ−ζ)2 +( 2τ )  Evaluating Equa-  πp  RD ≡ e− ωc τ . This result, referred to as the Dingle factor  11 ,  (2.58)  dampens the oscillatory amplitude by the  inverse of the product of the cyclotron frequency and the quasiparticle lifetime. Thus, the condition ωc τ > 1 is often quoted as necessary to observe oscillations; below this value the Landau levels become too broad to have an appreciable oscillatory effect when they pass through the Fermi surface.  2.4.3  Finite Temperature  Following a procedure similar to that in subsection 2.4.2, the broadened Fermi-Dirac distribution at finite temperature is treated as a weighted distribution of chemical potentials which the Landau levels pass through. There are rigorous ways of coming up with the correct distribution function, but intuition tells us it should be the derivative of the Fermi function:  1 1 D (µ) = β , 2 1 + cosh (β(µ − ζ))  (2.59)  11  The result was first derived in Dingle [21], and the scattering rate is often written in terms of a “Dingle temperature”: TD ≡ 2πpωc . This convention, however, obscures the simple result of damping by ωc τ , and so I will not use this convention.  32  2.4. Phase Smearing where β is  1 kB T .  Despite the fact that Equation 2.59 looks similar to the Lorentzian used  in subsection 2.4.2, the following calculation is not as simple as Fourier transforming a Lorentzian. I will present it here in full for the interested reader, since the result is always used but the derivation rarely given or cited. Equation 2.59 needs to be written in the form of Equation 2.57, and so for the case of finite temperature the phase shift is defined to be φ = 2πp (µ−ζ) ωc , with ζ being the real Fermi energy, and the width of the distribution is δ =  2πpkB T ωc .  Equation 2.57 can now be  evaluated; dropping the oscillatory term provides the damping factor, Ω∝  ∞ iδχ 1 −∞ e 1+cosh(χ) dχ ∞ 1 −∞ 1+cosh(χ) dχ  ,  (2.60)  where again χ = φδ . The denominator in Equation 2.60 will be calculated from the numerator by taking δ to zero at the end of the calculation. The integral ∞  I=  eiδχ dχ −∞ 1 + cosh (χ)  needs to be solved. Extend the integral into the complex plane, and noting that  (2.61) 1 1+cosh(χ)  goes to zero when χ goes to infinity, close the integral in a semicircle at infinity. The function  1 1+cosh(χ)  has poles at χ = (2n + 1)iπ for n ∈ {0, 1, 2, ..., ∞}, and, since the kernel  of Equation 2.61 is analytic everywhere else in the complex plane, the integral can be deformed so that it runs around all of these poles which are on the imaginary axis. Then the integral is simply 2πi times the sum of the residues of these poles. Taylor-expanding the denominator of Equation 2.61 about the nth pole gives 1 + cosh(χ) = 1 + cosh((2n + 1)iπ) + sinh((2n + 1)iπ)(χ − (2n + 1)iπ) 1 + cosh((2n + 1)iπ)(χ − (2n + 1)iπ)2 + ... 2 1 = − (χ − (2n + 1)iπ)2 + O(χ4 ). 2  (2.62)  Equation 2.62 shows that there are second-order poles in Equation 2.61. Using the fact that for second order poles, Res (f (χ), χ0 ) =  ∂ lim (χ − χ0 )2 f (χ) , ∂χ χ→χ0  (2.63)  the residue of the nth pole is given as Res (f (χ), χn ) = −i2δe−(2n+1)πδ .  (2.64)  33  2.5. Angle Dependence Writing the integral I as the sum of all of these poles times 2πi gives ∞  I = 4πδe−πδ  e−2nπδ .  (2.65)  n=0  Multiplying both sides by 1 + e−2πδ gives ∞  (1 + e  −2πδ  −πδ  )I = I + 4πδe  e  −2πδ  e−2(n−1)πδ  +  n=1 −πδ −2πδ  = 2I + 4πδe ∴I=  e  4πδ . sinh(2πδ)  (2.66)  We can now plug in the definition of δ, divide by the normalizing factor of lim I = 2, and δ→0  finally define the temperature damping factor as 2π 2 pkB T ωc 2π 2 pkB T sinh ωc  RT ≡  .  (2.67)  This damping factor depends on the ratio of thermal to cyclotron energy: intuitively, when kB T >  ωc , then the width of the Fermi-Dirac distribution is greater than the spacing  between Landau levels, and so the oscillatory effect is decreased. In the high temperature limit, this factor behaves very similarly as a function of field to RD .  2.4.4  Full Expression for the Oscillatory Thermodynamic Potential  Using Equation 2.55, Equation 2.58, Equation 2.67, and Equation 2.52, the full expression for the oscillatory thermodynamic potential of a quasi-2D metal in a magnetic field can finally be written down as eB ωc Ω= π 2d  2.5  ∞  Rs RT RD p=1  (−1)p cos π 2 p2  2πpF B  J0  2πp∆F B  .  (2.68)  Angle Dependence  The derivation to this point has relied on having the magnetic field B aligned with the cˆ-axis of the sample. When the sample is oriented such that there is an angle θ between the cˆ-axis and the magnetic field, the quasiparticle orbits rotate in momentum space so that they remain perpendicular to the magnetic field. For an unwarped cylindrical Fermi surface the cross-sectional area increases as  Sk (θ=0) cos θ .  In this extreme case, where the quasiparticles have  no velocity in the kz direction, it is useful to think of rotating the sample with respect to the magnetic field as projecting a component of the magnetic field onto the conducting plane of 34  2.5. Angle Dependence the sample. For zero warping, you can simply replace B by B cos θ in Equation 2.68 and get the correct answer. However, the angle dependence is not giving you any extra information in this case, as rotating the magnetic field just projects the field to lower values. Things become more interesting when there is finite kz dispersion. The cross sectional area perpendicular to the field, Sk , now acquires an angle dependence, and integrating this across the full Brillouin zone gives an angle dependence to the amplitude structure of the oscillations. The angle dependence of the cross sectional area can be calculated with some trigonometry and by using cylindrical coordinates that are perpendicular to the magnetic field; this was first done by Yamaji [110] for the case of simple warping. The result for more general warping shapes is obtained by expanding the shape of the Fermi surface in cylindrical harmonics. A good resource for these expansions is Bergemann et al. [10], with the full gory calculation appearing in Appendix B of Mercure [64]. The end result is an expression for the Fermi surface area as a function of κz , which is kz normalized by the cˆ-axis lattice constant so that it runs from −π to π, and the angle of tilt θ. Given in terms of an integral of the Fermi wavevector kF over the in-plane angle φ, the area reads Sk (κz , θ) =  1 2 cos θ  2π  kF κz − k F sin φ tan θ  2  dφ,  (2.69)  0  where kF is a function of κz − k F sin φ tan θ , and k F is the average value of the in-plane wavevector [64]. An example of a simple-warping Fermi surface, with these angles labelled, is shown in Figure 2.2. The next step is to expand kF in terms of cylindrical harmonics, and then decide which terms to keep based on symmetry considerations or knowledge about the hopping matrix elements from band structure calculations. Finally, the area can be converted to a frequency via Equation 2.26, plugged into Equation 2.39, and integrated along κz . Doing this for the example band structure given by Equation 2.40 (free in the plane, weak coupling in the cˆ-axis direction) gives 2πpF B cos θ  Ω ∝ cos  J0  2πp∆F J0 dk F tan θ B cos θ  ,  (2.70)  where d is the cˆ-axis lattice constant—11.75 ˚ A for YBa2 Cu3 O6.59 [59]. For more complicated Fermi surface shapes, the integral must be evaluated along kz numerically, since there will be no analytic solution [10, 64]. Equation 2.69 can, however, be integrated over φ to obtain Sk as a function of only κz and θ. This calculation can be found in Mercure [64], and yields 2  Sk (κz , θ) =  πk F + cos θ  ∞ µ=0,ν=1  2πk F kµν Jµ νdk F tan θ cos (νκz ) . cos θ  (2.71)  35  2.5. Angle Dependence  Figure 2.2: A quasi-2D Fermi surface, with an extremal quasiparticle orbit shown as the thick black line. The orbit is perpendicular to the magnetic field, which is applied at an angle θ to the cˆ-axis. Equation 2.69 gives the area of this orbit in terms of an integral around the angle φ. Reproduced with permission from Bergemann et al. [10]. Equation 2.71 comes about by expanding the in-plane shape of the Fermi surface in terms of polar harmonics (with index µ), and expanding it into Fourier components in the κz direction (with index ν.) Terms with ν = 0 are omitted because they are invisible to quantum oscillation measurement techniques (they simply rescale the average area). There is also a φ0 dependence, which represents the in-plane field angle, but since our experiment keeps this angle fixed it simply amounts to a rescaling of the warping parameters, and so I omit it here. We will mostly be interested in the terms k01 and k02 , which represent warping of a cylinder along the cˆ-axis. Higher order terms may be present, but require a full φ dependent quantum oscillation measurement, or angular magnetoresistance oscillations (AMRO) measurements, to be determined accurately. A nice derivation of the angle dependence can also be found by starting with the exact wave function for free electrons in a magnetic field in two dimensions, Equation 2.7, and then performing first order perturbation theory around the chemical potential with the introduction of a finite t⊥ , which is at some angle θ to the magnetic field. This calculation gives the same results as Equation 2.70 above as long as the hopping energy is much less than the chemical potential—t⊥  ζ, and provides corrections when the inequality is only  36  2.5. Angle Dependence weakly held (if the inequality is not held at all then the perturbation theory fails [51]). The next question is: what happens to the damping factors when the field is tilted at an angle θ? I will assume that parameters like the scattering rate and the g-factor have a minimal angle dependence. While this is obviously an assumption, and will not hold in the most general case, estimates of the g factor from nuclear magnetic resonance (NMR) and magnetic susceptibility indicate that the anisotropy between the planar and cˆ-axis directions to be around 10% [106]. More complicated models for the scattering rate will be considered later on if angle-independent scattering does not describe the data well, but the first choice should always be the simplest one. With these assumptions in hand, we can look at section 2.4 again and incorporate the angle dependence. It turns out that the only real change to RT , RD , and Rs needed to account for the field angle is to look at how the cyclotron effective mass changes with angle. Equation 2.30 is the definition of this mass, and if Equation 2.41 is identified as the momentum-orbit area at zero angle, the known angle dependence of the area from Equation 2.69 can be used to get the angle dependence of the mass. As long as t⊥ m (θ) =  ζ the mass will evolve as  m (0) . cos θ  (2.72)  Corrections come in for large t⊥ because as the limit of an isotropic 3D Fermi surface is approached there should obviously be an isotropic mass, but Equation 2.72 has held up well for previously studied quasi-2D systems [10, 64, 93]. The upshot of this is that the damping factors change, with m →  m cos θ ,  to  Rs ≡ cos πpg RD ≡ e− RT ≡  ms 2m0 cos θ  (2.73)  πp cos θ ωc τ  2π 2 pkB T cos θ ωc 2π 2 pkB T cos θ sinh ωc  (2.74) .  (2.75)  Equations 2.74 and 2.75 have monotonic angle dependences; Equation 2.73, on the other hand, has a highly complex angle dependence, and, since it is independent of field, comparison of the total oscillation envelope with this function allows the extraction the product g ms [10, 64, 93, 110]. Specifically, the whole oscillation envelope should have zero amplitude whenever the argument of Equation 2.73 is an integer multiple of  π 2.  Angles  where this happens are known as “spin zeros” in the literature. This behaviour is best understood by looking at Figure 2.3. Exploiting this spin factor, Equation 2.73, will be one of the most important parts of this thesis (see section 5.3 and chapter 7), and analysis of this term for the angle dependence of YBa2 Cu3 O6.59 is published in Ramshaw et al. [79].  37  2.6. Shubnikov-de Haas  1.0  Rs Amplitude  0.5  ms  4 me  0.0  0.5  1.0 0  10  20  30  40  50  60  Angle degrees Figure 2.3: The Rs spin factor term, Equation 2.73, plotted as a function of angle. The first spin zero is around 28◦ . Here, the oscillation amplitude is zero, and, beyond this angle, the oscillation phase shifts by π with respect to its value before the spin zero. This can also be thought of as the oscillations picking up an overall minus sign at each spin zero.  2.6  Shubnikov-de Haas  The first quantum oscillations of any kind were observed by Shubnikov and de Haas in the resistivity of bismuth metal [85]. Unfortunately for Shubnikov, no connection to quantum mechanics was made, and this effect is actually very difficult to observe in pure metals (the exception being semimetals like bismuth, which have small Fermi surfaces). A few months later, de Haas and van Alphen discovered oscillations in the magnetization of bismuth, and the connection was made to Landau’s observation that Fermi-Dirac statistics should lead to oscillations of the magnetization in a changing magnetic field, and the de Haasvan Alphen (dHvA) technique soon entered a golden age for measuring the Fermi surfaces of metals. For YBCO, it turns out that the size of the Fermi surface in high fields is very small, enabling the observation of very high quality resistivity, or Shubnikov-de Haas (SdH), oscillations. The origin of the SdH effect comes from the fact that the quasiparticle scattering rate is tied to the DOS at the Fermi surface. I argued in section 1.2 that the probability of a quasiparticle scattering is proportional to the number of states that it can scatter into. Thus an oscillating DOS at the Fermi surface will result in an oscillatory scattering rate, and an oscillatory resistivity will follow. Therefore to calculate the resistivity in a magnetic 38  2.6. Shubnikov-de Haas field, the DOS at  F  should be calculated from the thermodynamic potential given by  Equation 2.70. A fully quantum mechanical treatment of electrical transport in a magnetic field was first given by Adams and Holstein [1] for a spherical Fermi surface, with a number of different scattering mechanisms considered. While the calculation is much too complicated to reproduce here (or, for myself, reproduced at all), their general result is as follows: the oscillatory part of the conductivity divided by the background zero-field conductivity has a main contribution that goes as the oscillatory density of states at the chemical potential divided by the zero-field density of states, plus a secondary contribution that goes as this same quantity squared. The exact result is 1 σ 5 D(ζ) 3 = + RT σ0 2 D0 (ζ) 2  D(ζ) D0 (ζ)  2  ,  (2.76)  where the subscript 0 indicates a quantity at zero field, and a tilde indicates the oscillatory contribution in field. The factor of RT comes in because the DOS at the chemical potential ζ is independent of the temperature, whereas the conductivity must account for scattering to and from states within an energy kB T of the chemical potential. Since the second term is usually quite small in amplitude relative to the first[92], this expression is qualitatively the same as just equating the oscillatory conductivity with the oscillatory DOS . At low Landau level number, less than 10 or so, the semi-classical picture breaks down, and then the full quantum mechanical treatment is needed. Taking the equation for the thermodynamic potential at zero temperature, Equation 2.34, and changing the sum over states to an integral over the DOS as a function of energy, we get ζ  ( − ζ) D (ζ) d .  Ω (ζ, T = 0) =  (2.77)  −∞  The DOS is calculated from this expression as the negative of the second derivative with respect to chemical potential. Since we are only interested in the oscillatory contribution to the DOS, we just need to differentiate Equation 2.68 without the RT term (since the DOS is only concerned with the number of states and not their occupation). Using the definition F ≡  ζm e  , the oscillatory DOS is D (ζ) =  2m πd 2  ∞  (−1)p Rs RD cos p=1  2πpF B  J0  2πp∆F B  .  (2.78)  If the non-oscillatory components of Equation 2.38 are included, and their contribution to  39  2.6. Shubnikov-de Haas the DOS is calculated as well, the total DOS at the Fermi energy is found to be  D (ζ) =  defining D0 ≡  m πd  2  m πd  2    ∞  (−1)p Rs RD cos  1 + 2  p=1  2πpF B  J0  2πp∆F  , B  (2.79)  to be the DOS in zero field. Using the first term of Equation 2.76 the  conductivity can be written as RT  D − D0 σ − σ0 = =2 D0 σ0  ∞  2πpF B  (−1)p RT Rs RD cos p=1  J0  2πp∆F B  .  (2.80)  This is essentially as far as I will go with the semi-classical calculation: Equation 2.80 describes a quasi-2D material quite well as long as the number of Landau levels below the Fermi surface does not become too small (specifically, as long as ωc < 2t⊥ .) The integral of the Fermi surface area along κz that gives the Bessel function in Equation 2.80 can be replaced by the more general integral for different Fermi surface shapes. The astute experimentalist will note at this point that it is the resistivity, not the conductivity, that is measured in experiments, and that it would be helpful to have Equation 2.80 in a more useful form. Writing the total conductivity as σ = σ0 1 +  σ σ0  , then in terms of  the measured resistivity ρ = 1/σ the oscillatory conductivity can be written as σ =− σ0 to first order in resistivity, and  ρ− 1 σ0  1 σ0  ,  (2.81)  σ σ0 . When analyzing experimental resistivity data, ρ is the total measured 1 σ0 is the non-oscillatory background that is fit to the data. The angle  dependence can be incorporated into this expression in the same way it was in section 2.5.  40  Chapter 3  Experimental Technique Quantum oscillations can be observed in everything from the size of the sample (magnetostriction) to the temperature (change in entropy in a thermally isolated sample) [92]. Not all techniques give equal magnitudes in their effect, however, and choosing which technique to use will depend on the type of sample and the way in which the field will be applied. YBCO is a high temperature superconductor, which means that most of the Fermi surface properties will be masked by the superconducting gap below Tc . When a magnetic field is applied to a superconductor, currents are set up along the surface of the material in accordance with the London equations [101]. These currents screen the magnetic field from the interior of the sample, and this phenomenon is known as the Meissner effect  12 .  A piece of superconductor that has an applied magnetic field fully  screened from its interior, except within a penetration depth λ of the surface, is said to be in the Meissner state. Above a critical field value, Hc1 , the field will penetrate the sample. For type-I superconductors, a class which includes most elemental superconductors, this is the only critical field scale and superconductivity is destroyed at this field  13 .  In a type-II superconductor,  such as YBCO, the magnetic flux penetrates the sample in quantized units, Φ0 =  h 2e .  These  regions of flux penetration, called vortices, form a lattice structure in a clean superconductor. When an electric field is applied to a sample and a current flows, the magnetic field produced by the current creates a force on the vortices, causing them to move and dissipate energy. However, the vortex lattice can be pinned in place by disorder, preserving the vanishing resistivity property of a superconductor in an applied electric field. When a second critical field scale is reached, Hc2 , the vortices overlap and the sample becomes a normal metal. 12  A perfect conductor (that is not a superconductor) will also screen applied magnetic fields from its interior, provided that the fields are zero at some point in the infinite past. If the fields are finite before “turning on” the perfect conductivity, then the fields will not be screened from the interior of the conductor. This lack of screening is the crucial difference between a perfect metal and a superconductor; a superconductor will always expel magnetic fields (lower than the critical field) from its interior, even if there are magnetic fields penetrating the material before it becomes superconducting, and this expulsion is the Meissner effect. 13 When a magnetic field is applied perpendicular to the face of a very thin superconducting plate, there is a very large demagnetizing factor that results in the effective fields at the corners of the sample being much larger than the applied field. If the sample is a type-I superconductor, then the edges will be driven into the normal state. Because of inhomogeneities in the thickness of thin plates, the magnetic field will not always enter the sample in a uniform way, and the result can be disordered patches of superconductor and normal metal. This is a very different scenario from the penetration of the magnetic field in a type-II superconductor.  41  3.1. DC Electrical Transport This simple picture is actually much more complicated for real materials, and the interested reader should consult Fisher et al. [26] and Blatter et al. [11] for further information. The important point for YBCO is that there is another critical field scale above Hc1 , which I will refer to as Bmelt , where the vortex lattice melts to a vortex liquid, and finite resistivity is measured. This field is below Hc2 at finite temperature, and the exact value of Bmelt can depend on the coherence length, the penetration depth, the anisotropy, the temperature, and the amount of disorder in the system. In YBCO there is a first-order phase transition at Bmelt , and Hc2 in YBCO is not a critical field but a crossover from the vortex-liquid to the normal metal [11]. Pulsed magnetic fields, obtained by discharging a large capacitor bank through a coil of copper wire, can regularly achieve fields of up to 70 tesla, with 100 tesla fields being realized in some cases. These fields are sufficient to melt the vortex lattice in underdoped YBCO, allowing Fermi surface properties to be measured that were masked by superconductivity in zero field. The time scale of these pulsed fields varies, but field pulses to 60 tesla typically have a full-width-at-half-maximum pulse time of about 100 millisecond (ms). This in turn means that thermal techniques, such as heat capacity oscillations which require fairly long averaging times and stable background temperatures, are ruled out or at least extremely difficult. Mechanical techniques that measure sound speeds and attenuation factors require a vibration-free environment, which is hard to achieve in a pulsed field. Two techniques that do lend themselves well to pulsed fields are torque magnetometry—the dHvA effect—and resistivity—the SdH effect. The following subsections will give a brief overview of the electrical transport setup used to collect the data in this thesis. I would like to point out that this setup was designed and built by the lab of Dr. Cyril Proust at the Laboratoire National des Champs Magn´etiques Intenses (LNCMI). While I did operate this setup during the experiments, and performed maintenance on it between experiments, I did not build this apparatus. I will therefore only give a very brief overview of the resistivity setup.  3.1  DC Electrical Transport  The first quantum oscillations observed in YBCO were seen in the Hall coefficient [22]. Specifically, this was YBa2 Cu3 O6.51 , and the oscillation amplitude had a maximum of 4 milliohm (mΩ) in Rxy on a background of about 55 mΩ. With a 5 mA excitation current this means that the voltage being measured was about 275 µV, or 55 mV after amplification with a Texas Instruments INA103. These measurements were made with the current running in the a ˆ-ˆb-plane of the sample. A typical YBCO sample measured with the current parallel to the cˆ-axis is around 50 to 100 µm thick, has a resistance of about one ohm in this geometry. This corresponds to a voltage of 5 mV even before amplification, or half a volt after amplification. Larger voltage drops across the sample are beneficial because most noise 42  3.1. DC Electrical Transport  1 kΩ Voltmeter 5 Vac Sample I 1 V1  V2 I 2  Figure 3.1: Circuit diagram for the measurement, with an in-plane-contact-geometry (as opposed to cˆ-axis) shown for clarity. The voltage across the 1 kΩ resistor is measured with a separate voltmeter, giving the exact current through the sample, which will be near 5 mA since the lead, sample, and contact resistance are all much less than 1 kΩ. As long as the contact resistances at V1 and V2 are not too large, then the potential drop measured by the voltmeter will be equal to the potential drop across the sample between the voltage contacts. sources—such as Johnson-Nyquist noise from the electrical leads (which are about 30 Ω), and noise due to vibrations of the wires when the field is pulsed—are largely independent of the sample resistance, and thus make up a smaller fraction of the total signal for a large-resistance sample. Most of the measurements in this thesis are cˆ-axis resistivity measurements, with the current injected through large gold pads on opposite faces of each crystal, and the voltage being measured across a smaller set of contacts on the same opposing faces. This contact geometry can be seen in Figure 4.6, and the method for creating these contacts is detailed in that chapter as well. The most accurate way to measure the resistance of a sample is with a four-point measurement, where current is passed uniformly through the sample, and the potential difference between two points along this current path is measured with a high-impedance voltmeter. This measurement is realized by making electrical contact to the sample with two current leads, and measuring the voltage across a second set of contacts, all in a well defined geometry. This setup is shown in Figure 3.1. The resistivity probe itself is very simple: twisted pair wires—for the sample current, sample voltage, and temperature regulation—run down the length of a support rod to the tip where the samples are attached. Carefully twisted wires, with minimal cross sectional area between the two wires, are essential in a pulsed-field environment where voltages are generated in any section of wire that presents a closed loop perpendicular to the magnetic field. The sample mount can be seen in Figure 4.8. The entire probe is immersed in  43  3.2. Pulsed Magnetic Fields liquid/gaseous helium in a vacuum cryostat, which insulates the helium from the liquid nitrogen that fills the bore of the magnet. There are coaxial connections to the wires at the top of the probe making connection with current source and data acquisition system, which is described in chapter 5.  3.2  Pulsed Magnetic Fields  The pulsed magnetic fields at the LNCMI are generated by discharging a capacitor bank through a resistive coil of copper wire. For a 60 tesla field pulse, the capacitor bank is charged to around 20 kV (depending on the exact specifications of the coil), and then electrical contact is made between the coil and the capacitors and the stored charge is discharged through the coil. The total energy discharged during the pulse is five Megajoules, which gives a total capacitance of 0.025 farads. Because the coil is resistive and inductive, the field profile starts off looking like a sine wave. At the peak field a mechanical switch disconnects the capacitor bank from the circuit and mechanically shorts the circuit with a shunt (a bar of copper), and the magnetic field created by the current in the coil decays as it would in an LR circuit. The actual field profile is a bit more complicated in shape than this because the coil heats up during the pulse: the coil is initially cooled to 77 Kelvin in liquid nitrogen, and heats up to around 300K, hence R is a function of time. Nevertheless, Figure 3.2 is fairly well described by a quarter sine wave followed by an exponential decay. Data is taken during both the rise and fall of the field, and it is important to compare the data from both the rising and falling fields. Resistivity should not depend on the sign of dB dt ,  and therefore the data should be the same on the rise and the fall of the magnetic field.  Figure 3.3 shows that this is not always the case; for temperatures above 4.2 kelvin, the sample is no longer in liquid helium during the pulse, but rather is in helium gas which is less effective at transporting heat away from the sample. Thus the sample heats up during the rising field due to eddy currents in the gold contacts, and the bath is unable to cool the sample back down fast enough compared to the time scale of the measurement. This is another reason to have high-quality electrical contacts; highly resistive contacts will cause excessive heating when a current is passed through them. The lower and broader resistive transition for the falling sweep in Figure 3.3 indicates that the vortex lattice is easier to melt on the down sweep than it was on the up sweep, and thus that the sample is at a higher temperature on the down sweep.  44  3.2. Pulsed Magnetic Fields  60  Magnetic Field Tesla  50  40  30  20  10  0 0  30  60  90  120  150  180  210  240  270  300  330  360  390  Time milliseconds Figure 3.2: Magnetic field as a function of time during a pulse. The capacitor bank is disconnected from the circuit and the coil shorted with a shunt near the peak field at around 80 ms.  45  3.2. Pulsed Magnetic Fields  1.0  Rising field Falling field  Resistance  0.8  0.6  0.4  0.2  0.0 0  10  20  30  40  50  60  Magnetic Field Tesla Figure 3.3: The resistance of a YBa2 Cu3 O6.56 sample during the rising and falling of the magnetic field, nominally at 8 kelvin. Note that the resistive transition happens at a lower field value on the down sweep. This is an indication that the sample has heated above the base temperature during the pulse.  46  3.3. Temperature Control  3.3  Temperature Control  The resistivity probe was inserted into a helium-4 cryostat, which itself was inserted into the bore of the magnet. The magnet coil is made of copper wire, and is cooled to 77 kelvin with liquid nitrogen. Since the cryostat for the experiment is inserted directly into the bore of the magnet, it is also cooled to 77 K. For temperatures below 4.2 K, the sample is immersed directly in the liquid helium. The liquid helium temperature is regulated by regulating the vapour pressure above the helium by pumping [37]. A minimum temperature of 1.5 K was achieved this way. Above 4.2 K the helium in the sample chamber is allowed to boil off, leaving only helium gas in the chamber. The temperature of the gas was regulated with a Lakeshore LTC-21 temperature controller, using a resistive heater and a Cernox thermometer attached near the sample. This setup provided temperature regulation up to 300 K.  3.4  Field Angle Control  For the angle dependence measurements, analyzed in chapter 7, the sample stage was mounted on a rotating platform. The neutral position of this platform is such that the cˆ-axis of the sample is aligned with the magnetic field. A loop of wire running down the length of the probe runs around a pulley attached to the rotator stage. The other end of the loop of wire runs around a pulley at the top of the probe, and by turning this pulley the sample’s cˆ-axis can be rotated by an angle θ with respect to the magnetic field. Details of how the angle was measured can be found in section 5.1.  47  Chapter 4  Sample Preparation Sample selection and preparation is an extremely important, and often ignored, step in experimental physics. A single growth of YBCO may produce hundreds of single crystals, but only a very small number (if any) are suitable for a particular experiment. Measurements of the critical field Hc1 require samples with very low demagnetizing factors [58]; in-plane penetration depth measurements require very thin samples to avoid cˆ-axis penetration depth contamination [32]; neutron-diffraction measurements need a large volume of sample, often provided by a mosaic of several aligned samples [97]; heat capacity measurements require the largest possible samples to overcome the background heat capacity of the apparatus [81]. An incorrectly prepared sample will not just give bad data: inhomogeneous oxygen, bad contacts, large contaminations from geometry effects, and innumerable other problems can produce data that will lead to interpretations unrelated to the underlying physics of the material being studied. If the data analysis techniques developed in chapter 5 and chapter 6 are the most important parts of this thesis, then attention to careful sample preparation is close behind.  4.1  As-Grown YBa2 Cu3 O6.x  The best YBa2 Cu3 O6.x samples are grown in home-made crucibles of BaZrO3 . This material is inert to the Y2 O3 -BaO-CuO melt in which the crystals are grown [56]. Because BaZrO3 is inert to the melt, the crucibles maintain their structural integrity at high temperature much longer than commercial yttria-stabilized-zirconia crucibles. This allows slower crystal growth, resulting in fewer structural defects in the crystals. Another advantage of the inertness of the BaZrO3 crucibles is that fewer impurities are incorporated into the YBCO, which leads to lower quasiparticle scattering rates. The growth flux is decanted at high temperature (960◦ C) after the growth, leaving behind hundreds of crystals stuck to each other, and to the crucible wall. A picture of a BaZrO3 crucible, with most of the crystals removed after a growth, is shown in Figure 4.1. Hundreds crystals, from a fraction of a millimetre up to almost a centimetre in size, can be found after a growth, but the large majority of the large crystals have cracks and holes which make them unsuitable for most measurements. All crystals that might be suitable for an experiment are extracted from the crucible and undergo high-temperature annealing 48  4.2. Selecting the Appropriate Sample  Figure 4.1: BaZrO3 crucible after a crystal growth. Most of the crystals attached to the walls have been removed. The crucible is a pale cream colour before the growth, but the Y2 O3 -BaO-CuO melt leaches into the walls during the growth, turning it black and eventually destroying it. in flowing oxygen (around 860◦ C) to relieve strains, which may be present from when the crystal was attached to the crucible wall or to another crystal, and to remove possible CO2 intercalation from the chains. The next step is to set the oxygen content. Selected crystals are sealed in a quartz tube with YBCO ceramic pellets, which have a mass about 100 times that of the crystals, with the oxygen content of the pellets set to the desired oxygen concentration for the crystals. For Ortho-II crystals, such as those used in most of the experiments in this thesis, the crystals and pellets are then annealed for a week at 460◦ C and then quenched quickly to low temperature to prevent loss of oxygen from the samples [55, 60]. The oxygen is also homogenized throughout the sample during this step, since the concentration of oxygen is higher near the edges of the crystal after growth. At this stage, samples suitable for cˆ-axis resistivity must be chosen, modified if necessary, and then detwinned (see subsection 4.2.1 for a discussion of twins) before they can have electrical contacts evaporated onto them.  4.2  Selecting the Appropriate Sample  The following considerations should be taken in to account when picking samples of YBCO for cˆ-axis transport measurements:  49  4.2. Selecting the Appropriate Sample • Sample size. Very large samples of YBCO will give larger signals in most experiments, but are more likely to contain dislocations or flux inter-growth. A large sample will take longer to come to thermal equilibrium with its environment. This may become a problem in pulsed fields where the sample heats up due to the vortex lattice melting and eddy current heating, and must return to base temperature before the field reaches its maximum if data at a constant temperature is to be obtained. • Aspect ratio. The cˆ-axis of YBCO is approximately 4000 times more resistive than the planes. Maximizing the ratio of the cˆ-axis size to the in-plane dimensions ensures that the cˆ-axis resistivity gives the dominant contribution to the resistance, even if the contacts are misaligned and some in-plane component to the resistivity is picked up. • Surface quality. A relatively flat surface is needed to obtain absolute values of the resistivity, and if the surface is overly contaminated or degraded then the electrical contact will be poor even after diffusion of the gold contacts. Flux on the surface is another concern because it can pin the twin domains and prevent full detwinning. Some flux may be removed carefully with a sharp razor or needle if the crystal is thick enough. • Cracks, inter-growth, and stacking faults. Cracks in the sample are not always easily visible, and often look like step edges on the surface. A crack will often cause the sample to break when pressure is applied to it during detwinning, but this is not always the case. Cracks will force electrical currents to take convoluted and unpredictable paths through the sample, giving data that is not representative of the bulk material. Inter-growth is a layer of another YBCO compound or the growth flux sandwiched between layers of crystal. cˆ-axis stacking faults and inter-growth can usually be detected by examining the twins along the cˆ-axis of the sample (see Figure 4.2.) If one of these flaws is present, the twins will not run the whole cˆ-axis of the crystal, but will be interrupted and begin a new twinning pattern on the other side of the flaw.  50  4.2. Selecting the Appropriate Sample  Figure 4.2: a ˆ-ˆ c face of a YBa2 Cu3 O6.x crystal showing twin boundaries. In the bottom half of the picture the twins run continuously from left to right; in the top half, there are several places where the twins run half way across the sample, and then abruptly change pattern on the right hand side. This is either a cˆ-axis stacking fault or an internal crack. The upper right of the picture also shows flux attached to the surface of the crystal. This crystal is approximately 70 µm thick in the cˆ direction, and approximately 1 mm by 1 mm in-plane.  51  4.2. Selecting the Appropriate Sample  Figure 4.3: An a-b face, showing the twin domains. Applying force to the two parallel edges at high temperature (200-300◦ C) will force the sample into one domain, provided that internal flaws are not so great as to pin the domains. This crystal’s dimensions are approximately 600 by 750 by 50 microns.  4.2.1  Sample Modification  Subsections of a crystal will often be suitable for a particular experiment, but will have other crystals attached to it, have a rough edge, a crack, flux, or other flaws. This is especially true for thick crystals—roughly those greater than 50 µm. There are a number of techniques for removing undesired parts of the sample, and for cleaning up the remaining faces and edges. At high temperatures YBCO is tetragonal, but as the temperature is lowered after crystal growth the sample passes through a tetragonal to orthorhombic phase transition. In the orthorhombic phase, YBCO’s a ˆ and ˆb-axis lattice constants are 0.3843 and 0.3871 nm, respectively [55]. As the sample is cooled through the transition different regions of the sample choose a different direction for the longer ˆb-axis, resulting in domains of different alignment for the a ˆ and ˆb axes. This is referred to as “twinning”, and a sample with more than one domain is “twinned.” Cutting Samples of YBCO thinner than about 50 µm can be cleaved along the a ˆ or ˆb directions using a razor blade. For thicker samples, cleaving is much more difficult, and usually results in the sample shattering into smaller fragments. Cutting with a wire saw is more appropriate 52  4.2. Selecting the Appropriate Sample for these thicker samples. For cˆ-axis resistivity experiments the alignment of the crystal edges with the a ˆ and ˆb axes is not of crucial importance, and so cutting need only be “good enough” so that the sample is detwinnable (see the next paragraph on detwinning). The samples are attached with a thermoplastic (Crystalbond #509) to a silicon or graphite base, and then cut using a K.D. Unipress wire saw. The cut is aligned by eye with the aid of a microscope, using an as-grown crystal edge for reference. Detwinning YBCO is tetragonal at high temperature, changing to orthorhombic at a doping-dependent temperature [105]. The oxygen in the chains is randomly distributed in the tetragonal phase, but as the temperature is lowered preferential directions are chosen and the oxygen forms chains. These chains run along the ˆb-axis of the crystal, and give rise to an orthorhombic distortion. As described at the beginning of this section, this structure tends to result in twinned samples. By applying a uni-axial pressure of about 100 atmospheres across two opposing edges of a crystal, the shorter a ˆ-axis is forced to align with the axis of the pressure—detwinning the crystal [57]. The temperature at which the dewinning happens depends on how strongly the twins are pinned with disorder, and on the oxygen content. As sample purity has improved over the past 20 years, the detwinning temperature has decreased to the point where most samples can be detwinned around 200◦ C. Samples are detwinned under a dry argon/oxygen atmosphere to prevent surface degradation. Figure 4.4 shows the detwinning apparatus: the black triangular block in the middle is made of YBCO ceramic, on which a sample is placed for detwinning. The two gold pads on either side of the block are moved right-to-left with the manipulator arms (outside of this image), and are used to transmit force to parallel edges of the sample. The stainless steel stage is heated with a resistive coil, and the tube entering from the upper-mid-left flows argon/oxygen over the sample to keep it dry. The two thermocouple wires attached to the gold pad on the right monitor the temperature: if the temperature gets too high then the sample will start to lose oxygen. Once the sample is detwinned, it is ready for electrical contacts.  53  4.3. Electrical Contacts  Figure 4.4: Detwinning stage showing the YBCO ceramic sample platform (the grey triangular sliver in between the gold pads) and the gold pads which apply pressure to parallel edges of the sample. The entire apparatus is housed in a vacuum chamber. The sample is placed across the end of the sample platform, as shown in the inset (rotated 90◦ from the main figure). When the gold pads are moved together, they make contact with the sample and detwin it (the top and bottom edges of the sample in the inset will have pressure applied to them when placed between the gold pads, orienting the crystallographic a ˆ-axis with the long-dimension of the sample).  4.3  Electrical Contacts  A four-point resistance measurement, described in section 3.1, is designed to measure a sample’s resistance without contamination from the resistance of the electrical leads and contacts. This relies on the assumption that the internal impedance of the voltmeter is large in comparison to all other resistances in the circuit. If this is the case, then almost all of the current in the circuit passes through the sample (rather than the voltmeter), and the potential drop in the voltmeter is the same as the potential drop across the sample between the voltage contacts (see Figure 3.1). Simple silver-painting of wires directly onto a YBCO surface results in contacts in the tens to hundreds of mega-Ohms; a typical input impedance for a voltmeter is 1 MΩ. Thus, proper contact preparation is important. Especially, high resistance contacts are found to be unstable when the temperature is cycled up and down, whereas low resistance “Ohmic” contacts tend to be reproducible from run to run. When YBCO is exposed to carbon dioxide, particularly CO2 dissolved in water, the outer crystal surface reacts with the CO2 to form barium carbonate—BaCO3 . Barium carbonate is insulating, and this layer prevents good electrical contact from being made 54  4.3. Electrical Contacts with the YBCO underneath. This can be overcome by evaporating gold contacts onto the crystal surface, and then diffusing the gold through the BaCO3 layer to make electrical contact with the YBCO.  4.3.1  Masking and Gold Evaporation  Lithography is a commonly used technique for defining electrical contacts, but it does not lend itself well to three-dimensional geometries (needed for in-plane transport, see chapter 9), nor does it work particularly well on small samples where the photoresist’s surface tension causes it to pull away from the corners of the sample. This has led to the development of shadow masking techniques, where custom masks for each sample are hand-made from 10 µm aluminum foil, and gold is evaporated on to the sample through the mask. This results in well defined contacts that can be on the scale of tens of micrometres, and that can wrap around the edges of the crystal. For preparation of the mask, the 10 µm foil is attached with Crystalbond adhesive to a copper sheet that has been sanded and polished flat, shown in Figure 4.5. This provides a hard surface where the foil cannot move while being cut, and can be easily peeled off once the mask is complete. A mask is cut in the desired shape with a scalpel, cleaned of crystal bond in acetone, and attached with small pieces of tape to a large square of the 10 µm aluminum foil. This large square of foil is attached to a copper finger (see Figure 4.6). The copper finger, with sample and mask attached, is put in a clip on a retractable arm in a small vacuum chamber which is evacuated to around 10−5 torr. A Variac (an autotransformer) is then used to control the current through a tungsten filament which is wrapped in gold wire, and the gold is evaporated on to the sample’s surface. Around 1000 ˚ A of gold are deposited in this manner, and the sample is then removed from the vacuum chamber and from the mask. The resistance of contacts at this stage is around 100 kΩ, because of the large Schottky barrier that forms between high-conductivity gold and poorly conducting surface of the YBCO crystal. The gold needs to be diffused into the surface in order to make good contact, and this is done in conjunction with the oxygen homogenization [60].  55  4.3. Electrical Contacts  Figure 4.5: Copper sheet with 10 µm aluminum foil attached with Crystalbond. This provides a hard and flat surface for preparing masks. Several masks have been cut from this piece of foil already: the bow-tie shaped masks are for a ˆ-ˆb-plane contacts, and the square sections are where cˆ-axis masks were made. A typical YBCO sample for cˆ-axis measurements is shown in the middle, and in the zoomed-in inset. This sample is 650 µm long, 350 µm wide, and about 30 µm thick. This is the same sample that is shown in Figure 4.6.  56  4.3. Electrical Contacts  Figure 4.6: Top left: The foil mask is shown taped down to a larger sheet of backing foil. The sample— 650 µm x 350 x 30 µm—is shown above the mask, and will be slid underneath (the same sample was shown in Figure 4.5.) Top right: The sample has been slid between the mask and the backing foil, and the edges of the mask have been crimped around the sample, using tweezers, to prevent the sample from moving during the evaporation. Mid left: The backing foil, the mask, and the sample are taped to a large copper finger which is held during the evaporation. Mid right: The sample after the gold has been evaporated onto one side. This image is taken with cross-polarized light which emphasizes the edge of the gold, and also reveals twin boundaries (this sample was prepared only for illustrative purposes, and was not detwinned). The crystal will be flipped over, re-positioned under the mask, and gold will be evaporated on the other side. Bottom left: In-plane contact geometry used in chapter 9. The contacts are prepared in a similar way to the cˆ-axis contacts, but they wrap around the sides of the crystal to prevent cˆ-axis contamination. 57  4.4. Oxygen Homogenization and Contact Annealing  4.4  Oxygen Homogenization and Contact Annealing  Twinning is a source of internal strain in a YBCO crystal, and oxygen tends to concentrate near the twin boundaries as a result of this strain[60]. After detwinning the strain is relieved but the inhomogeneity in the oxygen remains. For oxygen concentrations greater than 6.56, the oxygen can be homogenized by heating the sample to 570◦ C for one week. At this temperature the gold is also diffused into the sample. For oxygen concentrations below 6.56, the sample will be tetragonal at 570◦ C, and so the homogenization has to occur before detwinning (since entering the tetragonal phase and then cooling re-twins the crystal). To preserve oxygen homogeneity, the samples are quenched quickly from the high temperature tetragonal phase to around 0◦ C. This results in twinning, but the oxygen is not mobile enough at room temperature to significantly cluster at the twin boundaries before the sample is detwinned. The gold is then diffused in at a lower temperature, around 500◦ C, which takes longer but prevents re-twinning. After annealing the contact resistance of large contacts is usually at most one ohm, often lower. Very small (of order 10 µm by 10 µm) contacts are inconsistent in their quality, and contact resistance does not appear to scale linearly with contact area. This has led me to suppose that there are “hot spots” on the crystal surface that are capable of making good contact with the gold, which gives large contacts a better chance of hitting a good area. If the distance between hot spots is greater than the size of the very small contacts, then even doubling the size of the small contacts does not guarantee that a hot spot will be hit.  4.4.1  Citric Acid Etch  In order to improve contact reliability, I have tried etching YBCO crystals in citric acid before gold deposition. The procedure followed was as follows (following Ginley et al. [29]): • 500 ml of 0.1 molar citric acid was boiled for 2 hours in a condenser, with N2 gas bubbling through it to remove CO2 . • The solution was cooled to around 70◦ C with nitrogen gas still flowing. • Crystals of YBCO were etched for 1, 2, and 5 minute lengths in the citric acid solution, and then immediately transfered to vacuum for gold evaporation. • Gold wires were attached to the contacts with Epo-Tek H20E silver epoxy, and cured for 7 minutes at 150◦ C. Several (6 to 8) small contacts were made on the face of each sample, and all resistances were measured pair-wise between the contacts. Contacts deposited on etched surfaces were in the tens of kilohm range, with a few contacts as low as a few hundred ohms. This is a success, because contacts applied to an un-etched surface are in the megohm range. Perhaps even more importantly, the etched 58  4.5. Oxygen Ordering contacts were stable with time, whereas contacts on un-etched surfaces seem to “drift” in resistance value over several hours. It is important to note that none of these contacts were diffused into the surface at high temperature (the procedure described in section 4.4.) If contacts on un-etched surfaces can be brought from megaohm values down to around an ohm with the gold diffusion procedure, then presumably contacts on etched surfaces will be even better after diffusion. My hypothesis is that as the citric acid etches away the barium carbonate [29], more “hot spots” are created on the surface of the crystal. This etching procedure should be tested in a more systematic way, and the study should include contact diffusion at high temperature. The best etched surfaces appeared to result from one minute of etching, with longer etching times producing worse contacts. However, the temperature of the citric acid solution was not the same for each test case, and temperature affects the etch rate. Thus more careful control of the temperature should be included in future studies of etching.  4.5  Oxygen Ordering  YBa2 Cu3 O6.x at high temperatures—how high depends on the doping—is tetragonal, and the oxygen in the chains is randomly distributed [105]. As the material is cooled it undergoes a structural transition to the orthorhombic phase (at around 600◦ C for YBa2 Cu3 O6.59 ). The initial orthorhombic phase below the orthorhombic-tetragonal transition is always ortho-I, which means that there is a preferred direction in the chain layer, but that the oxygen is still homogeneously distributed between each chain. As the sample is cooled further, between 50 and 150◦ C depending on the doping, the sample may enter another ordered orthorhombic state. For oxygen concentrations from around 6.30 to 6.60, this is the ortho-II phase; just above oxygen 6.60 the ortho-V phase is stable; near oxygen 6.67 the ortho-VIII state is stable; from oxygen 6.70 to around 6.85 it is the ortho-III state that stabilizes; above oxygen 6.85, YBCO stays in the ortho-I state [105]. These oxygen structures are shown in Figure 4.7. Bobowski et al. [12] have shown that the source of impurity scattering in clean YBCO is disorder in the chains. In subsection 2.4.2 it was demonstrated that quantum oscillations are damped out exponentially with decreasing mean free path  14 ,  so it is clear that chain  disorder should be minimized in order to maximize oscillation amplitude. By heating YBCO to just below the transition temperature to the desired oxygen-ordered state, and annealing at that temperature for a few days up to a few weeks, the ordered chain state becomes much cleaner (fewer vacancies and longer coherence length) than if the sample had only 14 Equation 2.55 is written in terms of ωc and τ , which hides the mean free path. We can use the results eF of section 2.2 to write ωc = m = 2m kF2 . Using kF = m vF , where vF is the Fermi velocity, and defining the mean free path to be l ≡ vF τ , we see that ωc τ = m vF l, which makes the dependence of RD on the mean free path explicit.  59  4.5. Oxygen Ordering  Figure 4.7: Clockwise from top left: No chain oxygen, ortho-II, ortho-V,ortho-I, ortho-VIII, and ortho-III. The a ˆ − ˆb crystallographic directions are shown in the first panel. Copper atoms are shown in red, and oxygen atoms, which have a larger ionic radius than copper, are shown in blue. The black arrows encompass one period of order, with the label of that state printed above the arrow. undergone the higher-temperature homogenization step [60].  60  4.6. Attaching Wires  4.6  Attaching Wires  Ten micron platinum or gold wires, whichever was available, were attached to the electrical contacts in one of two ways: with silver paint, which sets at room temperature, or with Epo-Tek H20E silver epoxy, which dries in five minutes at 150◦ C. Silver paint does not form a very strong mechanical bond with the crystal surface or with the gold contacts. This can result in electrical contacts that change as the temperature is cycled up and down. In the worst case, the silver paint cracks off at low temperature and the experiment is over. Once silver epoxy has been cured, it cannot be separated from the YBCO without destroying the sample. Epoxy bonds also seem to survive temperature cycling better than silver paint bonds. The disadvantage of epoxy is that the sample must have the oxygen re-ordered (see section 4.5) after the epoxy is cured at 150◦ C. Once the wires are attached to the crystal, the crystal is attached with General-Electric G-9700 varnish (G.E. varnish) to a block of sapphire. This sapphire block allows the sample to be positioned in the probe without handling the sample itself (which often results in wires being broken off), and also provides a large thermal mass that helps maintain the sample temperature during the pulse. Attaching the sample to the block is also important because of the large torques that may be generated on the superconducting sample during the field pulse. Because of the large demagnetizing factor of a thin plate with a magnetic field applied perpendicular to the surface, a sample in this geometry wants to align its long axis with the magnetic field. This has resulted in samples being completely torn off of the sapphire block during the pulse when not enough varnish was used. The entire resistivity probe tip, with two samples mounted, is shown in Figure 4.8. Once the sapphire blocks are varnished into the sample holder, the electrical connections between the probe-body wires and the wires attached to the sample can be made with silver paint. Any connected wire path that has open loops perpendicular to the magnetic field will pick up a voltage proportional to  dB dt  during the field pulse. If the areas of these loops  remains constant during the pulse, then the offset voltage induced by the field is removed by a lock-in amplifier  15 .  Problems can arise, however, because during the field pulse the  cryostat experiences mechanical vibration due to the enormous internal torques in the fieldgenerating coil. If the coil doesn’t tear itself apart and explode, then the next source of concern will be the changing pickup voltages due to vibrating wire loops. The closer the vibration frequency (or its harmonic components) is to the measurement frequency, the more noise that will leak into the data, even with the use of a lock-in amplifier. To minimize this problem, it is important to minimize the size of any loops of wire. The wiring in the probe body is made up of twisted-pairs, which have minimal cross sectional area, but where the probe wires attach to the sample there can be loops of significant size. The wiring in 15  A lock-in amplifier is essentially a very sharp bandpass filter, and the characteristic frequency of the offset voltages induced by vibrating loops of wire is unlikely to be the same as the measurement frequency.  61  4.6. Attaching Wires  Figure 4.8: The probe tip with two samples mounted on sapphire blocks. The wires are attached to the samples with silver paint in this image. The samples have been G.E.varnished down to sapphire blocks to minimize vibration and movement due to torque during the pulse. The loose wire ends at the top and bottom of the picture will be trimmed and connected to the probe wiring. Figure 4.8 was done with the shortest lengths of wire possible, and all wires are varnished down or coated in grease (which freezes at low temperature) to minimize vibration.  62  Chapter 5  Data Acquisition and Analysis During a magnetic field pulse, four National Instruments PCI-5911 data acquisition cards record the voltage across the sample, the voltage across two pickup coils (one to measure the field and one to measure the angle), and the voltage from the output of the lock-in amplifier (which acts as a current source with the use of a 1 kΩ series resistor, and to provide a reference signal for the software lock-in.) Using this data, along with time stamps provided by the digital acquisition system, the resistance of the sample as a function of time can be reconstructed, as well as the magnetic field as a function of time. This data is combined to obtain the resistance as a function of magnetic field. The voltage measured across a pickup coil mounted on the rotating sample stage is used to calculate the angle between the sample stage and the magnetic field. When the capacitor bank is discharged through the magnet, a trigger signal is sent to the data acquisition cards. One of the cards measures the 5 volts peak-to-peak (Vpp ) signal supplied by the Stanford Research Systems SRS850-DSP lock-in amplifier, and another measures the voltage across the sample (amplified with a Texas Instruments INA103 amplifier). The 5 Vpp signal drops mainly across the 1 kΩ series resistor, providing an approximately 5 mA constant current (see section 3.1.) After the experiment, both of the measured voltages are input into custom-written digital lock-in software analysis. A digital lock-in multiplies the reference and sample voltages together and then integrates the resulting signal over a user-defined time constant (usually around 150 µs for these experiments.) This acts as a very low pass filter, rejecting all frequency components that are not at the reference frequency. The same multiplication and integration is performed with a  π 2  phase  shift applied to the reference. In this way both the in-phase and out-of-phase components of the measured signal are recovered. The frequencies used in these experiments varied between 40 and 60 kHz, depending on the quality of the contacts at low temperatures (bad contacts are often highly capacitive). The advantage of digitizing the full waveform and using a software lock-in over a standard physical lock-in is that the data can be analyzed multiple times using different time constants, phase shifts, line-filters, etc. The voltage drop across the 1 kΩ is also measured to give the exact current through the sample. The voltage drop across the sample, divided by this measured current, gives the resistance of the sample as a function of time during the pulse, as shown in Figure 5.1. The data discussed in this chapter was taken on a YBa2 Cu3 O6.59 sample, and the full analysis is performed in chapter 7. Other samples were measured as well, but the O6.59 experiment 63  5.1. Angle and Field Measurement  Sample Resistance Ohms  1.  In Phase Out of Phase  0.8  0.6  0.4  0.2  0. 0  30  60  90  120  150  180  Time milliseconds Figure 5.1: Resistance of a YBCO sample as a function of time during the magnetic field pulse. The data is asymmetric about the maximum because the field rises faster than it falls. is used as an example in this chapter.  5.1  Angle and Field Measurement  The Maxwell-Faraday equation, ∇ × E =  ∂B ∂t ,  describes how a time dependent magnetic  field induces an electric field. Applying Stoke’s theorem to this equation gives E · dl = − ∂A  ∂ΦA , ∂t  (5.1)  where ΦA is the magnetic flux through a surface A, and ∂A is the boundary of this surface. If a loop of wire is perpendicular to a changing magnetic field, the right hand side of Equation 5.1 quantifies the change in the magnetic flux through the loop as a function of time, and the left hand side is an instruction to integrate along the loop of wire to find the voltage between the two ends. This gives Vf ield = N A  ∂B , ∂t  (5.2)  where N is the number of turns in the loop and A is its area. This pickup voltage, as measured during a typical magnetic field pulse to 60 tesla, is plotted in Figure 5.2. 64  5.1. Angle and Field Measurement  Pickup Coil Voltage Volts  2.0  1.5  1.0  0.5  0.0  0.5 0  30  60  90  120  150  180  210  240  270  300  330  360  390  Time milliseconds Figure 5.2: Voltage in the pickup coil used to measured the field, as a function of time. From Equation 5.2, the pickup coil voltage integrated up to time t is proportional to the magnetic field (assuming B = 0 when no current is flowing through the pulsed field coil). Rather than measure the exact area of the loop to calibrate the field value, which would be very hard, a calibration factor is measured by using a Hall sensor to measure the actual field. Alternatively, SdH peaks in a well-characterized material can be used to calibrate the field. Once the calibration constant Cf ield is known, the magnetic field is calculated as B(t) = −  1  t  Cf ield  −∞  Vcoil (t )dt .  (5.3)  Integrating the pickup voltage shown in Figure 5.2 gives the magnetic field shown in Figure 5.3. Once the field is determined as a function of time, each resistance value in Figure 5.1 can be correlated with a field value in Figure 5.3, giving resistance as a function of magnetic field, shown in Figure 5.4. The full up sweep from 0 tesla to the maximum field (near 58 tesla in Figure 5.3) occurs in about 55 ms, whereas the down sweep is an exponential decay with a time constant of 54 ms. This means that higher resolution data can be collected on the down sweep, especially at lower field values where the field is changing more slowly in time. Heating may occur during the up sweep, or the sample may be torqued slightly out of position. It is expected that the sample will come to thermal and mechanical equilibrium by  65  5.1. Angle and Field Measurement 60  Magnetic Field Tesla  50  40  30  20  10  0 0  30  60  90  120  150  180  210  240  270  300  330  360  390  Time milliseconds Figure 5.3: Magnetic field as a function of time during a pulse, obtained by integrating the data in Figure 5.2. the time the field reaches the maximum value, providing additional incentive to concentrate on the data from the down sweep of the field  16 .  A second coil is attached to the rotating sample stage in order to measure the angle of the sample stage with respect to the magnetic field. When the sample stage is perpendicular to the field in the “neutral” position (aligning the cˆ-axis of the sample with the field), this pickup coil is rotated 90 degrees with respect to the main field pickup coil, giving maximum angle-sensitivity at high angles. The voltage picked up by this coil during the pulse will be Vrotator = Crotator  ∂B sin θ, ∂t  (5.4)  where θ is the angle of rotation from the neutral position.  ∂B ∂t  can be solved for in Equa-  tion 5.2 and Equation 5.4: setting these two equations equal to each other and then solving for θ to gives θ = arcsin  Cf ield Vrotator Crotator Vf ield  .  (5.5)  The final problem is to accurately determine the constants Crotator and Cf ield , which depend on the areas of the two coils and the number of turns of wire they contain. Passing 16  66  5.1. Angle and Field Measurement  Rising field  1.0 Falling field  Resistance  0.8  0.6  0.4  0.2  0.0 20  30  40  50  60  Magnetic Field Tesla Figure 5.4: Raw resistance versus magnetic field for both the rising and falling field. Note the difference between the two: the sample was torqued to a slightly different angle during the rise of the field. a small AC current through the large pulsed field coil generates a small AC magnetic field: maximizing the resultant voltage across the rotator coil determines the 90◦ position (90◦ from the “neutral” position). Then a moderate field of about 10 telsa is generated in the field coil, and the pickup voltages are measured for the field and rotator pickup coils. Since the angle is known (90◦ ), the slope of Vrotator plotted against Vf ield will be equal to  Crotator Cf ield .  This number was determined to be 0.0964. This is a much more accurate way of calibrating the pickup coils than trying to measure the pickup coil areas under a microscope. By fitting the slope m of Vrotator vs Vf ield as measured during a field pulse (shown in Figure 5.5), the angle of the sample stage is determined to be θ = arcsin  m . 0.0964  (5.6)  Note that this gives only the angle of the sample stage: to get the absolute angle of the sample itself, the melting transition of the flux lattice needs to be compared at positive and negative angles of rotation for the sample stage. This is done in the next subsection. The pickup voltages plotted against each other in Figure 5.5 show a double line at around 1 volt on the x-axis, indicating that the angle is changing during the rising field. The data shown in Figure 5.5 were chosen to illustrate this particular point; the angle on the up and down sweeps is usually constant within the noise of the pickup coil voltages. 67  5.1. Angle and Field Measurement 0.1  Rising Field  Rotator Coil Voltage Volts  0.08  Falling Field  0.06 0.04 0.02 0. 0.02 0.04 0.5  0.  0.5  1.  1.5  2.  Field Coil Voltage Volts Figure 5.5: Voltage across the rotator pickup coil during the pulse plotted against the voltage across the field pickup coil. The slope is related to the angle of the rotator via Equation 5.6  5.1.1  Calculating the Angle  Typically, only the data taken during the down sweep of the magnetic field is used for analysis. There is heating when the vortex lattice first melts, and the sample needs to come to equilibrium with the bath before reliable data can be taken. The up sweep, shown in red in Figure 5.5, shows a lot of noise on the rotator coil near the peak field. Because this noise only appears on the rotator coil, it presumably has to do with vibrations of the sample stage. A thin superconducting platelet with its long axis perpendicular to a magnetic field experiences a strong torque and wants to align its long axis with the field 17 . This is because of the huge demagnetizing factor of a thin superconducting plate in this orientation: the field lines have to “stretch” all the way around the thin sample, effectively screening a spherical volume whose radius is of order the size of the long axis of the sample. A thin superconducting plate with the long axis parallel to the field, on the other hand, screens a volume roughly equal to the actual sample volume, since the field lines do not have to “stretch” very far to go around the sample. The torque generated by this anisotropy changes Strictly speaking, if the the angle between the long axis and the field is 90◦ then the torque is zero. However, this is an unstable point and any small angle between the field and the long axis will result in a torque on the sample. 17  68  5.1. Angle and Field Measurement  Rotator Coil Voltage Volts  0.02  0.  0° 10° 20° 30° 30° 35° 40° 45° 47° 50° 55° 57°  0.02  0.04 0.6  0.4  0.2  0.  Field Coil Voltage Volts Figure 5.6: The voltage-voltage curves for each of the angles of rotation during the angular dependence of YBa2 Cu3 O6.59 experiment. The nominal angles—the angles which were “aimed” for—are given in the legend. violently as the flux lattice melts, allowing field to penetrate the sample and reducing the torque. This effect appears to be much smaller on the down sweep of the field, where the field has already penetrated the sample, and the time scale is longer. The rotator coil voltage plotted against the field coil voltage is shown in Figure 5.6 for all angles measured during the YBa2 Cu3 O6.59 angle dependence experiment. Clearly even for the down-sweep data there is some vibration in the rotator pickup coil 18 . Fitting a straight line with zero intercept, since a finite intercept is unrealistic if the voltage is proportional to the field derivative for both coils, and applying Equation 5.6 gives the rotator angles displayed in Table 5.1. The angles of the rotator are determined accurately using this method, but misalignment of the sample with respect to the sample stage needs to be accounted for to obtain accurate angles for the sample. The field where the melting transition occurs should increase as cos1 θ with angle [11]. This is because the superconductivity in the a ˆ-ˆb-plane is so much stronger than it is along the cˆ-axis that it is only the projection of the field along the cˆ-axis that determines the melting transition. Even if this prediction is not exactly correct, we should at least expect a smooth evolution in the resistivity as a function of angle. 18  If the noise was in the field pickup coil then it would appear as noise in the x-axis direction rather than the y-axis as is the case in Figure 5.6.  69  5.1. Angle and Field Measurement Nominal  Calculated  0 10 20 -30 30 35 40 45 47 50 55 57  -0.67 9.82 19.03 -29.07 28.89 33.83 38.74 44.35 46.26 50.40 53.39 56.28  Table 5.1: The rotator angle: nominally, and as calculated using Equation 5.6 and fits to the data in Figure 5.6  Data was taken at both ±30◦ , and these two data sets should have almost identical melting transitions once any offset angle has been accounted for. Figure 5.7 shows how this was done, and an offset angle of θ0 = 1.06◦ was obtained. Figure 5.8 shows all of the raw data, the data scaled using the rotator angles, and finally the data scaled with the rotator angles plus the offset. The evolution is smooth once the offset is accounted for. The final values for the angle of the sample, including the offset, are displayed in Table 5.2.  70  5.1. Angle and Field Measurement  1.  0.8  30°  Resistance  30° 0.6  0.4  0.2  0.  0.02  0.03  1 BcosΘ Tesla  0.04 1  1.  0.8  30°  Resistance  30° 0.6  0.4  0.2  0.  0.02  0.03  1 BcosΘ Tesla  0.04 1  Figure 5.7: Top: The resistance curves at the nominal angles of ±30◦ , plotted against 1 B cos θ . The angles used in the scaling are those from Table 5.1, and the temperature is 4.2 K. Bottom: The same data as above, but now plotted using an offset angle of 1.06◦ , which accounts for the fact that the sample is not perfectly aligned when mounted on the rotator.  71  5.1. Angle and Field Measurement  1.  Resistance  0.8  0.6  0° 10° 20° 30° 30° 35° 40° 45° 47° 50° 55° 57°  0.4  0.2  0. 20  25  30  35  40  45  50  55  60  Magnetic Field Tesla  0° 10° 20° 30° 30° 35° 40° 45° 47° 50° 55° 57°  1.  Resistance  0.8  0.6  0.4  0.2  0.  0.02  0.03  1 BcosΘ Tesla  0.04 1  0° 10° 20° 30° 30° 35° 40° 45° 47° 50° 55° 57°  1.  Resistance  0.8  0.6  0.4  0.2  0.  0.02  0.03  1 BcosΘ Tesla  0.04 1  Figure 5.8: Top: All of the raw resistance data as a function of angle and field. Middle: 1 The same data shown in the top panel, but with the x-axis scaled by B → B cos θ , with the angles taken from Table 5.1. Bottom: The same data as the upper two panels but with the added offset of 1.06◦ . Note that the transition from the vortex solid to the vortex liquid now occurs at the same field value, as predicted in Blatter et al. [11]. 72  5.2. Background Removal Nominal  Sample angle  0 10 20 -30 30 35 40 45 47 50 55 57  0.37 10.86 20.07 -28.03 29.93 34.87 39.78 45.39 47.30 51.44 54.43 57.32  Table 5.2: The sample angle: nominally, and as shown in Table 5.1 plus an offset of 1.06 degrees  5.2  Background Removal  In section 2.6 it was demonstrated that the experimental quantity of interest is  σ σ0 ,  and  Equation 2.81 is an instruction on how to extract it from the measured resistivity. It is often much more useful to simply work with ρ rather than  1 ρ,  since  1 ρ  diverges when the  sample becomes superconducting. As long as the oscillatory part of the resistance is small compared to the background [14, 31], the resistance is ρ(B) = ρ0 (B)(1 −  σ(B) ). σ0  (5.7)  This equation implies that we should fit the background, divide the total resistivity signal by this background to normalize it, and then subtract off the background from the normalized data to obtain  σ σ0 .  In the superconducting state below the flux lattice melting transition the sample has zero resistance (see Figure 5.4). Near 25 tesla in Figure 5.4 the vortex lattice melts and the sample starts to become resistive. At higher fields the vortices are fully mobile, and the shape of the background is dominated by the intrinsic magnetoresistance of the material. The boundary between where the vortex lattice melts and where remnant superconductivity no longer contributes to the conductivity is a smooth crossover. Rather than try to fit the strange shape of the background to some functional form, which would be purely phenomenological, the easiest way to start is to remove the background by hand.  73  5.2. Background Removal  5.2.1  User-Controlled Background Removal  The best first attempt at subtracting a background can be done by hand, using intuition about where the background is with respect to the oscillating signal. I wrote Mathematica code, found in Appendix B, that allows the background to be adjusted by hand, and the output to be visualized in real time. The Manipulate command in Mathematica can be used to interact with data, and allow the presentation and manipulation of the data to both occur in real time. An interface allowing for creating and manipulating control points for a B-spline curve is overlayed on the resistance versus field data (such as is shown in Figure 5.4.) This B-spline curve is a parametric curve that lies inside a bounding polygon (a convex hull) defined by the user-specified control points. By moving the control points around with the mouse and adding/removing control points, a background curve can be drawn through the raw data. The residual between the background and the data (actually the data is divided by the background, and then 1 is subtracted, in accordance with Equation 5.7) is plotted in real time in an inset in the interface, allowing for detailed changes in the background to be made by hand until a suitable oscillatory part can be extracted. A snapshot of this interface, with a background drawn and the oscillatory part extracted, is show in Figure 5.9.  5.2.2  Fitting the Background  Once an initial fit of the background-subtracted data has been performed, this information about the shape of the oscillations can be used along with a polynomial in magnetic field to fit the background plus the oscillations. Because the background is both temperature and angle dependent, and because the dependence of the background on these parameters is not known, it is difficult to fit all of the data and the background at once. A polynomial surface in field, angle, and temperature can be used, but this results in a huge number of free parameters that frustrate even the genetic algorithm developed in chapter 6. The best procedure in this case is an iterative one, where the background is removed by hand, then the oscillatory data is fit, and then this fit is used along with a polynomial to fit the raw data itself again, and the procedure is repeated until it converges (typically 2 iterations is plenty). The derivative of the background is checked at each step to make sure that oscillatory components are not being introduced into the data. The data shown in the inset of Figure 5.9 can be fit with Equation 2.80, and for all angles it seems that at least two pieces of Fermi surface are required to get a good fit. At this point the more powerful technique of fitting a complicated single Fermi surface sheet could be employed using Equation 2.69, but this is numerically much more difficult and we are only interested in extracting oscillatory components for further analysis at this point. The actual fit parameters parameters obtained during the background subtraction are not of interest, and so it suffices to use two simply warped Fermi surfaces to extract the oscillatory  74  5.2. Background Removal  d  1.0  0.8  Resistance  0.6 0.06  0.04  0.4 0.02  0.00  0.02  0.2 0.04  0.06 25  30  35  40  45  50  55  0.0 25  30  35  40  45  50  55  Magnetic Field Tesla  Figure 5.9: The Mathematica graphical interface for removing a background from the raw resistivity data. The grey circles are the control points for the red B-spline curve, which can be moved with the mouse. The inset shows the data with the background removed, which is updated in real time. The slider bar labelled ‘d’ sets the order of the B-spline: the default is a cubic spline. data. One of the benefits of working at 4.2 kelvin rather than at the base temperature of 1.5 kelvin is encountered here: because the cyclotron mass m enters the temperature dampening factor RT for each harmonic p as the product  pm T cos θ ,  harmonics higher than the  first are highly suppressed at 4.2 K, and therefore the p = 2 term only needs to be included at the angles 0◦ and 10◦ . For angles higher than 10◦ , the p = 1 term alone is sufficient. This greatly simplifies the fitting procedure. The upper panel of Figure 5.10 shows the oscillatory component after subtracting the background using the manual procedure shown in Figure 5.9. The fit to this data is then combined with an 8th order polynomial in field, and the raw data is fit. This is shown in the middle panel of Figure 5.10. Note that the green background shown in the lower panel of this figure is monotonic and non-oscillatory: it is important to check this, as oscillatory components can be induced when none truly exist when fitting with a high order polynomial. The oscillatory component obtained by removing this fitted background is then fit again to Equation 2.80, and the procedure iterates again, resulting in the data in the bottom panel of  75  5.2. Background Removal Figure 5.10. This procedure is especially important for the low field data where estimating the background by hand is difficult.  76  5.2. Background Removal 0.06  Oscillatry Resistive Ratio  0.04  0.02  0.00  0.02  0.04  0.06  35  40  45  50  55  Magnetic Field Tesla 1.0  0.8  Resistance  0.6  0.4  0.2  0.0  35  40  45  50  55  Magnetic Field Tesla 0.06  Oscillatry Resistive Ratio  0.04  0.02  0.00  0.02  0.04  0.06  35  40  45  50  55  Magnetic Field Tesla  Figure 5.10: Top panel : The data at 40◦ after the background has been removed by hand. Middle panel : The fit to the data in the upper panel is combined with an 8th order polynomial to fit both the background and the data at the same time. The total fit is shown in red, and the background by itself is shown in green. Bottom panel : The oscillatory component after a second iteration. Note the improvement at lower field values compared to the first iteration in the top panel. 77  5.3. Building a Model  0° 10° 20° 30° 30° 35° 40° 45° 47° 50° 55° 57°  0.1  Resistance  0.05  0.  0.05  0.1 0.02  0.03  1 BcosΘ Tesla  0.04 1  Figure 5.11: All of the data from 0◦ to 57◦ with the background subtracted.  5.3  Building a Model  While many data sets were taken on many different dopings over the course of my doctoral studies, the data analysis technique was developed specifically to analyze the angle dependence of YBa2 Cu3 O6.59 . This section will make use of the data taken on this particular sample. Similar analysis has been done for different doping levels, although only O6.59 has the angular dependence, and the results for the other doping levels are shown in chapter 8. Before delving into the complexities of fitting all of the data in Figure 5.11 to a model, much can be understood just by looking at the data and applying what was learned in chapter 2. The field dependence is the simplest place to start, and it can tell us about the overall size of the Fermi surface and its degree of warping in the kz direction. The temperature dependence is also particularly simple, since the temperature only enters once into the oscillation formula—in the expression for RT . Finally, the more complicated angular dependence can be incorporated once a feeling is developed, based on qualitative grounds, for what the Fermi surface should look like.  5.3.1  Field Dependence  The absolute value of the oscillatory part of the resistance at zero angle is shown in Figure 5.12. Plotting the absolute value better highlights the features of interest in the envelope of the oscillations. The most basic piece of information that can be extracted is the average 78  5.3. Building a Model  0.1  Oscillatory Resistive Ratio  0°  0.05  0. 28  33  38  43  48  53  58  Magnetic Field Tesla Figure 5.12: The absolute value of the oscillatory part of the resistance. The features of central importance are the beats near 32 and 47 tesla, and the fact that the beat does not drive the amplitude fully to zero. Fermi surface size: take the distance in inverse field between the peaks in Figure 5.12 at 55 tesla and 31.5 tesla, divide by the 7 periods of oscillation between them, and invert this to get a frequency of F ≈ 510 tesla. Using Equation 2.26 gives a Fermi surface area of Sk = 4.9 nm−2 . kF can also be calculated, and is kF ≈ 1.25 nm−1 . The second piece of useful information is extracted from the minimum in the amplitude, or “beat”, in the data. According to Equation 2.80, the oscillations are modulated in field by the factor J0  2πp∆F B  , where ∆F ≡  2t⊥ m e  . Approximately one half of a period  of modulation is seen between 31 and 47 tesla, which leads to a warping of ∆F ≈ 32 tesla. Using estimates of the effective mass from previous studies on YBCO [22], or looking ahead to subsection 5.3.2, t⊥ is calculated to be approximately 1.1 meV. For the sake of completeness, Equation 2.44 is used to calculate the chemical potential from F , and the result is ζ ≈ 35 meV. This demonstrates that t⊥  ζ, and so the quasi-2D results of  chapter 2 should be valid.  5.3.2  Preliminary Temperature Dependence  Temperature appears in only one place in the Lifshitz-Kosevich formula: in the RT factor, which reduces the amplitude of the oscillations as the width kB T of the Fermi-Dirac distribution becomes broad with compared to the Landau level spacing ωc . It is important to 79  5.3. Building a Model 0.15  Oscillatory Resistive Ratio  0.12 1.5 K 2.5 K 3.1 K 4.2 K  0.09 0.06 0.03 0. 0.03 0.06 0.09 0.12 30  35  40  45  50  55  Magnetic Field Tesla Figure 5.13: The temperature dependence at an angle of −30◦ . The angular dependence of m m for a quasi-2D Fermi surface is just cos θ , and so we can measure the effective mass at any angle and get a consistent result. remember that this damping is not due to electron-phonon scattering, which does not have a pronounced effect on quantum oscillations [24]. The standard approach to extracting m from oscillatory data, such as the data shown in Figure 5.13, is to Fourier transform the data and fit the amplitude of a peak as a function of temperature to the expression for RT . Looking at the expression for RT , however, we see that the field value B is also a parameter in RT , while a Fourier transform amplitude contains information from a whole range of field values. It is not clear which field value should be used for the fit, since the amplitude of the oscillations have a strong field dependence. Mercure [64] explains this problem in some detail, and resolves it for the case of Sr3 Ru2 O7 . The technique developed there, however, relies on having a very large number of oscillations with relatively constant amplitude over some field window, which we do not have in the case of YBCO. A best first guess at the mass can easily be obtained by fitting the amplitude of an oscillation peak in Figure 5.13 as a function of temperature to the form of RT , Equation 2.67. The result of this procedure is shown in Figure 5.14 for four of the peaks seen in Figure 5.13. For the peaks at 40, 43, 46, and 50 tesla, we get masses of 1.66, 1.71, 1.71, and 1.80 me , respectively. Rather than indicating an actual field dependence of the mass, which is  80  5.3. Building a Model 0.15  40 tesla 43 tesla 46 tesla 50 tesla  Peak Amplitude  0.12  0.09  0.06  0.03  0. 0.  0.5  1.  1.5  2.  2.5  3.  3.5  4.  4.5  5.  Temperature kelvin Figure 5.14: The amplitude of four of the peaks seen in Figure 5.13 as a function of temperature. These amplitudes have been fit to the expression for RT to obtain an estimate for m . possible but not common [92], the slight change in mass between peaks probably indicates that there are two pieces of Fermi surface with slightly different masses contributing to the whole of the amplitude. This mass roughly agrees with the mass found near this doping in earlier works [22, 88]. Thus I will take 1.7 me as the initial estimate for the cyclotron mass, keeping in mind that if there is more than one piece of Fermi surface then this estimate represents some sort of average of the two masses. The mass will be fit more accurately in section 7.3.  5.3.3  Preliminary Angle Dependence  The expression for the angle dependence in Equation 2.70 shows that when J0 dk F tan θ = 0, the modulation or beating disappears. Using k F ≈ 1.25 nm−1 from above and d = 1.175 nm from Liang et al. [59] for YBa2 Cu3 O6.59 , the first “Yamaji angle” [10, 110] where the beating should disappear is estimated to be 58.6◦ . Note that this only depends on the average Fermi surface size and the cˆ-axis lattice parameter, and not on the size of the warping. Because the first Yamaji angle should not appear before around 60◦ , it is surprising that in Figure 5.15 the beat structure almost fully disappears around 35◦ before returning at higher angles. There are two possible simple explanations for this: either there are two 81  5.3. Building a Model 0.08  Oscillatory Resistive Ratio  35° 0.06  0.04  0.02  0.  33  38  43  48  53  58  Magnetic Field Tesla  Oscillatory Resistive Ratio  0.015  47°  0.01  0.005  0. 38  43  48  53  58  Magnetic Field Tesla  Figure 5.15: Top panel : The beat structure, easily visible in Figure 5.12, has almost completely disappeared near 35◦ . Bottom panel : The beat structure has begun to return by 47◦ . pieces of Fermi surface, one warped and one unwarped with the warped one having a spin zero around 35◦ , or there is a single Fermi surface with a very complicated warping structure such that the simple Yamaji-angle picture does not apply. Further information is needed to disentangle these two possibilities. Up to about 35◦ the phase of the oscillations remains fairly constant when plotted versus 1 B cos θ ,  as shown in Figure 5.16. There is a bit of a shift near the Bessel function zero (which  induces a phase flip of 180◦ ), but the constant phase recovers on either side. Beyond θ = 35◦ , however, the phase shifts by about 90◦ (Figure 5.17), and then again by another 90◦ near θ = 50◦ (Figure 5.18.) The simplest explanation for this is the presence of the Rs factor on two separate sheets of Fermi surface: with slightly different g ms values on each surface, the surfaces go through their spin zeros at different angles (near 35◦ and 50◦ according to the data in Figure 5.17 and Figure 5.18.) Since these phase flips occur at different angles, 82  5.3. Building a Model  Oscillatory Resistive Ratio  0.1  0° 10° 20° 30° 30° 35°  0.05  0.  0.05  0.1 0.02  0.03  1 BcosΘ Tesla  1  Figure 5.16: The first six data sets, up to 35◦ . Note that the phase remains fairly constant, except perhaps near the node around 0.03 tesla−1 where the Bessel function changes sign. The constant phase recovers on either side of the Bessel function node. the amplitude never truly goes to zero because one surface always has finite amplitude. Additionally, the 180◦ phase flip will not be seen before both surfaces have gone through their first spin zero. This is exactly what is seen in Figure 5.19.  5.3.4  Preliminary Model  Without any actual calculations or fits to the data a lot of information has been gained about the Fermi surface of YBCO. The field dependence at zero angle describes a Fermi surface that is approximately 510 tesla in area  19 ,  and has a kz warping that is of order 32  tesla. The field dependence also says that there is either more than one warping parameter or more than one piece of Fermi surface because the beat in the data does not drive the amplitude to zero. The product of the cyclotron frequency and the scattering time ωc τ must be approximately 1 or greater, otherwise oscillations would not be visible due to the Dingle factor. From the temperature dependence the cyclotron effective mass m is estimated to be approximately 1.7 times the free electron mass me . If there are two pieces of Fermi surface then 19  The standard way to report Fermi surface areas in quantum oscillation experiments is in units of tesla, since this is the natural Fourier transform variable when analyzing data. Additionally, a frequency can be F easily converted to a Landau level number via n = B .  83  5.3. Building a Model  30° 35° 40° 45° 47°  Oscillatory Resistive Ratio  0.06  0.02  0.02  0.06 0.02  0.03  1 BcosΘ Tesla  0.04 1  Figure 5.17: The five data sets from 30◦ to 47◦ . The phase between 30◦ and 35◦ is constant, and is constant between 45◦ and 47◦ , with an offset from the previous two angles. The data at 40◦ seems to bridge the gap between them. their masses are similar, otherwise one frequency component would die out much quicker than the other as a function of temperature—an effect that is not observed. The angle dependence requires more careful analysis in order to extract accurate information, but the phase shifts seen in Figure 5.19 indicate two pieces of Fermi surface that have their first spin zeros at around 35◦ and 50◦ . The fact that beats in the data die out and then return at higher angle indicates either high order warping components on a single surface, or that there is one warped and one unwarped piece of Fermi surface, with the warped surface having a spin zero near 35◦ .  84  5.3. Building a Model  Oscillatory Resistive Ratio  0.03  45° 47° 50° 55°  0.02  0.01  0  0.01  0.02  0.03  0.025  0.03  1 BcosΘ Tesla  0.035  0.04  1  Figure 5.18: The four data sets from 45◦ to 55◦ . The phase between 45◦ and 47◦ is constant, as is the phase between 50◦ and 55◦ , with an offset from the previous two angles.  85  Oscillatory Resistive Ratio  5.3. Building a Model  35° 45° 50° 0.02  0.02  0.025  0.03  1 BcosΘ Tesla  0.035 1  Figure 5.19: Three representative data sets showing the phase offset in each of the 3 regimes shown in Figure 5.16, Figure 5.17, and Figure 5.18. The phase shifts a bit when the spin zero is passed through for one Fermi surface just above 35◦ , and then the full 180◦ phase shift is seen near 50◦ once both surfaces have gone through the spin zero. Note that the data at 45◦ and 50◦ have had their amplitudes scaled in order to be visible when plotted with the data at 35◦ .  86  5.4. Limitations of the Fourier Transform Technique in YBa2 Cu3 O6.x  5.4  Limitations of the Fourier Transform Technique in YBa2 Cu3 O6.x  Before describing the data analysis technique I developed for YBCO, I will first motivate why I did not use standard Fourier transform techniques. In essence, it is because the small size of the Fermi surface of YBa2 Cu3 O6.59 gives such few periods of oscillation that the Fourier peaks are poorly resolved. It needs to be emphasized right from the start that the Fourier transform of an oscillatory signal contains the exact same information that the untransformed signal does, and that I am not implying that the two representations of the same data are somehow not equivalent. The problem arises when only the absolute value of the Fourier transform of the signal is analyzed, as is most often the case in quantum oscillation studies. When the frequency components are closer together than the frequency resolution (set by the size of the field window), the relative amplitudes of the peaks in the absolute value plots are not representative of the relative amplitudes of the frequency components in the input signal. This statement is made much clearer by looking at Figure 5.21. In order to illustrate the difficulties encountered when using Fourier transform amplitudes to analyze YBCO quantum oscillation data, a comparison can be made with the more favourable situation in the ruthenates, for which the data is shown in Figure 5.20 [10].The smallest piece of Fermi surface in Sr2 RuO4 , the alpha pocket, has a frequency of 2.6 kT. Because the critical temperature (Tc ) is so low—1.43 Kelvin—the oscillations are visible at very low fields, and there are about 800 visible oscillations between 3 and 33 tesla; the larger Fermi pockets have several thousand oscillations in this same range. In this case, Bergemann et al. [10] Fourier transformed data over a field window that contained a large number of oscillations at a relatively constant amplitude. This results in well defined and well separated peaks for each piece of Fermi surface. These peaks can be seen in the lower inset of Figure 5.20. There is no problem with analyzing the amplitudes of the Fourier transform peaks in this case because they are separated enough in frequency that there is no interference between them. To quantify the problem in YBCO, note that the frequency resolution of a power spectrum is limited by the number of oscillation periods measured. With a field range of 30 to 60 tesla, one is limited to a resolution of  1 1/30−1/60  = 60 tesla. This is larger than the size of  the kz warping in YBa2 Cu3 O6.59 , as found in section 5.3. Increasing the upper field range of the measurement to 80 tesla only increases the resolution to 48 tesla because the oscillations are periodic in 1/B, so only 2.2 periods of oscillation are gained with the additional 20 tesla. Thus the frequency components will overlap each other in an amplitude plot, and the relative phases between oscillatory signals will cause interference. Figure 5.21 illustrates this problem with simulated data for frequencies similar to those seen in YBa2 Cu3 O6.59 . The three simulated Fermi surfaces have areas of 450, 510, and 530 tesla, and the simulated data is taken from 20 to 70 tesla. The oscillations are of 87  5.4. Limitations of the Fourier Transform Technique in YBa2 Cu3 O6.x  Figure 5.20: The relatively large Fermi surface of Sr2 RuO4 (shown in the upper inset) yields several thousand periods of oscillation between about 3 and 33 tesla. This large number of oscillations results in sharp peaks when the data is Fourier transformed (the lower inset). Figure reproduced with permission from [10]. the form sin  2πF B  , and the damping factors are suppressed. Fourier transforming each  peak separately gives the top panel of Figure 5.21. If the three peaks are instead added together before Fourier transforming, the amplitudes looks very different. This is because while Fourier transforming is a linear operation, taking the absolute value of it is not. The phase of the 510 tesla peak is allowed to shift with respect to the others, and the result is plotted in the bottom panel of Figure 5.21. Even though the amplitudes of each individual frequency component remain constant as a function of this phase, the absolute value of the Fourier transform of the sum of the frequency components depends strongly on the phase. It should be clear from this exercise that when pockets of Fermi surface have areas that are closer together in frequency than the frequency resolution provided by the field window, looking at only the magnitude of the Fourier transform is misleading. Fitting Lorentzians to the Fourier transform magnitude should not be considered valid at all when the peaks overlap because, as can be seen in Figure 5.21, the positions of the peaks do not accurately 88  5.4. Limitations of the Fourier Transform Technique in YBa2 Cu3 O6.x reflect the actual frequency components. Fourier transforms of actual YBCO data are shown for comparison in Figure 5.22  20 .  What makes quantum oscillation techniques so powerful is their ability to provide information about the scattering rate, effective mass, Fermi surface warping, and quasiparticle g factors, and all of this information is tied up in the shape of the oscillation envelope as a function of angle, field, and temperature. When the peaks are well separated these parameters can be accurately extracted from the angle, field, and temperature dependences of the Fourier transform amplitudes [10, 64]. When the Fourier transform peaks are broad and overlap each other, as in some of the preliminary analysis of quantum oscillation data in YBCO Audouard et al. [7], then it becomes difficult to unambiguously extract the interesting parameters. While in principle one could fit both the magnitude and the phase of the Fourier transformed data, this is a needless complication that provides no advantage at all over fitting the actual data.  20  A good review of Fourier transform windows, including the relative effects of different window types, can be found in Harris [33]. Mercure [64] describes, in great detail, the specific problems related to Fourier transforming quantum oscillation data.  89  5.4. Limitations of the Fourier Transform Technique in YBa2 Cu3 O6.x  450 T 510 T 530 T  Amplitude  F F F  200 250 300 350 400 450 500 550 600 650 700 750 800  Frequency tesla  Amplitude  Φ Φ Φ Φ Φ  0 Π8 Π4 3Π 8 Π2  200 250 300 350 400 450 500 550 600 650 700 750 800  Frequency tesla Figure 5.21: Top panel : Individual Fourier transforms for Fermi surface pockets with areas of 450, 510, and 530 tesla. The data is taken between 20 and 70 tesla. Bottom panel : The magnitude of the Fourier transform of the sum of the three frequency components that were shown in the upper panel. The relative heights of the peaks depends strongly on the phase of the oscillations, which is allowed to vary between 0 and π2 for the 510 tesla signal. Also note that for a phase of π4 , the upper peak in the green curve is at 550 tesla: a frequency that is not present in the actual data.  90  5.4. Limitations of the Fourier Transform Technique in YBa2 Cu3 O6.x  1.1  O6.51 O6.59 O6.67  Amplitude Arbitrary Units  1. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0. 0  200  400  600  800  1000  1200  1400  1600  1800  2000  Frequency Tesla Figure 5.22: Fourier transforms of the oscillatory magnetoresistance for O6.51 , O6.59 , and O6.67 . The field ranges for the transforms are 34.5→59.8 T, 27.5→69.2 T, and 36.5→60.0 T, respectively. The data has been apodized with a Kaiser window with α = 2.5 [33], and the amplitudes have all been normalized to 1.  91  Chapter 6  Genetic Algorithm Given the difficulties noted in section 5.4, a method for fitting the data directly is desired. Such fits can provide highly constrained parameters if the signal to noise ratio is high. In the fits presented in this thesis, even the simplest model has many parameters, and so must be fit to the entire data set in order to provide tight constraints. A very useful tool for this is a genetic optimization algorithm inspired by evolution as it occurs in nature, which takes an heuristic approach to finding the optimal fit rather than a deterministic one [8]. The problem is finding the best set of parameters for a model given a certain set of data points, and the optimal solution is the set of parameters that gives the smallest least squares value. A gradient based search algorithm such as Newton’s method may become stuck in local minima if the parameter “landscape” is very rough, making the fit obtained very sensitive to the initial conditions chosen. The genetic algorithm does not take a user-specified starting point, but rather a range for the parameters in which the solution is thought to lie, and then the algorithm randomly generates a large number of starting points within the bounds specified. Local minima can be “escaped” by virtue of the design of the algorithm, and the propensity of the algorithm to search outside of the minimum to which it is converging is a controllable parameter. The algorithm used in this thesis is based on the “differential evolution” algorithm of Storn and Price [98], and proceeds as follows: 1. A set of a large number (N ) of initial parameter vectors is created, with the ith one referred to as xparent . There are about 10 times as many parameter vectors in this i set as there are parameters. In my case the parameters are the average frequency, the warping size, the quasiparticle lifetime, an amplitude factor, a phase factor, and g ms ; there are two each of these parameters—one for each piece of Fermi surface. Bounds are specified for each parameter in xparent , and the initial values are randomly generated to lie between those bounds: for example, the average size of the first cylinder could be restricted to lie between 400 and 700 tesla. 2. A second new set of parameter vectors (the “mutated” vectors) are created by taking three parent vectors at random and creating a new vector via xmut = xparent + i k s(xparent −xparent ), where s is a user defined scaling factor, and k, l, and m are random m l integers between 1 and N . Note that each xi is a vector of parameters, so the algebra is performed on each parameter within the vector. If any parameter ends up outside its original specified bounds, then its value can be set to the boundary value. This is 92  6.1. C++ Implementation repeated until there are N new “mutated” parameter vectors. 3. Crossing is then performed by taking the ith vector from the parent parameter set and the ith vector from the mutated parameter set. Starting with the first parameter (say the average frequency of the first piece of Fermi surface), a new set of parameter vectors, xnew is created by selecting the first parameter from the parent vector with i probability p, or from the mutated vector with probability 1 − p. This is repeated for each parameter in the vector until xnew is completed. This is then repeated for all N i parameter sets—for i = 1 to N . 4. Competition and selection then take place by computing the sum of squared residuals between the model and the data using the parameters from the vectors xnew and i xparent , and then keeping the parameter vector that has the lowest residual. This is i repeated for i = 1 to N . 5. If the lowest residual value is less than the tolerance specified, the algorithm terminates. Otherwise, the algorithm returns to step 2 with the new set of parent vectors being the vectors that were selected in step 4. The crossing parameter p and scaling parameter s are chosen for fast and accurate convergence, and their values depend on the exact nature of the optimzation problem. Figure 7.1 shows the minimum residual value at each generation step for different s and p values when analyzing YBCO data. Various adjustments can be made to the algorithm, including (but not limited to): always keeping a few of the best parameter vectors from each generation (called “elitism”), penalizing parameters that are sitting on the boundary (effectively increasing their least squares value in some artificial way), or even changing the order of some of the steps. Banzhaf [8] is a good resource for learning about genetic algorithms, and the original paper on differential evolution by Storn and Price [98] is quite educational as well.  6.1  C++ Implementation  One of the advantages of genetic algorithms is the ease of their implementation in any programming language. The algorithm is fairly straightforward, with no knowledge of the Jacobian of the fitting function needed, unlike other gradient based fitting methods. This is especially helpful when the model is an integral over kz , and the Jacobian has many terms to be integrated over. There is an implementation of the Differential Evolution algorithm in Mathematica, which can be used with the FindFit function. A quick test of this algorithm, however, runs into great difficulties: even with a population of only 10 it takes 93 seconds for the algorithm to complete two generations fitting two simply warped Fermi surfaces to data for the first three angles (211 data points). This performance is not acceptable, as 93  6.1. C++ Implementation with this small of a population there is very little hope of ever converging on a good fit. A population of the order 10 times the number of fit parameters (12) is needed, and two generations are not nearly enough to converge on a good fit. The situation becomes even more hopeless when fitting to a complicated Fermi surface shape which requires integration. The problem is that the Bessel functions in Equation 2.70 are very costly to evaluate. Exponentials, cosines, and sines also take up a lot of CPU cycles. This can be alleviated somewhat by creating lookup tables for the cosines and tangents of the angles the data was taken at, but the Bessel functions still need to be evaluated at every data point, for each parameter vector in the population, for each generation. Despite the problems with the Differential Evolution implementation in Mathematica, promising initial results were obtained by running the fit over several days, and this motivated writing a more optimized algorithm in C++.  6.1.1  Optimization Goals  The limiting step in a direct search algorithm, such as a genetic algorithm, is almost always the evaluation of the “fitness function” [98]. In my case, the fitness function is the sum of the square of the residual difference between the data points and the model. The data points are known and stored in memory, and so the computationally expensive step is evaluating the model at each data point. On modern computers, with several computational pipelines per core, several cores per processor, and lots of fast on-chip memory, there are several techniques to explore for speeding up the computation: • Parallelization – Evaluating the function at each data point for each parameter vector is an independent operation that does not require any information about other parameter vectors or data points. Thus, the total population can be divided in to a number of groups equalling the number of CPU cores available, and the residuals can be calculated for each vector separately. – In a similar way, individual cores can process several calculations in parallel. Modern compilers often take care of this automatically, but care must be taken to achieve optimal efficiency, making sure that loops do not contain dependencies that prevent them from being parallelized. • Lookup tables – Computing the trigonometric functions is relatively costly, as they usually involve checking the size of the argument and then calculating the relevant Taylor expansion to sufficient order. In Equation 2.70 we need the cosine and the tangent of the angle the sample is at with respect to the magnetic field; since these  94  6.1. C++ Implementation angles were fixed by the experiment, the cosines and tangents of these angles can be calculated once and for all and then stored in a table. Other calculations can be done ahead of time as well, such as calculating the value of 2π. – Integrating the oscillations along the kz direction as in Equation 2.49 can be expensive, but I will show later that, when integrating, the function will always be evaluated at specific kz points—the zeros of a Legendre polynomial—which can be evaluated ahead of time and stored in memory, since cos (kz ) will not change when the fit parameters do. • Compiler Some of the largest improvements in speed can be realized by using a compiler optimized for speed. The Intel compiler is optimized for Intel processors, and is much faster than the default Visual Basic compiler and the gcc compiler. All of the C++ code used in this thesis was written in the Microsoft Visual Studio Professional development suite, which is free for students from Microsoft [45]. The code for reading the input data file is contained in section A.1. The data is imported into a 2D array of double precision floating point numbers (double), and in the order {field, amplitude, angle number, temperature}. Only the angle number is specified (i.e., 0,1,2 etc...) because we have the values of the cosines and the tangents of the angles are already computed and stored in an array. The genetic algorithm was run on an Intel Core 2 Quad Q8300 (Yorkfield), overclocked to 2.80 GHz. All benchmark numbers reported in the thesis are for execution on all four cores of this processor. The central data object around which the program is built is the Parameters struct, shown in section A.2. It contains two sub structures: fitParameters, and arrayBounds. A fitParameters struct is created for each vector in the population, with the trial fit parameters being stored in fitParameters, and information relevant to multi-threading is stored in arrayBounds (explained in greater detail in section 6.1.3.)  6.1.2  Initialization  Upon creation of the GeneticAlgorithm21 object, the user specifies the population size N , the scale factor s, and the crossing probability p. The program then initializes the random number generators, and then generates the initial values for the parameter population via the initializeParameters method (see subsection A.3.1.) The Mersenne Twister random number generator from the Boost library is used to initialize the parameters between the specified bounds [107]. As this initialization is only done once per run of the program, the speed of this random number generator is not a particular issue here. The program also calculates the sum of the square of the residuals, or chiSq, value at this point for each member of the population (this step will be detailed later). 21  I will always omit the function arguments for clarity, unless needed.  95  6.1. C++ Implementation The program finds the best initial parameter vector from the population (by looking for the smallest chiSq value, see subsection A.3.2), and prints out the corresponding best parameter values after each generation.  6.1.3  Algorithm Iteration  The function calculateNewGenerations is the heart of the algorithm, and can be found in subsection A.3.3. It loops over the number of generations specified by the user, running through the algorithm and creating the threads for evaluating the fitness function (i.e., determining chiSq). This function first has to create the N “mutant” parameter vectors via xmut = xk + s (xl − xm ) ,  (6.1)  where k, l and m are random indexes, and s is the scale factor. To this end, three vectors of random integers from 1 to N are created first using the viRngUniform function from the Intel Math Kernel Library [46], which is optimized to create vectors of pseudo-random numbers very quickly. The function then implements Equation 6.1 for each variable in each parameter vector. Parameter bounds (specified when the initial parameter vectors are created) are not strictly enforced during this step, as little seems to be gained from fixing them. The only exception is that the phase is kept between 0 and 2π, since other values are redundant. Next, the function creates the xnew population vectors by taking a parameter from the mutant with specified probability p, or from the parent with probability 1 − p. This whole operation, from creating the vectors of random numbers, to creating the mutant population vectors and then the new child vectors, takes under 1 microsecond per parameter vector  22 ,  which is much faster than the time-dominant step of calculating the  residuals (which we will see is on the order of tens to hundreds of microseconds per parameter vector). Thus, while this step could be optimized further by spiting it into multiple threads, this extra work is unnecessary as the relative gains would be negligible. The steps executed during each generation are detailed below. After the residuals have been calculated for each parameter vector, the algorithm proceeds as described at the start of this chapter until the user-specified number of generations have been executed. Execution of the algorithm using YBCO data is described in chapter 7. Multi-threading After a new generation of parameter vectors has been created, the algorithm has to calculate the fitness of the parameter vectors, which involves calculating the model at each data point 22  The QueryPerformanceCounter function available in the windows.h library [65] allows for efficient counting of the number of clock cycles elapsed since program execution. By taking clock cycle stamps before and after sections of code are executed, and by obtaining the time per clock cycle via QueryPerformanceFrequency, sections of code can be timed. All timing information reported in this thesis was obtained this way.  96  6.1. C++ Implementation and then calculating the residual between the model and the data. As this is the most costly step by far in the entire algorithm, the program creates four new threads at this point (since the computer being used has four cores) and assigns one quarter of the population to each of them. This results in a speed up of almost four times, since the overhead associated with creating threads is very small in comparison to the time needed to evaluate the fitness function. Four arrayBounds structures are created (see section A.2), one for each thread, and the indexes for the portions of the population that each thread is responsible for are specified in this structure. Each thread is also assigned a handle, which lets the program know whether the thread has finished executing or not; a time storage variable, which lets each thread report how long it took to execute; a threadID integer, which is used to direct each thread toward which section of memory it should use for storing temporary calculations (so that the threads do not overwrite each other’s information). The threads are then started with the AfxBeginThread command. The execution of each thread is described in section 6.1.3. The WaitForMultipleObjects command is then called to watch all of the thread handles to report that their thread is finished, at which point execution of the program can continue. Residual Calculation There are some subtleties with using functions from afxwin.h which require a static function call for launching a thread, the details of which can be seen in subsection A.3.4 and subsection A.3.5. I will skip to the interesting part, which is the calculateResidual function (see subsection A.3.6). The calculateResidual function iterates over every data point and calculates the model value at that point. If there is only one warping parameter and the explicit expression for the oscillations is known (i.e., Equation 2.70), the function getYValue is called (subsection A.3.7). In the case that there is more than one warping parameter, the function integrateLegendre is called(subsection A.3.8). Since the expression  1 B cos θ  appears so many  times, it is calculated once per function call and the value is stored for further use. The function then takes the difference between the model and the data point, for each data point, and returns the sum of the squared residuals. Numerical Integration The following section describes the numerical integration scheme implemented in the fitting program. There are many software packages that already implement similar routines, and I am not pretending that I invented numerical integration. However, the integration step in the genetic algorithm is by far the most costly, and blindly using the first integration routine within reach, without understanding the specifics of the integral being evaluated, results in poor performance. An integration scheme was developed to achieve high enough precision  97  6.1. C++ Implementation  Amplitude arb. units  1.0  0.5  0.0  0.5  1.0 3  2  1  0  1  2  3  kz normalized Figure 6.1: A plot of cos e Sk (κBz ,θ) , which is the argument of integral Equation 6.2. Typical values needed to fit the YBa2 Cu3 O6.59 data were used. Judging by the number of times the derivative changes sign, one might expect to need a polynomial of at least order 18 to roughly approximate this curve. to avoid the accumulation of rounding errors while, at the same time, maximizing the speed at which the specific type of integral encountered in this problem can be evaluated. The oscillatory contribution to the resistivity comes from the following integral: π  I=  cos −π  Sk (κz , θ) e B  dκz .  (6.2)  Equation 2.71, which gives Sk (κz , θ), can be inserted into Equation 6.2, and the integration performed. As seen in subsection 2.3.1, this integral can be done exactly in the case where there is only one warping parameter (k01 , for example). When there is more than one warping parameter then the integral must be performed numerically. Figure 6.1 shows a typical example of the function to be integrated. One can immediately note, from inspecting Equation 6.2 or looking at Figure 6.1, that only half of this integral needs to be done, since it is symmetric under κz → −κz . Gaussian quadrature is an approximation scheme for evaluating integrals by sampling the integrand at specific points, multiplying the integrand values by specified weights, and then summing it all together. In general, the points at which the integrand is evaluated, referred to as the “nodes”, are the roots of a set of orthogonal polynomials. The weights are 98  6.1. C++ Implementation also determined as functions of the same polynomials. The integration scheme used in this thesis is based on Legendre polynomials, whose roots can be calculated as the eigenvalues of the matrix M=                0    1 4·12 −1  1 4·12 −1  0 2 4·22 −1  2 4·22 −1  0 ..  3 4·32 −1  ..  .  .  ..  (n−2) 4·(n−2)2 −1  0  .  (n−1) 4·(n−1)2 −1             (n−1)  2 4·(n−1) −1  0  (6.3)  where n is order of the highest order Legendre polynomial being used, and M is an n × n matrix. The eigenvalues and eigenvectors of Equation 6.3 can easily be calculated in Mathematica. When using Legendre polynomials for Gaussian quadrature, the weights at each node are two times the square of the first element in each eigenvector (the eigenvectors must be normalized). Once the nodes and weights are calculated they can be stored in the fitting program once and for all, and will not have to be computed again unless higher accuracy is needed (which requires higher-order Legendre polynomials.) With appropriate scaling of the integration variable, since the Legendre polynomials are orthogonal on the interval -1 to 1 whereas Equation 6.2 runs from −π to π, the integral Equation 6.2 is approximated as n 2  I ≈ 2π  wi cos i=1  Sk (πxi , θ) e B  ,  where the wi ’s are the weights, the xi ’s are the nodes, and the factor of 2 and the  (6.4) n 2  come  about because the function is symmetric and only half of it needs to be integrated. The factor of π comes from the rescaling of kz . After some testing it was found that a value of n = 60 gives a relative error, when compared to solving the integral exactly in the case of one warping parameter, of around 10−14 for any reasonable set of warping parameters, fields, and angles that will be encountered when fitting YBCO data.  99  Chapter 7  Angle Dependence of YBa2Cu3O6.59 With the background subtracted data shown in Figure 5.13 and Figure 5.11, the theory developed in chapter 2, the estimates of what the Fermi surface should look like from section 5.3, and the genetic algorithm developed in chapter 6, the Fermi surface of YBa2 Cu3 O6.59 can finally be analyzed and discussed in a quantitative way.  7.1  Angle Dependence Up to 35 Degrees  The first subset of the data to fit is the angle dependence up to 35◦ , as seen in Figure 5.16. The reason for stopping at 35◦ is that beyond this angle there is a shift in the phase, as seen in Figure 5.17. Since the Rs factor changes very rapidly as a function of angle near a spin zero, any slight problem with the model will be amplified there and will likely result in a poor fit. Thus the global fit will initially be confined to the data from the first six angles. Only the first harmonic of the data will be fit, as it overwhelmingly dominates the signal (especially at high angles), and I will initially not fit m , but will take the estimate of 1.7 me from subsection 5.3.2  23 .  Finally, I will assume g to be 2 for now, and just fit the spin  mass ms . Since g and ms enter only as a product, this is not actually a restriction on the fit. Estimates of g will be discussed in section 7.4. The model is σ = σ0  2  Ai Rs (msi ) RD (τi ) RT cos 2π i=1  J0  Fi + φi × B cos θ  2π∆Fi J0 (κF i tan θ) B cos θ  (7.1) ,  (7.2)  where the index i runs from 1 to 2 representing the two pieces of Fermi surface being fit to. The phase φi is introduced into the model because the result of Equation 2.43, which gives a phase factor of  −2πp 2 ,  comes from the bands being exactly parabolic, and deviations are  expected when the bands are not parabolic [92]. κF is the Fermi wave-vector kF multiplied by the cˆ-axis lattice parameter d. It is calculated from the frequency F using Equation 2.26, 23 The Rs and RT terms have such similar field and angle dependences that any small error in the m estimate can be compensated for by a change in the scattering rate τ . This can be corrected for in an iterative manner once the temperature dependence, and thus m , is analyzed accurately in section 7.3.  100  7.1. Angle Dependence Up to 35 Degrees Parameter  Surface 1  Surface 2  A F (tesla) ∆F (tesla) τ (picoseconds) φ ms (me )  0→5 460 → 480 0 → 50 .1 → 1 0 → 2π 0→3  0→5 500 → 550 0 → 20 1 → 10 0 → 2π 0→3  Table 7.1: The ranges for the fit parameters for the two pieces of Fermi surface. These estimates are based on the qualitative data analysis performed in section 5.3.  and from the cˆ-axis lattice parameter d = 1.175 nm for YBa2 Cu3 O6.59 from [59]: κF = d  2eF  .  (7.3)  From section 5.3, estimates can be made for the initial fit parameters, which are given in Table 7.1. These estimates warrant some further justification: • A: The amplitude should be of order 1 after normalizing by the background (see Equation 2.80). “Negative” amplitudes are included by letting the phase run from 0 to 2π. • F : Figure 5.22 shows that the oscillatory signal is dominated by a piece of Fermi surface with a θ = 0 area of F ≈ 530 tesla. This is especially apparent near 35◦ , where the other warped piece of Fermi surface disappears (see subsection 5.3.3.) If the 530 tesla component is removed by simple fitting and subtracting, components of equal amplitude near 440 and 520 tesla remain. This most likely corresponds to a warped Fermi surface with average area 470 tesla. • ∆F : The dominant surface, with average frequency near 530 tesla, appears to be unwarped, which is consistent with the observations of other groups [89]. Removing this main component indicates a warping of around 40 tesla for the other surface. • τ : Estimating τ is difficult, as there is more than one parameter controlling the envelope of the oscillations, and beats confuse things further. However, the general consideration that ωc τ ≈ 1 or greater—else the RD term suppresses the oscillations so much that they cannot be observed—allows estimates of at least the order of magnitude of τ . Using the size of background conductivity to estimate τ is probably not very productive, as there is likely more than one band contributing to the conductivity [53]. Further, it is it not clear what would be a valid theory for the resistivity to use in the melted vortex lattice regime.  101  7.1. Angle Dependence Up to 35 Degrees  0.100 s s s s s s s s s  0.050  .9 N .9 N .9 N .9 N .9 N .09 N .09 N .9 N .9 N  200 200 200 200 200 200 200 1000 1000  Χ2  0.020  .3 p .3 p .3 p .03 p .03 p .3 p .3 p .3 p .3 p  0.010  0.005  0.002 1  10  100  1000  104  Generation Number Figure 7.1: Plot of the residuals from the fit to Equation 7.2 for various parameter values, with a Log-Log scale. There are 3 trials, in shades of purple, with s = .3, p = .9, and N = 200, which reach a minimum after 300 iterations. Two trials, in shades of blue, with s = .03, p = .9, and N = 200, reach their minima after 30 iterations, but the residual is much larger than when s = .3. Two trials, in shades of green and yellow, with s = .3, p = .09, and N = 200, indicate that they may eventually reach the same minimum as the p = .9 trials, but the convergence will take more than 5000 generations. Two trials, in shades of red, with s = .3, p = .09, and N = 1000, reach the same minima as with N = 200, but take more generations to minimize. • φ: The phase is allowed to run from 0 to 2π. • ms : It is rather hard to pick an initial value for the spin susceptibility mass. One group proposed that the product gms is near zero [89], but their analysis only looks for the phase flip of the spin zero which can be masked when there is more than one piece of Fermi surface. I allow for the possibility of zero spin mass, and let it range up to around two times m since estimates of g for YBCO are close to 2 [106]. The progress of the fitting algorithm, with several different parameter values and a couple of trials for each, is shown in Figure 7.1. While this is not a completely exhaustive list of all the parameter values tried, it does a good job of representing the qualitatively different behaviours seen for different parameter values. Initial estimates for p, s and N were obtained with guidance from Storn and Price [98]. With the optimal parameters of s = .3 and p = .9 there are two qualitatively different minima, tabulated in Table 7.2. The 102  7.1. Angle Dependence Up to 35 Degrees  Minimum 1  Minimum 2  Parameter  Surface 1  Surface 2  Surface 1  Surface 2  A F (tesla) ∆F (tesla) τ (picoseconds) φ ms (me )  0.564 478.37 40.05 0.271 1.426 1.987  1.109 523.11 1.24 × 10−5 0.190 1.263 1.575  0.633 477.45 40.23 0.277 4.712 1.165  0.977 523.77 6.24 × 10−4 0.196 1.162 1.572  χ2  0.00267144  0.00267681  Table 7.2: The two sets of fit parameters corresponding to the minima reached by the purple and red curves in Figure 7.1. The warping on surface 2 is essentially zero. The only qualitative difference between the two minima is the value of ms on sheet 1, as well as the phase. The phase, however, is different by about π, which is equivalent to multiplying the amplitude by a negative sign, thus the only real difference lies in the ms value. This difference is plotted in Figure 7.3.  0.7 0.6  ms  1.987  ms  1.575  Amplitude  0.5 0.4 0.3 0.2 0.1 0.0 0  10  20  30  40  50  60  Angle degrees Figure 7.2: The absolute value of the spin factors (Rs ) for surface 1 and 2 from minimum 1 in Table 7.2. The blue curve is surface 1, and the red is surface 2. Note that there is no angle where the total amplitude is zero, thus searching for a “spin zero” is a hopeless task. The equivalent plot for minimum two would look very similar, except with slightly different behaviour for surface 1 beyond 35◦ (see Figure 7.3.)  103  7.1. Angle Dependence Up to 35 Degrees  0.6  Amplitude  0.4 0.2  ms  1.987  ms  1.165  0.0 0.2 0.4 0.6 0  10  20  30  40  50  60  Angle degrees Figure 7.3: The two spin factors (Rs ) for surface 1 for the two different minima tabulated in Table 7.2. They have both been multiplied by the amplitude, A, to show that they are almost identical up to 35◦ , which is the the limit of the data used in the fits. The negative sign is accounted for by the π phase shift between the two minima for surface 1. execution time per generation was approximately 33 milliseconds, using 400 data points and a population of N = 200. This should be contrasted with an execution time of 47 seconds per generation for the Differential Evolution algorithm in Mathematica, using a population of only N = 10 and only 211 data points. Scaling the execution time for the number of points and the population size, and noting that it takes 300 generations to reach a minimum in Figure 7.1, it would take the Mathematica implementation 134 hours to converge, whereas it takes my implementation around 10 seconds. Both minima in Table 7.2 have one warped Fermi surface with F = 478 tesla and ∆F = 40 tesla, and another unwarped with F = 523 tesla. The quasiparticle lifetimes are similar for each surface (and differences between them may be accounted for by differences in m not included in this iteration), but the ms values are quite different for the two surfaces, especially when one looks at the Rs terms plotted in Figure 7.2. Figure 7.3 shows the only substantial difference between the two minima in Table 7.2, which is a difference in the ms value for the warped surface (surface 1). The negative sign in the relative amplitude between the two Rs terms plotted in Figure 7.3 is accounted for by the nearly π phase shift between minimum 1 and minimum 2 for surface 1; higher angle data needs to be examined to determine which ms value describes surface 1 of YBa2 Cu3 O6.59  104  7.1. Angle Dependence Up to 35 Degrees best, as the χ2 values for the two minima differ by less than 0.2%. The fit from minimum 2 is compared to the data up to 35◦ in Figure 7.4. Even though, at this stage of fitting, the Fermi surface of YBa2 Cu3 O6.59 has not exactly been pinned down, there is already a lot to say about it. The model of two small pockets of Fermi surface appears to describe the data quite well, at least up to 35◦ ; we can also say that, at least for surface 2, ms is very close to m , meaning that g is close to two, in direct contradiction with [89]. The fit in general looks quite good, with a couple of problems: the fit seems to be worst around the beat in the data (e.g., near 47 tesla in pannel 1 of Figure 7.4), and there are deviations from the fit at the highest fields. The first of these problems is most likely due to the fact that only the first harmonic was fit: the second harmonic will have a beat in a completely different place, and thus may have its maximum relative contribution to the total signal at places where the first harmonic has a node. The second problem is easily explained by the fact that the background polynomial has a tendency to wander around near the boundaries of the data, which is why data above 55 tesla was not actually even used in the fit (the full field range is shown in Figure 7.4 for sake of completeness.) The next section will examine the higher angle data, and resolve which of the minima in Table 7.2 best describes YBa2 Cu3 O6.59 . Plots of what the Fermi surfaces look like are shown in Figure 7.5 and Figure 7.6.  105  7.1. Angle Dependence Up to 35 Degrees  0°  10 °  20 °  30 °  30 °  35 °  30 35 40 45 50 55  30 35 40 45 50 55  Magnetic Field Tesla Figure 7.4: The fit from minimum 2 in Table 7.2, plotted with the data, from 0◦ to 35◦ . The second harmonic of the oscillations, which can make up roughly 10% of the signal at high field, was not included in the fits. This accounts for the deviations near the node at low angle. The background subtraction is inaccurate at the boundaries when fitting with a polynomial, leading to deviations in the fit from the data around 57 tesla, especially at 10◦ and −30◦ .  106  7.1. Angle Dependence Up to 35 Degrees  Figure 7.5: Plots of the two pieces of Fermi surface, using parameters from Table 7.2, minimum 2. The relative dimensions are to scale. The colour scale on the warped surface has been used to highlight the warping.  107  7.1. Angle Dependence Up to 35 Degrees  Figure 7.6: The warped section of Fermi surface, with F = 478 and ∆F = 40, in the Brillouin zone of YBCO. The positioning is for illustrative purposes only, to show the relative size of the pocket with respect to the zone. This image should should not imply that this is where the Fermi pocket is located, nor should it imply that there is only one copy of this pocket. The kx and ky axes have been scaled to run from −π to π, and all dimensions are to scale.  108  7.2. Angle Dependence from 40 to 57 Degrees  7.2  Angle Dependence from 40 to 57 Degrees  The global fit starts to have problems around 40◦ near the first spin zero of surface 1, as predicted in section 7.1. It is not too surprising that the model breaks down eventually: a scattering rate independent of angle has been assumed, the possibility of field-dependent scattering from superconductivity has been ignored, and the oscillatory resistivity was calculated directly from the density of states (resistivity is a dynamical process, and proper treatment of scattering is a very difficult problem [1, 31].) The model will have to be relaxed somewhat if it is to describe the higher-angle data. Figure 7.7 shows the fit at 40◦ . At this angle, beyond the first spin zero of surface 1, the fit still looks qualitatively correct except for a small phase shift. The shape of the Fermi surface itself probably does not change with angle, although there are small corrections in a magnetic field for very low Landau level number [51]. The cyclotron mass and spin splitting factors may change with angle, but without an actual model for how they change it is not helpful to just guess at the angle dependence. As a first approximation, I will assume that any deviation from the model can be accounted for by letting the constant amplitude factor and the phase change as a function of angle. At this point it is the Rs term that is of most interest, and so the data at each angle will be fit  0.1  Amplitude  0.05  0  0.05  0.1  35  40  45  50  55  Magnetic Field Tesla Figure 7.7: The fit to the data (using parameters from minimum 2) at θ = 40◦ . The fit would line up if the phase was shifted slightly.  109  7.2. Angle Dependence from 40 to 57 Degrees separately, setting Rs to one and holding all other parameters constant (except for A and φ, which are being fit). The fit amplitudes as a function of angle can then be compared to the Rs function as it appears from the fit to the low-angle data. The reason for letting the phase run free is because small deviations in the frequency can be accounted for by a phase offset. This will account for the effects of higher order warping parameters which have not been included. Figure 7.8 shows the results of this amplitude extraction procedure. The two different ms values for surface 1 from minima 1 and 2 are plotted for comparison to the extracted amplitudes, since the low angle data alone could not distinguish between them. Figure 7.8 shows clearly that the global fit to the data up to 35◦ has correctly predicted the spin zero angles for both surfaces, even though the spin zeros occur at angles higher than 35◦ . While ms = 1.174 and ms ≡ 1.998 for surface 1 were indistinguishable for the low angle data, Figure 7.8 shows that ms = 1.998 predicts a second spin zero near 55◦ for surface 1 which not observed in the data. Additionally, the data for surface 1 is fit better at low angle by the ms = 1.174 value.  110  7.3. Temperature Dependence  ms 1.174 ms 1.998 ms 1.581 Surface 1 Surface 2  1.0  Amplitude  0.8  0.6  0.4  0.2  0.0  0  10  20  30  40  50  60  Angle degrees Figure 7.8: The oscillation amplitude as a function of angle, fitted individually without the Rs term, holding all other parameters except phase constant. The fit through the data is the Rs term, fitted to the first six data points (up to 35◦ .) The fit is quite good up to 40◦ , as might be expected since the model parameters (when fitting at each angle) came from a global fit to the data up to 35◦ . While the amplitude is overestimated at higher angles, the shape appears to be at least somewhat correct. The masses extracted from the fit (shown in the key) are very close to those in Table 7.2. Two values for ms for surface 1 are shown in blue: the smaller value fits the data better at low angles, and does not have the extra node around 55◦ as the higher ms value does. It appears that ms = 1.174 provides a better fit to the data.  7.3  Temperature Dependence  Using the results of section 7.1, the temperature dependence can be re-analyzed to obtain a more accurate value of m than was estimated in subsection 5.3.2. The previous m analysis was hampered by the assumption that both Fermi surfaces had the same mass: now that the frequencies and scattering rates for both Fermi surfaces are known, the m can be extracted by fitting the entire oscillatory resistance as a function of temperature. This is one of the many advantages that a direct fit to the data has over analysis of the Fourier transform amplitude: if the frequencies of different pockets overlap, then extracting the temperature dependent amplitudes for each surface becomes impossible. Because harmonic number p and the temperature T enter the dampening factor RT in the form  p T,  harmonics higher than the first can make up an appreciable fraction of the total  oscillatory amplitude at low temperatures. Therefore at least the second harmonic should be 111  7.4. Summary of YBa2 Cu3 O6.59 0.15  Oscillatory Resistive Ratio  0.12 0.09 0.06  1.5 K 2.5 K 3.1 K 4.2 K  0.03 0. 0.03 0.06 0.09 0.12 0.15 30  35  40  45  50  55  Magnetic Field Tesla Figure 7.9: Fits to the temperature dependence at θ = −30◦ , using m for each surface as the free parameter, and fixing the other parameters to the values obtained in minimum 2 of Table 7.2. The fits are worst at the highest field values, as expected, since the background subtraction is most inaccurate near the boundaries. These fits yield m1 = 1.52me and m2 = 1.65me . included in this temperature-dependence analysis. Performing this fit on the temperaturedependence data gives m1 = 1.52me and m2 = 1.65me , where i = 1, 2 indicates the surface number as per Table 7.2. These fits indicate that the initial estimate of m = 1.7me was not too bad. A plot of the fit at each temperature is shown in Figure 7.9. The fact that a good fit to the RT term can be obtained at all might be surprising, given the clearly non-Fermi-liquid behaviour of YBCO at high temperatures and the proximity at low doping to the Mott insulating state [19, 100]. This apparent agreement has been interpreted as Fermi liquid behaviour by Sebastian et al. [87].  7.4  Summary of YBa2 Cu3 O6.59  The fits have given useful information about the Fermi surface size, the cˆ-axis coupling, the quasiparticle g factor, the quasiparticle lifetime, and the cyclotron effective mass for YBa2 Cu3 O6.59 . The best fit to the data gives two similar pieces of Fermi surface with different cˆ-axis couplings. I will speculate about the location of the Fermi pockets in the Brillouin zone, and I will use the scattering rate from the Dingle factor to make a rough comparison to the transport quasiparticle lifetime. A more definitive statement about the cyclotron 112  7.4. Summary of YBa2 Cu3 O6.59 effective mass and its implications for interpretation of the specific heat measurements on YBCO is made, and finally the most robust conclusions are drawn from the analysis of the g factor.  7.4.1  Fermi Pocket Locations  Quantum oscillation techniques can only give the Fermi surface sizes and shapes—not their locations in the Brillouin zone. Comparisons with band structure calculations and/or angleresolved photoemission spectroscopy (ARPES) are usually needed to locate the pieces of Fermi surface in the zone, although symmetry arguments using the shape of the zone and the shape of the pockets can restrict where the pockets lie. Band structure calculations for YBCO do not give small pockets (at least without the artificial introduction of electronic order, which can always be used to produce a reconstructed Fermi surface), and ARPES gives small arcs instead of pockets. Therefore, Fermi surface information from overdoped YBCO (which has a Fermi surface in ARPES experiments), and other cuprates such as Tl2 Ba2 CuO6+x , must be used to come up with plausible locations for the pockets. The unreconstructed Fermi surface of overdoped Tl2 Ba2 CuO6+x has nodes in the cˆaxis hopping rate along the zone diagonals (the (0, 0) → (π, π) direction in Figure 1.4) [44]. If periodic in-plane order reconstructs this large Fermi surface into pockets on the underdoped side of the phase diagram, then pockets created from the Fermi surface near the nodal directions could result in the unwarped pocket measured in this experiment. Similarly, there is kz warping at the anti-nodal points (at (π, 0) and (0, π) in Figure 1.4) which could give rise to the warped pocket after reconstruction. The larger warping of the 478 tesla pocket should give it a larger contribution to the cˆ-axis conductivity than the 523 tesla pocket’s contribution [31]. However, both the pseudogap and the actual d-wave superconducting gap have a maximum amplitude at the anti-nodal points [100], and it is not exactly clear how they affect the density of states in this region at high field. Figure 7.10 shows the reconstruction of the Fermi surface when the ordering wave-vector is Q = (π, π), which is the antiferromagnetic wave-vector (although other types of order can have this wavevector as well.). Electron pockets form near (±π, 0) and (0, ±π) in this scenario, and hole pockets form along the zone diagonal. Multiple scenarios have been put forth for the origin of the pockets, including reconstruction from spin-density wave (SDW) order [69], reconstruction from charge-density wave (CDW) order and more exotic orders such as d-density wave order [28], and oscillations coming from the disconnected Fermi-arcs seen in ARPES without the need for closed orbits [76]. All of these theories require tuning parameters—the gap size of the order parameter and the chemical potential—to obtain the correct frequencies seen in experiment, and it is hard to distinguish between scenarios without being able to definitively locate the pockets in momentum space. Most of these reconstruction scenarios also predict the existence of additional pockets of Fermi surface of higher and lower frequencies, in order to match the 113  7.4. Summary of YBa2 Cu3 O6.59 doping in accordance with Luttinger’s theorem, and none of these extra pockets are seen in the cˆ-axis measurements. A second “β” frequency has sometimes been reported around 1500 tesla [89], but there is no sign of it in the Fourier transform shown in Figure 5.22.  7.4.2  Quasiparticle Lifetime  The quasiparticle lifetime τ from the fits is a different object than the lifetime that comes from a semi-classical model of the resistivity [92]. This is because there is a weighting factor of (1 − cos θ) in the integral that determines τ in resistivity which is not present in the calculation of the quantum oscillation τ : large-angle scattering is much more important than small-angle scattering for electrical transport, whereas scattering at any angle causes a quasiparticle to lose coherence when traversing a cyclotron orbit. Despite this difference, agreement between the two τ values is often better than a factor of two at low temperature[92], and so it is worth calculating the resistivity using the values of τ from the fit. It needs to be emphasized that even though the cˆ-axis resistivity is measured in this experiment, the cyclotron orbits are in the a ˆ-ˆb-plane of the material; the τ measured from the oscillation amplitude is the a ˆ-ˆb-plane quasiparticle lifetime, not the cˆ-axis lifetime. Assuming that there are two bands contributing to the conductivity, one electron-like and one hole-like, the two-band resistivity is ρ1 ρ2 , ρ1 + ρ2 mi ρi = , ni e2 τi ρ=  (7.4)  where ρi is the conductivity of a single band, and ni is the density of carriers (page 240 of Ashcroft and Mermin [6].) Luttinger’s theorem says that the Fermi surface volume is proportional to the density of electrons or holes in the material, even when interactions are present [61]. In the quasi-2D system of YBCO this means that the carrier density per copper oxygen plane is n=  F , Φ0  where Φ0 is the magnetic flux quantum and F is the oscillation frequency  (7.5) 24 .  This number  n must be multiplied by 2 for the two copper-oxygen planes in YBCO’s unit cell, and also by the number of pockets in the Brillouin zone. This is where a guess has to be made as to what type of Fermi surface reconstruction is ˆ = (π, π) reconstruction, as would be the case if there was static taking place. For a simple Q antiferromagnetism at this doping, there are four hole pockets and two electron pockets in the Brillouin zone [16]. There is no direct experimental evidence for such a reconstruction 24  h The flux quantum is Φ0 = 2e here, not he , even though these are charge e carriers. This is because of the factor of 2 from spin in the momentum density of states.  114  7.4. Summary of YBa2 Cu3 O6.59  Π  ky  ky  Π  0  Π  Π  0  Π  kx  0  Π  Π  Π  0  kx  Figure 7.10: Left: The blue lines are a schematic of the Fermi surface for a high Tc cuprate. With an ordering wave-vector Q = (π, π), shown in black, there is a second copy of the Fermi surface in the centre of the zone, shown in green. Right: An energy gap, proportional to the energy cost U of having two electrons doubly-occupy a copper site, breaks the degeneracy of the two bands and gives electron (red) and hole (black) pockets [16]. at O6.59 , but this is as good an example as any to work with for illustrative purposes. Figure 7.10 shows this reconstruction, and how it leads to electron and hole pockets. Associating the unwarped Fermi pocket with the four hole pockets near (± π2 , ± π2 ), and the warped Fermi pocket with the two electron pockets near (±π, 0) and (0, ±π), the resistivity can be calculated using the fit parameters from Table 7.2 and Equation 7.4. This gives ρ = 10.1 µΩ−cm.  (7.6)  The B → 0 extrapolated value for the a ˆ-ˆb-plane resistivity of YBa2 Cu3 O6.60 is 5-10 µΩ−cm [54]: it is expected that the ρ value calculated from the quantum oscillation parameters will be higher than the measured value, because the scattering rate for quantum oscillations is larger than it is for electrical transport. The hole doping of this Fermi surface is n = nholes − nelectrons = 0.084, where ni is calculated from the number of pockets and their areas. If Luttinger’s rule is to be obeyed then this number should be 0.11, which is not too far off. Other configurations using between 2 and 4 of each of the hole and electron pockets give resistivities between 5-20 µΩ−cm, all of which should be considered consistent with the measured value given the questionable validity of this calculation in the first place.  115  7.4. Summary of YBa2 Cu3 O6.59  7.4.3  Cyclotron Mass  The cyclotron effective masses of 1.52 me and 1.65 me are similar to those reported in other measurements near this doping, although only one total mass is reported in other publications, since the mass is obtained from the Fourier transform amplitude [7, 22, 88]. Heat capacity measurements on YBCO give a residual specific heat in zero field at zero kelvin of 1.85  mJ mol·K2  [81]. This residual specific heat can be related to the concentration of  ungapped quasiparticles (quasiparticles not taking part in superconductivity). The authors of Riggs et al. [81] interpret this relatively small value of 1.85  mJ , mol·K2  which is roughly  comparable to what a single pocket with an area near 500 tesla would give, as coming from barium-oxide/chain hybrid pocket that remains ungapped due to disorder. The main motivation behind this proposal is that there is only one layer of copper-oxygen chains per unit cell in YBa2 Cu3 O6.56 ; any Fermi pocket originating from the copper-oxygen planes will have two copies per unit cell due to bilayer splitting, and thus will give more residual specific heat than the value measured. There are a number of reasons why I believe that this scenario does not fit with existing experimental data: • There are two barium-oxide planes in the unit cell of YBCO, and band-structure calculations show that there are two barium-oxide/chain bands per unit cell [23]. It would require extreme fine-tuning of the chemical potential to have only one of the resulting bands cross  F.  • ARPES measurements find the barium oxide bands 600 meV below  F  [27]. Despite  the complications in ARPES on YBCO, it seems unlikely that the relative positions of the bands could be off by this much in energy. • The other YBCO compound—YBa2 Cu4 O8 —sees very similar oscillations to those seen in YBa2 Cu3 O6.59 . The barium-oxide/chain bands in this material are completely different than they are in YBa2 Cu3 O6.59 , because there are two chain layers per unit cell in YBa2 Cu4 O8 , and only one in YBa2 Cu3 O6.59 . • The chain periodicity and degree of oxygen order changes dramatically between orthoII at O6.50 and ortho-VIII at O6.67 , passing through ortho-V at O6.62 . If the oscillations are coming from a barium-oxide/chain hybridized band, then surely the oscillation frequency should change dramatically when the chain periodicity changes. • The chains are one dimensional objects that localize in underdoped YBCO at low temperatures, as shown by the (lack-of) resistive anisotropy [5]. The microwave conductivity also shows a very large scattering rate in the chains [12], which is inconsistent with the long mean free path needed for quantum oscillations. With these considerations in mind it seems more likely that the oscillations are coming from the copper-oxygen planes than from a pocket created by the chains. The residual 116  7.4. Summary of YBa2 Cu3 O6.59 specific heat can be explained by the presence of disorder in the sample, which produces quasiparticles near the nodes of the d-wave superconducting gap even at zero kelvin.  7.4.4  Quasiparticle Spin Magnetic Moment  In subsection 2.4.1 it was explained that the spin splitting factor πg  ms 2m0  does not contain  electron-phonon renormalizations, as shown in the quantum mechanical calculation of the damping factors by Engelsberg and Simpson [24]. The mass ms can then be related to the thermodynamic cyclotron mass m via its definition ms ≡  m . 1+λ  (7.7)  To obtain the fit values for ms in section 7.1, g was set equal to 2, because there is a redundancy if both g and ms are free parameters. Using Equation 7.7, 2ms = g  m , 1+λ  (7.8)  where the 2 on the left hand side of Equation 7.8 is the assumed value for g used in the fits. The measured m from the temperature dependence can then be used, along with the fact that the electron-phonon coupling constant λ is positive, to restrict g as g ≥  2ms . m  (7.9)  Using the values of ms from minimum 2 in Table 7.2, the limits for g are g ≥ 1.53 for the warped “surface 1”, and g ≥ 1.91 for the unwarped “surface 2”. This is in rough agreement with Walstedt et al. [106], where the in-plane g = 2.075. If the warped surface 1 is an electron pocket near (π, 0) in the Brillouin zone, as was supposed earlier, this is consistent with the prediction of Garcia-Aldea and Chakravarty [28] that the g factor at (π, 0) can be suppressed for certain types of density-wave order. The theoretical work of Norman and Lin [68] shows that spin zeros near 40 and 50 degrees are only consistent with a non-magnetic density wave state, or a SDW state with a significant portion of the spin canted in the direction of the magnetic field. The contrasting case of transverse SDW states give highly suppressed g factors and spin zeros do not appear before 60◦ . It should be noted that the estimates given above for g are lower bounds, and finite electron-phonon coupling λ will make the g estimates larger. The values for g obtained in this analysis are clearly in disagreement with measurements of the oscillatory skin depth in YBCO by Sebastian et al. [86], in which they concluded that g ≈ 0 because the spin zero phase flip was not observed up to 57◦ . The complicating factor of two Fermi surfaces, with different spin zeros for each, is likely what led to the lack of observed spin zero. More recently, currently unpublished measurements made by the same  117  7.4. Summary of YBa2 Cu3 O6.59 group up to higher fields and angles now claim similar results to what I have described here, with g ≈ 2 [90].  7.4.5  Parameter Uncertainties  The electrical resistance data is sufficiently dense and of such a high signal-to-noise ratio that the main sources of parameter uncertainties comes from the uncertainties in the sample angle, the field, or the temperature, and not from the statistical uncertainty of the fits that arises from the scatter of the data. In subsection 5.1.1, the angle offset was determined by comparing the melting transition at ±30◦ , and adjusting the angle offset until the melting transitions lined up. This gave an offset of 1.06◦ , with an uncertainty (due to the limited resolution of the resistance data near the melting transition where R → 0) of ±0.02◦ . It is useful to note that this is an extremely sensitive measure of the “offset” angle for a highly anisotropic superconducting sample, and this calibration technique could be used in other circumstances where the absolute angle is needed with great precision. The fits to the pickup coil slopes, also described in subsection 5.1.1, each have a statistical uncertainty that results in an angle uncertainty of about ±0.01◦ . The uncertainty here is from the noise in the pickup coil data. However, the resulting uncertainty is quite small because of the large number of points taken at voltage values relatively far from the origin. These two uncertainties in the angle, combined in quadrature, give a total uncertainty of ±0.022◦ . Offsetting the angles by this amount in either direction, and then performing the genetic algorithm minimization near the known global minimum, gives the uncertainty estimates under the column ∆θ in Table 7.3. The absolute value of the magnetic field is calibrated with a Hall-probe, and has a worst-case uncertainty of ±0.25%. Using the same method described above for the angle uncertainty, the field can be scaled by ±0.25%, and the resulting bounds on the fit parameters can be used to calculate an uncertainty (the column labelled ∆B in Table 7.3). The uncertainty in the temperature is about 10 mK due to the (in)stability of the pressure regulation. This adds a very small uncertainty to the m values (the column labelled ∆T in Table 7.3), and does not affect the other parameters. The three sources of uncertainty are independent of one another, and the total uncertainty in each parameter is the quadrature sum of each uncertainty (the column labelled ∆T otal in Table 7.3). The uncertainty in gmin can then be calculated, using Equation 7.8 with λ = 0: ∆gmin =  ms m  2 ms  2  (∆ms )2 +  2 m  2  (∆m )2 .  (7.10)  This formula was used to calculate ∆T otal for the gmin parameter in Table 7.3. It should be emphasized that the uncertainties in Table 7.3 are the uncertainties assuming the model of two warped Fermi surfaces, each with a single warping parameter, 118  7.4. Summary of YBa2 Cu3 O6.59 Surface 1  A F (T) ∆F (T) τ (ps) φ ms m gmin  0.564 478.40 40.15 0.271 1.426 1.165 1.516 1.537  Surface 2  ∆θ  ∆T  ∆B  ∆T otal  0.004 0.04 0.02 0.001 0.001 0.004 0.019 —  — — — — — — 0.001 —  0.005 0.18 0.01 0.002 0.005 0.006 0.002 —  0.006 0.18 0.02 0.002 0.005 0.007 0.019 0.021  1.109 523.12 ∼0 0.190 1.263 1.575 1.651 1.908  ∆θ  ∆T  ∆B  ∆T otal  0.021 0.16 10−4 0.001 0.002 0.001 0.003 —  — — — — — — 0.001 —  0.036 0.04 10−6 0.003 0.001 0.001 0.002 —  0.042 0.16 10−4 0.003 0.002 0.001 0.004 0.005  Table 7.3: The Fermi surface parameters, and their uncertainties, for YBa2 Cu3 O6.59 . The parameter gmin is the minimum value for g assuming zero electron-phonon coupling, as described in subsection 7.4.4. The uncertainties labelled ∆θ , ∆T , and ∆B come from the uncertainty in angle, temperature, and field, respectively. The column ∆T otal is the quadrature sum of all of the uncertainties.  and assuming that the formalism developed in chapter 2 is correct. Ways in which the model can be extended include accounting for non-linear spin splitting, including a fielddependent scattering rate, and including the contributions from the second (and higher) oscillation harmonics. The model of the Fermi surface shape can also be modified by including higher warping harmonic parameters into Equation 2.71. Thus, while the relative uncertainties in Table 7.3 are small, they should not be over-interpreted, since the greater uncertainty lies in the assumptions made about the model in the first place. Despite this caveat, the simplest model is still the correct starting point; including the additional effects listed above requires more fit parameters, and more fit parameters will always provide a better fit (lower residual), even when they may not be physically relevant. A solution to this problem lies in the use of Bayesian statistics, which provides a method for discriminating between different models that accounts for both the quality of the fit to the data and penalizes needlessly-complicated models.  119  Chapter 8  Other Dopings The cˆ-axis resistances of YBa2 Cu3 O6.47 , O6.51 , O6.67 , and O6.75 were measured as a function of magnetic field and temperature, but not as a function of angle. The detailed data analysis for YBa2 Cu3 O6.59 can not be reproduced for these doping levels without the angular dependence, but a quick survey of the field and temperature dependence gives some useful information. In this chapter I will first present the data for all five doping levels, and then discuss the trends that come out of the doping dependence. In addition to the quantum oscillations, I will also look at how the doping affects the vortex-lattice melting transition.  8.1  Oxygen 6.47  YBa2 Cu3 O6.47 was measured with the hope that the doping level would be below the apparent Lifshitz transition, where the Fermi surface changes topology from pockets to bands that run across the entire Brillouin zone [54]. If this were the case, then no oscillations would be present due to the lack of closed Fermi surface orbits. Resistance as a function of magnetic field, at temperatures from 1.5 to 120 K, is shown in Figure 8.1. No oscillations are visible in the data. This fact does not preclude the existence of closed Fermi surface pockets on its own, since increased scattering would also damp out oscillations: below O6.50 the ortho–II order in YBCO is less robust than it is at higher doping levels. The hole doping 0.002 , as determined from the T [59]. of this particular sample is p = 0.088±0.005 c  Figure 8.2 shows the cˆ-axis resistance as a function of temperature. The resistance values in red in this plot are taken at 50 Tesla. Note the fairly good agreement at high temperature between the zero field data and the high field data: when ωc τ  1 due to phonon scattering  at high temperatures, then the magnetoresistance is negligible [6]. Despite the complication of the magnetoresistance at low temperatures, it is clear that the resistance is not diverging to infinity as T → 0, and so YBa2 Cu3 O6.47 is certainly not insulating in the same sense that a material with a large band-gap is insulating [6]. The field value at which R > 0 is significant because at this field the vortex lattice is melted, and dissipation of energy through vortex motion can occur [26, 101]. I have defined Bmelt to be the field where the resistance is 0.1% of its value at 50 tesla. This is very similar to the definition used in a study of the melting transition in optimally doped YBCO by Ando et al. [4]. These Bmelt values are plotted up to near Tc in Figure 8.3. This value should be comparable with Birr , the irreversibility field seen in torque magnetometry 120  8.1. Oxygen 6.47 2. 1.5 K 4.2 K 10 K 20 K 30 K 40 K 50 K 60 K 70 K 80 K 95 K 120 K  1.8  Resistance Ohms  1.6 1.4 1.2 1. 0.8 0.6 0.4 0.2 0. 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 8.1: The magnetoresistance of YBa2 Cu3 O6.47 , at temperatures ranging from 1.5 to 120 K. No oscillatory behaviour is seen; the small bumps in the data ( near 57 tesla at 1.5 K, for example) are due to movement of the whole and cryostat during the field pulse. measurements (the irreversibility field is the field value where the magnetic torque measured is independent of the sign of  dB dt ,  and is equal to Bmelt : see for example Vignolle et al. [103]).  A comparison of the shape of Bmelt as a function of temperature to the theory of vortex lattice melting is presented in section 8.6.  121  8.1. Oxygen 6.47  1.8  50 Tesla 0 Tesla  1.6  Resistance Ohms  1.4 1.2 1. 0.8 0.6 0.4 0.2 0. 0  20  40  60  80  100  120  140  160  180  200  220  240  260  Temperature Kelvin Figure 8.2: The resistance as a function of temperature for YBa2 Cu3 O6.47 in the cˆ-axis direction. The red dots are taken at 50 tesla, and the black curve is in zero field. It should be noted that there is strong magnetoresistance below about 30 K (see Figure 8.1 for the magnetoresistance).  122  8.1. Oxygen 6.47  40  Vortex Melting Field Tesla  35 30 25 20 15 10 5 0 0  5  10  15  20  25  30  35  40  45  50  55  Temperature Kelvin Figure 8.3: The magnetic field where the vortex lattice melts and a resistive signal is first seen, plotted as a function of temperature. Models of this transition depend on the London penetration depth, the superconducting coherence length, the dimensionality of the superconductivity, and assumptions about how melting takes place [26].  123  8.2. Oxygen 6.51  8.2  Oxygen 6.51  Figure 8.4 shows the resistance as a function of magnetic field for YBa2 Cu3 O6.51 , from 1.5 to 60 K. The data is qualitatively similar to that seen in YBa2 Cu3 O6.47 , except that there are oscillations at the highest fields and lowest temperatures. This is the lowest doping presented in this thesis in which oscillations have been seen in the cˆ-axis resistivity. The 0.001 , as determined from the T . hole doping of this sample is p = 0.092±0.006 c  Figure 8.5 shows the resistance as a function of temperature, both in zero field and at 50 tesla for temperatures below Tc . The increase in resistance with lowered temperature is similar to that seen in Figure 8.2, except that in YBa2 Cu3 O6.51 it appears to turn over at the lowest temperatures. This may be an indication of the conduction in the cˆ direction becoming more coherent as the temperature is lowered and the scattering rate decreases [104]. Figure 8.6 is the vortex melting field as a function of temperature. The data are similar to that for YBa2 Cu3 O6.47 , except that the maximum field at 1.5 K is about 10 tesla lower for YBa2 Cu3 O6.51 , even though Tc is higher. A discussion of why this might be so is given in section 8.6. Figure 8.7 shows the oscillations seen in YBa2 Cu3 O6.51 with the background removed and the data fit to the two Fermi surface model used in chapter 7. The fit parameters are  1.5 K 2.5 K 3K 3.5 K 4.2 K 10 K 20 K 30 K 40 K 50 K 60 K  Resistance Ohms  1.2 1. 0.8 0.6 0.4 0.2 0. 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 8.4: Magnetoresistance of YBa2 Cu3 O6.51 at temperatures ranging from 1.5 to 60 K. Oscillations can be seen above about 40 tesla, and persist up to around 10 kelvin.  124  8.2. Oxygen 6.51  50 Tesla  1.  Resistance Ohms  0 Tesla 0.8  0.6  0.4  0.2  0. 0  20  40  60  80  100  120  140  160  180  200  220  240  260  Temperature Kelvin Figure 8.5: The resistance as a function of temperature for YBa2 Cu3 O6.51 in the cˆ-axis direction. The red dots are taken at 50 tesla, and the black curve is in zero field. The abrupt dip below about 10 kelvin is right where quantum oscillations become visible (the resistance value is taken from the background average between the oscillation peaks.) given in Table 8.1. There are similarities to the Fermi surface found in section 7.1: one surface is warped and the other is not, and the m s are similar. There is also a decrease in quasiparticle lifetime, as extracted by the RD term, which is not too surprising since the Ortho II order is actually better in YBa2 Cu3 O6.59 than YBa2 Cu3 O6.51 due to the way that excess oxygen is incorporated into the chains  25 .  The unwarped Fermi surface has almost the same area at this doping as that in YBa2 Cu3 O6.59 , but the warped surface is smaller by about 10%. The cyclotron effective mass m , as extracted via the RT term, has increased for both surfaces from its value at YBa2 Cu3 O6.59 . This is consistent with the effective mass increase with lowered doping observed by Sebastian et al. [88], which they associate with a quantum critical point near O6.45 .  25  Excess oxygen (beyond 6.50) allows chains to become completely filled, whereas right at O6.50 , entropy dictates that some oxygen atoms will be sitting in nearly empty chains, producing vacancy defects in the nearly full chains. It was explained in section 9.1 why the ends of broken chains, and single oxygen atoms in empty chains, do not dope the planes. Therefore there should be a larger disorder imprint on the planes from a small number of missing oxygens in full chains than there is with the same number of extra oxygen atoms in empty chains.  125  8.2. Oxygen 6.51  30  Vortex Melting Field Tesla  25  20  15  10  5  0 0  5  10  15  20  25  30  35  40  45  50  55  Temperature Kelvin Figure 8.6: The vortex lattice melting field Bmelt as a function of temperature up to Tc .  Parameter  Surface 1  Surface 2  A F (tesla) ∆F (tesla) τ (picoseconds) φ m (me )  0.776 429.59 51.36 0.141 3.060 2.06  0.896 530.57 ∼0 0.138 5.661 2.20  Table 8.1: Fit parameters for the temperature and field dependence of the oscillatory magnetoresistance of YBa2 Cu3 O6.51 . ms is not reported because extraction of that parameter requires the angular dependence of the oscillations.  126  8.2. Oxygen 6.51  Oscillatory Resistive Ratio  0.04  0.02  1.5 K 2.5 K 3K 3.5 K 4.2 K  0.00  0.02  0.04 35  40  45  50  55  60  Magnetic Field Tesla Figure 8.7: Quantum oscillations for YBa2 Cu3 O6.51 with the background subtracted. The fits are made to the same model used in section 7.1. The fit is worse at high fields and low temperatures because only the first harmonic is fit, and background subtraction is difficult near the boundaries.  127  8.3. Oxygen 6.59  8.3  Oxygen 6.59  This is the same sample that was used in the angular dependence measurements in chapter 7, but measured during a different pulsed field run at zero angle for temperatures from 1.5 to 200 K. The fit to the Fermi surface will not be repeated here, as the angular dependence gives a much more complete and accurate picture. Since this sample has already received considerable discussion in previous chapters (chapter 5, chapter 7), I will only present the data here, and leave the discussion for section 8.6. The hole doping of this sample is p = 0.105±0.001 0.002 as determined via Tc . Figure 8.8 shows the oscillatory magnetoresistance, Figure 8.9 shows resistance as a function of temperature, and Figure 8.10 shows the vortex lattice melting transition.  1.4  1.5 K 2K 2.5 K 3K 3.5 K 4.2 K 6K 8K 10 K 15 K 20 K 30 K 40 K 50 K 60 K 90 K 128 K 150 K 200 K  Resistance Ohms  1.2 1. 0.8 0.6 0.4 0.2 0. 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 8.8: Magnetoresistance of YBa2 Cu3 O6.59 up to 60 tesla, from 1.5 to 200 K. At the highest fields, the peak-to-peak oscillation size is 50% of the size of the background. 128  8.3. Oxygen 6.59  1. 0.9  50 Tesla  Resistance Ohms  0.8  0 Tesla  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0. 0  20  40  60  80  100 120 140 160 180 200 220 240 260  Temperature Kelvin Figure 8.9: Resistance of YBa2 Cu3 O6.59 in zero field from Tc = 60 K to 270 K, and at 50 tesla below Tc . As was seen in O6.51 , the resistance begins decreasing with temperatures below about 8 kelvin.  129  8.3. Oxygen 6.59  Vortex Melting Field Tesla  25  20  15  10  5  0 0  5  10  15  20  25  30  35  40  45  50  55  60  65  Temperature Kelvin Figure 8.10: The magnetic field value Bmelt where the vortex lattice melts. This shape is similar to what was seen in O6.47 and O6.51 .  130  8.4. Oxygen 6.67  8.4  Oxygen 6.67  When YBa2 Cu3 O6.67 is annealed just below 50◦ C for several weeks, ortho-VIII order forms in the chains [60, 105]. The second to last panel of Figure 4.7 shows what this order in the chains looks like. The coherence length of the chain order is much shorter at this doping than it is at ortho-II (in the directions perpendicular to the chains, the a ˆ-axis) [105], so a higher scattering rate for the conduction electrons in the planes is expected. The hole doping of this sample is p = 0.116±0.001 0.001 , as determined by Tc . Figure 8.13 shows the vortex lattice melting transition as a function of temperature. Figure 8.11 shows the magnetoresistance up to 60 tesla, from 1.5 to 60 K. Oscillations are only visible at 1.5 and 4.2 K, and only above 40 tesla. The strange shape of the background magnetoresistance, seen clearly at 10 kelvin where Bmelt is lower than at the lower temperatures, could be due to the appearance of a very small Fermi pocket with a very light m . It could also be due to “slow” oscillations that appear in a fully quantum mechanical derivation of cˆ-axis resistivity in a magnetic field, as shown by Grigoriev [31]. These slow oscillations are not damped out with temperature the same way that regular quantum oscillations are, which could explain why they are still visible at 10 kelvin. Because the period of oscillation is so long, measurements up to much higher fields need to be performed to confirm if this really is the phenomenon described by Grigoriev [31].  Resistance Ohms  0.6  1.5 K 4.2 K 10 K 20 K 30 K 40 K 50 K 60 K  0.4  0.2  0. 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 8.11: Magnetoesistance from 1.5 to 60K. Quantum oscillations are visible at 1.5 and 4.2 K above ˜ 45 tesla.  131  8.4. Oxygen 6.67  50 Tesla  0.5  Resistance Ohms  0 Tesla 0.4  0.3  0.2  0.1  0. 0  20  40  60  80  100  120  140  160  180  200  220  240  260  Temperature Kelvin Figure 8.12: cˆ-axis resistance at 0 and 50 tesla up to 270 K for O6.67 . The rollover at low temperature seems to be associated with the appearance of quantum oscillations, in this material as well as in O6.51 and O6.59 . Figure 8.12 shows the superconducting transition and the B = 50 tesla resistance value down to low temperature for YBa2 Cu3 O6.67 . A turn-over in the resistivity, similar to that seen in O6.51 and O6.59 , is seen at the lowest temperature. The background-subtracted oscillatory resistance at 1.5 and 4.2 K is shown in Figure 8.14, along with fits to the data. The data is much noisier than it was at O6.59 , due to the decreased oscillation amplitude with respect to the size of the background. Using the genetic algorithm from chapter 6, a fit to the two Fermi surface model was performed. Comparing the fit parameters in Table 8.2 to those for O6.59 in Table 7.2, it is evident that the unwarped surface has increased in area by roughly 15%, and that the warped surface has increased in average area by roughly 5%. The cyclotron effective mass is similar to what was seen in O6.59 , and the quasiparticle lifetime has decreased, as expected, by a factor of two or three.  132  8.4. Oxygen 6.67  Vortex Melting Field Tesla  25  20  15  10  5  0 0  5  10  15  20  25  30  35  40  45  50  55  60  65  Temperature Kelvin Figure 8.13: The vortex lattice melting transition in YBa2 Cu3 O6.67 , as determined by the onset of a finite resistance, up to Tc . The maximum Bmelt at low temperature is much lower for O6.67 than it is for both O6.59 and O6.75 .  Parameter  Surface 1  Surface 2  A F (tesla) ∆F (tesla) τ (picoseconds) φ m (me )  0.518 500.55 48.88 0.084 5.152 1.52  0.447 611.38 ∼0 0.065 5.174 1.52  Table 8.2: Fit parameters for the temperature dependence of YBa2 Cu3 O6.67 . ms is not reported because extraction of that parameter requires the angle dependence of the oscillations.  133  8.4. Oxygen 6.67  Oscillatory Resistive Ratio  0.003 0.002  1.5 K 4.2 K  0.001 0.000 0.001 0.002 0.003 40  45  50  55  60  Magnetic Field Tesla Figure 8.14: The oscillatory component of the magnetoresistance at 1.5 and 4.2 K. The data is shown in shades of purple, and the fits are shown in shades of orange. The higher frequency component visible near the oscillation peaks at 1.5 K may come from a Fermi pocket, but the signal is near the noise level and so higher field measurements are needed to confirm this.  134  8.5. Oxygen 6.75  8.5  Oxygen 6.75  YBa2 Cu3 O6.75 can be annealed into the ortho-III oxygen structure, shown in Figure 4.7. This chain structure should have a longer coherence length than ortho-VIII does, due to the much simpler structure for ortho-III, and this is borne out in experiment [105]. With quantum oscillations visible in both the ortho-II and ortho-VIII superstructures, oscillations should be seen in ortho-III if chain-order is the limiting factor. The increase in the vortexlattice melting transition field is somewhat compensated for by making use of the new 70 tesla pulsed field coil at the LNCMI. The hole doping of this O6.75 sample is p = 0.132±0.001 0.001 . Figure 8.15 shows the magnetoresistance up to 70 tesla, from 1.5 to 30 K. No quantum oscillations above the noise level are seen in this data, even after background subtraction. The magnetoresistance shows slow, broad oscillations at 20 K—qualitatively similar to the slow oscillations seen in O6.67 . Quantum oscillations are damped out by short-range impurity scattering and by longer range defects such as sample inhomogeneity, whereas the slow oscillations described in Grigoriev [31] are only scattered by short-range impurities.  Resistance Ohms  0.2 0.18  1.5 K  0.16  4.2 K  0.14  20 K  0.12  30 K  0.1 0.08 0.06 0.04 0.02 0. 0  5  10  15  20  25  30  35  40  45  50  55  60  65  70  Magnetic Field Tesla Figure 8.15: Magnetoresistance of YBa2 Cu3 O6.75 to 70 tesla, from 1.5 to 30 K. No quantum oscillations are visible. The small bumps near 65 tesla are due to vibrations of the cryostat during the magnetic field pulse. 135  8.5. Oxygen 6.75  Resistance Ohms  0.18 0.16  60 Tesla  0.14  0 Tesla  0.12 0.1 0.08 0.06 0.04 0.02 0. 0  20  40  60  80  100 120 140 160 180 200 220 240 260  Temperature Kelvin Figure 8.16: The zero field resistance from 4.2 to 270 K, and the 60 tesla resistance value at 1.5, 4.2, 20, and 30 kelvin. The resistance at low temperature does not turn over—a feature this sample shares with the other doping with no oscillations: O6.47 .  136  8.5. Oxygen 6.75  40  Vortex Melting Field Tesla  35 30 25 20 15 10 5 0 0  5  10  15  20  25  30  35  40  45  50  55  60  65  70  75  Temperature Kelvin Figure 8.17: The vortex-lattice melting transition field as a function of temperature up to Tc . The maximum field value is greater than it was for O6.67 , reversing the trend of decreasing Bmelt with increasing doping from O6.47 to O6.67 .  137  8.6. Conclusions from the Doping Dependence  8.6  Conclusions from the Doping Dependence  In this chapter three different material properties have been tracked as a function of doping: the Fermi surface as measured by quantum oscillations, the magnetoresistance as a function of temperature, and the behaviour of the vortex-lattice as it melts with applied field. The Fermi surface area increases with increased hole doping, while the cyclotron effective mass is seen to decrease. For doping levels that show quantum oscillations, the background magnetoresistance (at 50 tesla) increases as the temperature is decreased down to around 10 to 20 kelvin (depending on the doping), at which point the resistance “turns over” and begins decreasing as the temperature is lowered. This behaviour has been associated with the crossover between incoherent and coherent transport along the cˆ-axis. A more comprehensive examination of the magnetoresistance can be found in Vignolle et al. [104]. The vortex lattice melting transition was studied near optimal doping by [4], but I could find no systematic doping study on the underdoped side of the phase diagram. A preliminary analysis of the melting transition data is presented in subsection 8.6.2.  8.6.1  Doping Dependence of the Fermi Surface  All of the fit parameters for all of the samples that showed quantum oscillations are compiled in Table 8.3. It is worthwhile checking if the change in oscillation frequencies between O6.51 and O6.67 agrees with change in doping, as it should if Luttinger’s theorem is valid. The larger unwarped pocket, surface 2, increases in area from 530 tesla to 611 tesla from O6.51 to O6.67 . The smaller warped pocket, surface 2, increases in average area from 470 telsa to 501 tesla over the same range in doping. Using the same model for the Fermi surface used in subsection 7.4.2, with four unwarped hole pockets and two warped electron pockets in the Brillouin zone, the hole doping corresponding to these Fermi surface areas can be calculated. For O6.51 this gives p = 0.091, and for O6.67 this gives p = 0.10376. This is a hole doping increase of about 13%. If Luttinger’s theorem holds, then this should be equal to the actual hole doping values of p = 0.092 and p = 0.116, for O6.51 and O6.67 respectively, which is an increase of 23%. The calculated doping is off by about 10% for O6.67 , and the relative change in doping is off by even more. This is of course not the only possible model for the Fermi surface of YBCO, and there could be pockets of Fermi surface with high scattering rates that can’t be seen by quantum oscillation measurements. It is also possible that Luttinger’s theorem does not hold for underdoped YBCO. Luttinger’s theorem is quite general, and states that if the electron density in a material is held constant, then the volume enclosed by the Fermi surface if the electrons are noninteracting is the same as the volume enclosed by the Fermi surface of the quasiparticles when interactions are turned on [61]. This theorem relies on the system not undergoing a phase transition when the interactions are turned on: if the system becomes gapped— to an antiferromagnetic insulator, for example—then there is no longer a Fermi surface and 138  8.6. Conclusions from the Doping Dependence YBa2 Cu3 O6.51  YBa2 Cu3 O6.59  YBa2 Cu3 O6.67  Parameter  Surface 1  Surface 2  Surface 1  Surface 2  Surface 1  Surface 2  A F (tesla) ∆F (tesla) τ (picoseconds) φ ms (me ) m (me ) gmin  0.776 429.6 51.4 0.141 3.060 — 2.06 —  0.896 530.6 ∼0 0.138 5.661 — 2.20 —  0.564 478.4 40.1 0.271 1.426 1.987 1.52 1.53  1.109 523.1 ∼0 0.190 1.263 1.575 1.65 1.91  0.518 500.1 48.9 0.084 5.152 — 1.52 —  0.447 611.4 ∼0 0.065 5.174 — 1.52 —  Table 8.3: The Fermi surface parameters for YBa2 Cu3 O6.51 , YBa2 Cu3 O6.59 , and YBa2 Cu3 O6.67 . The parameter gmin is the minimum value for g assuming zero electronphonon coupling, as described in subsection 7.4.4.  Luttinger’s theorem obviously no longer applies. In YBCO the copper-oxygen chains are conducting near O7 and this leads to a large resistive anisotropy [5, 34, 43]. At lower doping levels, however, the resistive anisotropy decreases as vacancies are introduced into the chains, and at low temperatures the chains appear to localize and are no longer conducting [5, 20]. It is not clear in this whether Luttinger’s theorem still applies, since the chains have undergone a sort of metal-to-insulator transition as the disorder potential is introduced into the chains. The best way to resolve questions about the number of Fermi surface pockets and the validity of Luttinger’s theorem in underdoped YBCO is to perform quantum oscillation measurements across the entire phase diagram at higher fields and including the angle dependence. If all pockets can be identified and tracked as a function of doping then Luttinger’s rule can be checked. AMRO measurements would also be extremely helpful, since they provide much better resolution of the in-plane Fermi surface shape than quantum oscillation measurements. More accurate Fermi surface shape information can help locate the pockets in the Brillouin zone based on symmetry considerations, and different reconstruction scenarios for the Fermi surface can then be ruled out.  8.6.2  Doping Dependence of the Vortex-Lattice Melting Behaviour  A calculation of Bmelt —the vortex lattice melting field—can be performed using the Lindemann criterion, which states that a crystal melts when the average thermal displacement is some fraction of the lattice spacing [11, 26]. The exact value of this fraction depends on the dimensionality of the fluctuations governing the melting of the lattice (which can vary between 2D and 3D in the cuprates). For vortices in a high temperature superconductor, two factors determine which dimensionality regime the vortex lattice is in: the anisotropy  139  8.6. Conclusions from the Doping Dependence parameter ≡ ter,  Φ0 B /d  values of  λab λc ,  [26].  and the ratio of the vortex lattice constant to the cˆ-axis lattice parameis a measure of how two dimensional the superconductivity is, with small  for layered superconductors coupled weakly along the cˆ-axis. Since vortices in  quasi-2D superconductors can be thought of as occurring mostly in the planes with couΦ0 B /d  pling along the cˆ-axis,  is a measure of whether vortices will interact more with their  in-plane neighbours, or with the vortices they are coupled to along the caxis. The product of these two parameters determines whether the vortex lattice melting will be two or three dimensional. ˚ ˚ = 0.026 [39, 96]. Thus the = 1699A/65200 A  For YBa2 Cu3 O6.59 , the anisotropy is inequality from Fisher et al. [26], 1  Φ0 /d, B  (8.1)  is satisfied for YBa2 Cu3 O6.59 whenever B > 1.05 tesla. This means that for almost all of the field range up to 60 tesla, it is the two dimensional fluctuations of the vortex lattice that determine the melting temperature. In this two dimensional regime, the vortex-melting field is given as [11] 4Θ2  Bmelt (T ) ≈ Hc2 (T = 0) 1+  2.  (8.2)  1 + 4ΘTs /T  Hc2 (T = 0) is the zero temperature mean-field critical field value, and Θ and Ts are temperature scales given by Θ = c2L Ts = Tc c2L  Tc −1 , T  βm /Gi  (8.3)  βm /Gi.  (8.4)  Gi is the Ginsburg number, which characterizes the width of the region around Tc where thermal fluctuations are important. It is given by 1 Gi = 2  kB Tc 4π 2 3 µ0 Hc ξ  2  ≈  16.7325 Tc λab λc ξ  2  ,  (8.5)  where λab and λc are the penetration depths parallel and perpendicular to the a ˆ-ˆb-plane at zero temperature, and I have used the definition of Hc to replace it with the penetration depths. The penetration depth values are taken from µSR, electron-spin resonance, and infrared reflectance measurements [39, 75, 96]. The other parameters in Equation 8.3 and Equation 8.4 are βm , a constant equal to 5.6 [11], and cL —the Lindemann number— which varies between 0.2 and 0.4 depending on the specific model of the melting [11, 26]. Equation 8.2 is not expected to be valid very close to Hc2 where the vortex cores strongly  140  8.6. Conclusions from the Doping Dependence  Vortex Melting Field Tesla  25  Tmin 6 K, Tmax 30 K Tmin 15 K, Tmax 30 K  20  15  10  5  0 0  5  10  15  20  25  30  35  40  45  50  55  60  65  Temperature Kelvin Figure 8.18: The vortex lattice melting field for YBa2 Cu3 O6.59 , with fits to Equation 8.2. The red and black curves correspond to different bounds on the data, as indicated in the legend. These fits are made with cL = 0.3 and a single fit parameter, ξ. The fit values are ξ = 37 ˚ A for the upper curve and ξ = 38.5 ˚ A for the lower curve. overlap. Fitting the melting transition data for YBa2 Cu3 O6.59 , shown in Figure 8.10, with the coherence length ξ as the only free parameter and cL = 0.3 gives ξ = 37 ± 6 ˚ A. Data below 5 tesla were excluded to ensure that the vortices are in the 2D regime. The error bars on ξ are dominated by the choice of cL , with the upper range of ξ coming from choosing cL = 0.2, and the lower limit from cL = 0.4. Equation 8.2 is not expected to hold when B ≈ Hc2 , but ξ is relatively insensitive to where this cutoff is made, as seen in Figure 8.18. The effects of disorder have been ignored; Blatter et al. [11] points out that the vortex lattice correlation length, which depends on the disorder, is much larger than the vortex lattice constant. Since the melting transition depends on the length scale of the vortex lattice constant, disorder should only weakly affect the value of the melting transition. At very low temperatures it is possible that the vortex lattice first depins before melting, and the onset of resistivity is actually the depinning line. Because thermal motion causes the vortices to effectively average the disorder in an area around the vortex, increasing temperature actually increases the depinning field [11]. Rather than try to counter every possible argument against using Equation 8.2, for which there are probably many, I point to the quality of the single-parameter fits shown in Figure 8.18 as evidence that this avenue at  141  8.6. Conclusions from the Doping Dependence Oxygen Content 6.47 6.51 6.59 6.67 6.75  Doping  ξ (˚ A)  0.088 0.092 0.105 0.116 0.132  28 ± 0.5 33 ± 0.5 37 ± 0.5 38 ± 0.5 32 ± 0.5  Table 8.4: The superconducting correlation length as obtained by fitting the vortex lattice melting curves to Equation 8.2, with cL = 0.3. There is an absolute uncertainty of ±6 ˚ A for the ξ values, coming from the choice of cL between 0.2 and 0.4, but for a fixed cL the uncertainty is relatively small.  ˚ is in reasonable agreement least warrants further exploration. The fit value of ξ = 37 ± 6 A ˚ coherence length measured in the underdoped regime by Ando and Segawa with the 20-40 A [3] via fits to the high temperature magnetoresistance. Table 8.4 gives the correlation lengths for each of the dopings measured by fitting the melting transition to Equation 8.2. An uncertainty of about ±6 ˚ A can be associated with these values because ξ obtained from the fit depends on the choice of cL . The relative change as a function of doping, however, is preserved as long as the same cL value is ˚, with this used for each doping. For a fixed cL , the uncertainty is approximately ±0.5 A uncertainty coming from the temperature range over which the data is fit. The values for λab were taken from Pereg-Barnea et al. [75]and Sonier et al. [96], and the values for λc were taken from Homes et al. [39]. Interpolated values for the penetration depth when the exact doping levels were not available. Penetration depths as a function of hole doping, for both the cˆ-axis and the a ˆ-ˆb-plane, are plotted in Figure 8.19. The relatively small uncertainty of ±0.5 ˚ A reflects the fact that the fits are quite insensitive to the choice of λab , λc , and the temperature window. Figure 8.20 shows these coherence lengths plotted as a function of hole doping, along with the deviation of Tc from the function 1−Tc /Tc,max = 82.6(p−0.16)2 [59]. The maximum coherence length roughly corresponds to the maximum suppression in Tc from the parabolic form, and this suppression of Tc is generally thought to be due to the competition of some sort of order (e.g., stripe order) with superconductivity [54]. Since the penetration depth is used in the extraction of the coherence length (see Equation 8.2), it is important to check that there is no anomaly in λ inducing the peak seen in ξ. Figure 8.19 shows that there is no such anomaly in λ for both the cˆ-axis and the a ˆ-ˆb-plane, giving support to the argument that the peak seen in ξ around 1/8th hole doping is real. It is clear that there is interesting physics to be explored here, and a more complete doping and temperature dependence should be measured along with a more careful examination of whether the use of Equation 8.2 is valid.  142  8.6. Conclusions from the Doping Dependence  9  18  Λc  7  Λc Μm  16  Tc  14  6  12  5  10  4  8  3  6  2  4  1  2  0  0  0.08  0.1  0.12  0.14  0.16  Tc K  8  0.18  Hole Doping p 190  18 16  Tc  170  Λab ESR  14  160  Λab ΜSR  12  150  10  140  8  130  6  120  4  110  2  100  0  90 0.08  0.1  0.12  0.14  0.16  Tc K  Λab nm  180  0.18  Hole Doping p Figure 8.19: Top: The cˆ-axis magnetic penetration depth for YBCO as a function of hole doping, shown in red, as measured by infrared reflectance [39]. The dashed red line is a guide to the eye. The deviation in Tc from the parabolic form (see Figure 1.3) is also shown, in order to illustrate that the penetration depth shows no anomaly centred around 1/8th doping. Bottom: The a ˆ-ˆb-plane penetration depth, as measured by electron-spin resonance (ESR) and muon-spin rotation (µSR) [75, 96]. The dashed red line is a guide to the eye. Although the data is sparse near 1/8th , it is clear that there is no peak in λab , as there is in ∆Tc .  143  8.6. Conclusions from the Doping Dependence  40  40  Tc Ξ  30  20  20  10  10  0  0  Ξ  Tc Kelvin  30  0.06  0.08  0.1  0.12  0.14  0.16  0.18  Hole Doping p Figure 8.20: The superconducting coherence length as a function of doping, along with the deviation in transition temperature from the parabolic form of Tc vs p (the parabola can be seen in the phase diagram in Figure 1.3). The peak in coherence length roughly corresponds to the maximum in the suppression of Tc , which is thought to be caused by the competition of superconductivity with some other form of electronic order [54].  144  Chapter 9  In-Plane Transport Quantum oscillations were first seen in the underdoped cuprates in Rxy , the Hall channel of the resistivity, of YBa2 Cu3 O6.50 [22]. The fact that the Hall coefficient is negative at low temperature and high fields implies that electron-like quasiparticles, not holes, are the dominant carrier type in this field-temperature regime. This is in contrast with the hole pockets that are expected if the large hole Fermi surface seen in overdoped Tl2 Ba2 CuO6+x is extrapolated down to the doping level of YBa2 Cu3 O6.50 . Various scenarios of how the Fermi surface can be reconstructed to give electron pockets have been put forth, but almost all rely on some sort of symmetry breaking order (rotational or translational in most cases), which folds the Fermi surface into new shapes and opens up gaps. The negative Hall coefficient disappears at high temperatures [53], implying either that the mobility of the electron pocket is strongly temperature dependent (see problem 4 in chapter 12 of Ashcroft and Mermin [6]), or that the Fermi surface reconstruction only exists at low temperature (below some onset temperature of the symmetry-breaking order). Whatever the case may be, by measuring a large number of samples from oxygen content 6.47 to 6.85, the temperature at which the Hall coefficient becomes negative at high field was mapped out. This temperature is designated T0 . The entirety of this work was published in LeBoeuf et al. [54]; I will only discuss the dopings which I was directly involved in measuring (I prepared the samples for the entire doping range), and the unique procedure by which the doping was fine-tuned. In this chapter I will first take a look at the mechanism behind the doping of YBCO, and see how it can be exploited to finely tune the hole doping without changing the oxygen content of a crystal.  9.1  Order-Disorder Doping  YBCO has two very different copper-containing layers: the copper oxygen planes, which are ubiquitous among the cuprates, and the copper oxygen chains, which are unique to YBCO and provide its unique doping mechanism. At oxygen 6.0 the copper in the plane is in a 2+ oxidation state, with a nominal electronic configuration of [Ar]3d9 (see Damascelli et al. [19] for a brief review). The five 3d orbitals are split into two higher energy eg and three lower energy t2g orbitals, which then hybridize with the six oxygen p orbitals (three from each oxygen, two oxygen per planar copper) and form bands. The naive end result is that 145  9.1. Order-Disorder Doping the hybridized dx2 −y2 orbital ends up with the “ninth electron”, resulting in a half filled band and a metallic state. Conductivity, however, requires a charge carrier to hop from one copper to another, and the resulting double occupancy is too energetically “expensive” for this orbital. This results in localization of the holes and the material is an insulator [72]. Doping holes into the copper-oxygen plane produces empty sites on some of the copper atoms, and then double occupancy can be avoided while the holes are free to move around. The mechanism for doping in YBCO is to introduce oxygen into the chain layer, whose coppers are in the 1+ oxidation state at oxygen 6.0. The simplest picture is that the addition of the first oxygen atom to a copper chain results in the oxygen taking an electron from each neighbouring copper, putting the oxygen atom in its desired 2- oxidation state and the two copper atoms in the 2+ state. When the next oxygen atom is added to this “chainlet” on the other side of the copper, another electron cannot be taken from the copper since it is already in the 2+ state. An electron instead comes from the copper oxygen plane, which dopes the plane with a hole. This is repeated as more and more oxygen is added to the chains, resulting in further hole doping of the planes. Instead of a random distribution of chainlets, these chains have a tendency to form ordered states in the a-b plane, resulting in alternating patterns of full and empty chains [59, 105]. For example, at oxygen 6.50, the chains alternate full and empty, producing what is known as the ortho- II structure. This is the oxygen structure of the YBa2 Cu3 O6.59 sample studied in chapter 7. A diagram of the ordered states is shown in Figure 4.7. The ordered oxygen superstructures are energetically preferred, but at finite temperature there will always be some oxygen atoms that are in a nominally empty chain, and there will be some vacancies in an otherwise full one. Further, YBCO can be produced at any oxygen concentration between 6.0 and 7.0, requiring extra oxygen atoms to go into the empty chains when not at an exact oxygen content for an ordered state (i.e., 6.51 has 1 extra oxygen per 100 unit cells, adding disorder to the Ortho-II structure by putting oxygen in some of the empty chains). It should be immediately apparent that the exact doping level in the planes is strongly dependent not only on the oxygen content in the chains, but also on the degree of order that they possess. In this somewhat simplified picture of doping in YBCO, each oxygen missing from an otherwise full chain results in two fewer holes in the plane. This is because a missing oxygen results in two dangling chain ends, where the coppers are content to remain in the 2+ since they each only have one associated oxygen atom. If the missing oxygen is sitting in an empty chain, then it dopes no holes into the plane because it satisfies its -2 charge by taking electrons from the empty-chain coppers. The details turn out to be more complicated: there are two planes to dope holes into per chain; each oxygen in a chain does not dope an integer number of holes into the plane; excess oxygen does not enter the empty chains completely randomly, but tends to form clusters. Nevertheless, it is clear that chain oxygen order does have a strong effect on the hole doping of the planes, and this is born out in experiment (see Bobowski et al. [12], Liang et al. [59].) 146  9.1. Order-Disorder Doping  YBa2 Cu3 O6.49 YBa2 Cu3 O6.51  Ordered  Disordered  Re-ordered  0.0922 0.0936  0.0861 0.0869  0.0875 0.0900  Table 9.1: The various hole doping values obtained during the order-disorder experiment. Note that “re-ordered” is not supposed to imply that the sample is as ordered as it was to begin with; this would require a much longer annealing period than was used to obtain doping levels in the third column.  YBCO samples used in transport experiments usually have their oxygen annealed into the oxygen superstructures to reduce quasiparticle scattering, but the order-dependent doping phenomenon can be used to finely control the hole doping. By heating YBa2 Cu3 O6.50 above 150 Celsius the ortho-I state is entered, where the oxygen is distributed randomly between all chains [105]. This was performed in a furnace with flowing oxygen gas to provide a dry environment (to prevent surface degradation). The samples and the quartz tube they were held in were then immersed in ice water, quickly bringing their temperature down to where the chain oxygen become relatively immobile. This quenching step “locks in” the disorder introduced at high temperature, giving the sample a lower hole doping while maintaining the same oxygen content. After measurement, the samples can then be brought up to just below the ortho-I → ortho-II transition temperature (85 Celsius is typical), and held there while the ortho-II chain order re-forms. This annealing procedure can be stopped at any time by cooling the sample again, allowing access to a range of dopings between the disordered and the fully ordered state. The size of this range depends on how quickly the sample can be quenched from high temperature: the quicker the quenching the more disorder that will remain, and the lower the doping. Since the oxygen is mobile even at room temperature, the sample must be mounted in the apparatus and brought down to liquid nitrogen temperature as quickly as possible (there is evidence that the chain oxygen in YBCO freezes out in a glassy transition just below room temperature, see Nagel et al. [67].) Figure 9.1 shows the effect this disorder and re-order procedure has on the Tc of O6.49 and O6.51 . The Tc was changed by about 4 K for each sample. The actual changes in hole doping, based on Liang et al. [59], are shown in Table 9.1. Below are figures showing the Hall coefficient data for the two samples, at each doping level in Table 9.1. Figure 9.2, Figure 9.3, and Figure 9.4 show the resistance and Hall data for O6.49 , and Figure 9.5, Figure 9.6, and Figure 9.7 show the resistance and Hall data for O6.51 .  147  9.1. Order-Disorder Doping  0.14  Ordered Disordered Reordered  0.12  Rxx Ohms  0.1 0.08 0.06 0.04 0.02 0. 40  45  50  55  60  65  70  75  80  70  75  80  Temperature Kelvin 0.14  Ordered Disordered Reordered  0.12  Rxx Ohms  0.1 0.08 0.06 0.04 0.02 0. 40  45  50  55  60  65  Temperature Kelvin Figure 9.1: Top panel : YBa2 Cu3 O6.49 in-plane resistance as a function of temperature for the ordered sample, the disordered sample, and the same sample again after reordering. Bottom panel : The same as above but for YBa2 Cu3 O6.51 .  148  Rxx Ohms  9.1. Order-Disorder Doping  0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0. 0  1.5 K 20 K 30 K 100 K  5  10  15  20  25  30  35  40  45  50  55  60  RH x 10  3  m3 C  Magnetic Field Tesla 1.2 1. 0.8 0.6 0.4 0.2 0. 0.2 0.4 0.6 0.8 1. 1.2 1.4 1.6 1.8 0  1.5 K 20 K 30 K 100 K  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 9.2: Top panel : YBa2 Cu3 O6.49 in-plane resistivity as a function of magnetic field at 1.5, 20, 30, and 100K. Note the strong magnetoresistance at low temperature. Bottom panel : The hall coefficient, Rxy /B, as a function of magnetic field at the same temperatures shown in the top panel. Quantum oscillations are barely visible at the highest field and lowest temperature. The Hall data has been truncated around 20 tesla because otherwise noise (from dividing by small field values to obtain RH ) obscures the plot. 149  Rxx Ohms  9.1. Order-Disorder Doping  0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0. 0  1.5 K 4.2 K 10 K 30 K 50 K 85 K 100 K 150 K 200 K 250 K  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla 1.5 K 4.2 K 10 K 30 K 50 K 85 K 100 K 150 K 200 K 250 K  1.  0.6  RH x 10  3  m3 C  0.8  0.4 0.2 0. 0.2 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 9.3: The same YBa2 Cu3 O6.49 sample shown in Figure 9.2, except that now the sample has been disordered and the doping is lower. Note that the Hall coefficient is now positive at high fields, at all temperatures.  150  Rxx Ohms  9.1. Order-Disorder Doping  0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0. 0  1.5 K 10 K 20 K 40 K 100 K  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla  1.  0.6  RH x 10  3  m3 C  0.8  0.4 1.5 K  0.2  10 K  0.  20 K 40 K  0.2 0.4 0  100 K  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 9.4: The same YBa2 Cu3 O6.49 sample as the previous two figures, but with the chain oxygen partially reordered. The negative Hall coefficient at high fields and low temperatures has returned.  151  9.1. Order-Disorder Doping  0.2 0.18  1.5 K  0.16  20 K  0.14  30 K  Rxx Ohms  100 K  0.12 0.1 0.08 0.06 0.04 0.02 0. 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla  0.4  0.  RH x 10  3  m3 C  0.2  0.2 1.5 K  0.4  20 K  0.6  30 K 100 K  0.8 1. 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 9.5: Top panel : YBa2 Cu3 O6.51 in-plane resistance as a function of magnetic field at 1.5, 20, 30, and 100 K. Bottom panel : Hall coefficient. Quantum oscillations are clearly visible at the lowest temperature and highest field.  152  9.1. Order-Disorder Doping  0.22 0.2 0.18  Rxx Ohms  0.16 0.14 0.12 1.5 K 4.2 K 10 K 30 K 50 K 85 K 100 K 150 K 200 K 250 K  0.1 0.08 0.06 0.04 0.02 0. 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla 1.5 K 4.2 K 10 K 30 K 50 K 85 K 100 K 150 K 200 K 250 K  RH x 10  3  m3 C  0.4  0.2  0.  0.2 0  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 9.6: Top panel : The same YBa2 Cu3 O6.51 sample as Figure 9.5, but with the oxygen disordered. Bottom panel : The Hall coefficient is now positive at high field, at all temperatures. The noise has also increased due to sample contact degradation while in the furnace.  153  9.1. Order-Disorder Doping  Rxx Ohms  0.26 0.24  1.5 K  0.22  10 K  0.2  20 K  0.18  40 K  0.16  100 K  0.14 0.12 0.1 0.08 0.06 0.04 0.02 0. 0  5  10  15  20  25  30  35  40  45  50  55  60  RH x 10  3  m3 C  Magnetic Field Tesla 1.4 1.2 1. 0.8 0.6 0.4 0.2 0. 0.2 0.4 0.6 0.8 1. 1.2 1.4 0  1.5 K 10 K 20 K 40 K 100 K  5  10  15  20  25  30  35  40  45  50  55  60  Magnetic Field Tesla Figure 9.7: Top panel : The in-plane magnetoresistance of the same YBa2 Cu3 O6.51 sample as above, but with the oxygen partially reordered. Bottom panel : The Hall coefficient, which is negative again at high fields and low temperatures.  154  9.2. Conclusions from the Hall Experiment  9.2  Conclusions from the Hall Experiment  The figures above show a clear trend: the temperature at which the Hall coefficient changes sign (defined as T0 in accordance with LeBoeuf et al. [54]) increases with increased hole doping. The extraction of the T0 value is a bit difficult with data taken at only a few temperatures, but linear extrapolation was used and error bars are included accordingly. If T0 is then plotted as a function of doping— Figure 9.8—along with data from different dopings as reported by LeBoeuf et al. [54], a peak in T0 is seen around 1/8th hole doping. This doping value is also at the point of maximum Tc suppression from the otherwise parabolic shape of Tc vs p [59]. The fact that La1.8−x Eu0.2 Srx CuO4 also sees similar behaviour in the Seebeck and Nernst coefficients (as does YBCO), as well as exhibiting charge stripe order, has led to the supposition that YBCO also exhibits stripe order. This would lead to a breaking of translational symmetry, reconstructing the Fermi surface and opening up gaps, resulting in electron pockets which give the negative Hall coefficient [52]. There are many more details to this argument, but it is not the subject of this thesis and is fully discussed in Laliberte et al. [52], Chang et al. [15], and references therein. There is always the concern that when the sample disorder is increased, the resulting physics observed is due to the increase in disorder. In the case of this experiment, for example, it is possible that the increased disorder selectively inhibits the mobility of the electron pockets (relative to the mobility of the hole pockets.) This would result in the observed trend in T0 below oxygen 6.50, as well as the disappearance of quantum oscillations near this oxygen concentration. Above 0.11 holes per copper-oxygen plane, however, YBCO crosses through the ortho-V, ortho-VIII, and finally ortho-III states [105]. These states all have lower correlation lengths in their oxygen order than ortho-II does, yet T0 increases all the way up to the ortho-VIII state (which has the shortest coherence of all the ordered states.) Then T0 decreases as YBCO enters the ortho-III state, which has longer-range order than ortho-VIII. Thus the behaviour of T0 with doping does not support the idea that the electron pocket is selectively suppressed with increased disorder. The behaviour of T0 with doping does, on the other hand, support the idea that the stripe potential centred around 1/8th doping, known to exist in other cuprates, is responsible for the Fermi surface reconstruction that gives rise to the electron pocket seen in YBCO.  155  9.2. Conclusions from the Hall Experiment  100  Tc T0 this work T0 other work  90 80  Tc,T0 Kelvin  70 60 50 40 30 20 10 0 0.  0.02  0.04  0.06  0.08  0.1  0.12  0.14  0.16  0.18  0.2  Hole Doping p Figure 9.8: The temperature T0 where the Hall coefficient changes sign at high field as a function of hole doping. Tc is also included in this plot for reference. The green data points are from this work, and the blue data is taken from [54].  156  Chapter 10  Conclusion This thesis studied the electrical transport properties of YBCO in high magnetic fields, focusing on Fermi surface properties measured by the SdH effect. An emphasis was placed on careful sample preparation and customized data analysis, both of which were very important in the extraction of the underlying physics of YBCO. The extension of the Lifshitz-Kosevich model to quasi-2D systems presented in chapter 2 is essentially a free-electron-gas model, with renormalized parameters put in by hand to account for interactions. While Luttinger showed that the frequency of the oscillations should remain un-renormalized with the introduction of electron-electron interactions [62], strong interactions could lead to appreciable changes in the functional form of the amplitude factors for scattering, temperature, and spin, especially in the strongly-two-dimensional limit [99]. The fact that the formulas presented in section 2.6 fit the data so well over a wide range of magnetic field, angle, and temperature seems to indicate that underdoped YBCO is well described by a Fermi liquid picture—at least above 24 tesla and below 10 kelvin. Analysis of the angular dependence of the oscillations in YBa2 Cu3 O6.59 found the Fermi surface to be comprised of quasi-2D cylinders, as is expected for a strongly anisotropic material. The genetic algorithm developed specifically for the analysis of this data found two Fermi pockets—one with kz dispersion, and one without. The most important result to come out of this analysis is the value for g for the two surfaces, which is g ≥ 1.53 for the warped surface and g ≥ 1.91 for the unwarped surface. These numbers suggest that strong in-plane spin order is not present at this doping level, and that the Fermi surface reconstruction may instead be due to charge or d-density wave order [28, 68]. Spin density wave order with the spin moments aligned with the magnetic field are also still a possibility [68]. The genetic algorithm itself is an important advancement in the analysis of quantum oscillations. For very closely spaced frequencies, or for small pieces of Fermi surface that give very few periods of oscillation over available magnetic fields, this algorithm allows a unique determination of the Fermi surface geometry. This algorithm works even better when the angular dependence of the oscillations is available, since Fermi pockets that have similar sizes at zero angle may still have very different angular dependences due to different warping sizes or different g factors. By fitting the temperature, field, and angle dependence with a single set of parameters, the cyclotron effective mass m can be extracted separately 157  10.1. Looking Forward for each piece of Fermi surface, even if they overlap in frequency space. The algorithm offers a 5×105 improvement in speed over the implementation in Mathematica, and makes the fitting of large data sets to complicated models actually feasible. The preparation of sub-millimetre gold electrical contacts helped produce the high level of signal to noise and reproducibility of the data required for the detailed analysis. For the in-plane geometry used in chapter 9, the three-dimensional nature of the contacts ensures that there is no cˆ-axis contamination to the in-plane signal. For the cˆ-axis geometry, the same is true but for a ˆ-ˆb-plane contamination. The techniques described in chapter 4 can easily be transferred to other condensed matter systems. The observation of quantum oscillations in the ortho-VIII state is important not only because it gives information about the Fermi surface’s evolution with doping, but also because it suggests that the Fermi pockets are not created by a potential induced by ortho-II order. These oscillations in the ortho-VIII state, along with the reasons given in subsection 7.4.3, combine to give compelling evidence that the quantum oscillations seen in the two YBCO materials are not just a product of their unique chain structures, but are in fact an inherent property of the copper-oxygen planes contained in all cuprates. Oxygen order-disorder doping was shown in chapter 8 to be a valuable technique for tuning the doping in fine steps around a point of interest. If a quantum critical point near optimal doping is ever verified, where T goes to zero, this technique in combination with pressure doping could allow the critical point to be traversed with a single sample.  10.1  Looking Forward  There are three main questions with respect to quantum oscillation measurements in the cuprates that still need to be addressed: • How do the oscillations evolve with doping, and do the small pockets turn into the large hole pocket somewhere near optimal doping? • Can quantum oscillations be seen in other underdoped cuprate materials, or are they unique to the YBCO compounds? • Are the pockets just a high field phenomenon, and perhaps more importantly, will studying them help lead to a microscopic understanding of high-Tc ? These are tough questions, and it is easy to get lost performing experiments on a very narrow region of the underdoped side of the phase diagram without thinking of this big picture. The first of the questions will be answered by going to higher magnetic fields. The LNCMI in Toulouse recently started operating an 80 tesla magnet, and the National High Magnetic Field Lab at Los Alamos recently test fired a magnet at 100 tesla. The hope is that 158  10.1. Looking Forward eventually oscillations will be observable at these fields in ortho-III ordered YBa2 Cu3 O6.75 . Ortho-I ordered YBCO should be the cleanest and have the lowest quasiparticle scattering, but with a mean-field Hc2 near 200 tesla it may never be possible to melt the vortex lattice at low temperature. A promising area of research could be using pressure to increase the hole doping in ortho-II YBCO without increasing the oxygen (and the disorder). Recent preliminary measurements in collaboration with Cyril Proust’s group in Toulouse have shown some promising results, but it is difficult to produce quality electrical contacts that survive the application of pressure. The application of pressure can also produce cleaner oxygen ordered states, which may enable higher quality oscillations to be observed in ortho-VIII at O6.67 . The current challenge is building a pressure probe that can survive the annealing temperature of 80◦ C. With respect to contacts, I think that the citric acid etch described in subsection 4.4.1 is quite promising, and should give higher-stability contacts that will survive the application of high temperatures and pressures. The second question is a frustrating one because it is the unique doping mechanism of YBCO that makes them clean enough to observe quantum oscillations in the first place. Other cation-substituted cuprates have very strong disorder from the large rare-earth elements introduced right next to the planes, and it is not clear that oscillations will ever be visible in these materials. Tl2 Ba2 CuO6+x is probably the most promising place to look given that oscillations have already been seen in this material. What has not yet been studied in Tl2 Ba2 CuO6+x are compounds close to optimal doping where the Fermi surface should reconstruct. Obtaining high-quality low-doping Tl2 Ba2 CuO6+x samples should be a high priority for continuing the quantum oscillation story in the cuprates. The first half of the third question can be addressed by high-field x-ray spectroscopy and neutron diffraction measurements, looking for the onset of spin and/or charge order in a magnetic field. Quantum oscillations have been observed in the vortex state of more classic type-II superconductors (Nb3 Sn, NbSe2 , V3 Si, and URu2 Si2 , for example [35, 47, 70]) , and similar measurements in YBCO could either reveal a transition in field where the oscillations disappear, or could at least put an upper bound on where such a transition could occur. Extremely high-sensitivity dHvA measurements are required if a signal is to be seen in this type of a measurement, and would represent a significant experimental triumph if oscillations could be seen in the vortex-solid state of YBCO. The second half of the third question is the toughest of all, and in all likelihood will not be answered until the mechanism of high-Tc is discovered. It is possible that the quantum oscillations are simply a distraction in the underdoped regime, and that superconductivity does not care about what ground state the cuprates happen to chose when a magnetic field is applied. However, the fact that quantum oscillations reveal quasiparticles at all is somewhat of a miracle given the strongly correlated physics seen by ARPES in the pseudogap region. If understanding the pseudogap really is the key to understanding high-Tc , then the quantum oscillation measurements need to be reconciled with all of the spectroscopy measurements 159  10.1. Looking Forward in the pseudogap region before progress on high-Tc can be made.  160  Bibliography [1] EN Adams and TD Holstein. Quantum theory of transverse galvano-magnetic phenomena. Journal of Physics and Chemistry of Solids, 10(4):254–276, 1959. ISSN 0022-3697. doi:10.1016/0022-3697(59)90002-2. [2] Pw Anderson. New approach to the theory of superexchange interactions. Physical Review, 115(1):2–13, 1959. ISSN 0031-899X. doi:10.1103/PhysRev.115.2. [3] Yoichi Ando and Kouji Segawa. Magnetoresistance of untwinned yba2 cu3 Oy single crystals in a wide range of doping: Anomalous hole-doping dependence of the coherence length. Phys. Rev. Lett., 88:167005, Apr 2002. doi:10.1103/PhysRevLett.88. 167005. URL http://link.aps.org/doi/10.1103/PhysRevLett.88.167005. [4] Yoichi Ando, G. S. Boebinger, A. Passner, L. F. Schneemeyer, T. Kimura, M. Okuya, S. Watauchi, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, and S. Uchida. Resistive upper critical fields and irreversibility lines of optimally doped high-Tc cuprates. Phys. Rev. B, 60:12475–12479, Nov 1999. doi:10.1103/PhysRevB.60.12475. URL http://link.aps.org/doi/10.1103/PhysRevB.60.12475. [5] Yoichi Ando, Kouji Segawa, Seiki Komiya, and A. N. Lavrov. Electrical resistivity anisotropy from self-organized one dimensionality in high-temperature superconductors. Phys. Rev. Lett., 88:137005, Mar 2002. doi:10.1103/PhysRevLett.88.137005. URL http://link.aps.org/doi/10.1103/PhysRevLett.88.137005. [6] N.W. Ashcroft and N.D. Mermin. Solid state physics. Holt-Saunders International Editions: Science : Physics. Holt, Rinehart and Winston, 1976. ISBN 9780030493461. [7] Alain Audouard, Cyril Jaudet, David Vignolles, Ruixing Liang, D. A. Bonn, W. N. Hardy, Louis Taillefer, and Cyril Proust.  Multiple quantum oscillations in the  de haas˘van alphen spectra of the underdoped high-temperature superconductor yba2 cu3 o6.5 . Phys. Rev. Lett., 103:157003, Oct 2009. doi:10.1103/PhysRevLett. 103.157003. URL http://link.aps.org/doi/10.1103/PhysRevLett.103.157003. [8] W. Banzhaf. Genetic Programming: An Introduction on the Automatic Evolution of Computer Programs and its Applications. Morgan Kaufmann Publishers, 1998. ISBN 9783920993584.  161  Bibliography [9] JG Bednorz and KA Muller. Possible high-tc superconductivity in the ba-la-cu-o system. Zeitschrift fur Physik B-Condensed Matter, 64(2):189–193, 1986. ISSN 07223277. doi:10.1007/BF01303701. [10] C Bergemann, AP Mackenzie, SR Julian, D Forsythe, and E Ohmichi. Quasi-twodimensional fermi liquid properties of the unconventional superconductor sr2ruo4. Advances in Physics, 52(7):639–725, Nov 2003. ISSN 0001-8732. doi:10.1080/ 00018730310001621737. [11] G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur. Vortices in high-temperature superconductors. Rev. Mod. Phys., 66:1125–1388, Oct 1994. doi:10.1103/RevModPhys.66.1125. URL http://link.aps.org/doi/10. 1103/RevModPhys.66.1125. [12] J. S. Bobowski, J. C. Baglo, James Day, Lynne Semple, P. Dosanjh, P. J. Turner, R. Harris, Ruixing Liang, D. A. Bonn, and W. N. Hardy. Oxygen chain disorder as the weak scattering source in yba2 cu3 o6.50 . Phys. Rev. B, 82:134526, Oct 2010. doi:10. 1103/PhysRevB.82.134526. URL http://link.aps.org/doi/10.1103/PhysRevB. 82.134526. [13] A. Carrington and E. A. Yelland. Band-structure calculations of fermi-surface pockets in ortho -ii yba2 cu3 o6.5 . Phys. Rev. B, 76:140508, Oct 2007. doi:10.1103/PhysRevB. 76.140508. URL http://link.aps.org/doi/10.1103/PhysRevB.76.140508. [14] J  Caulfield,  SJ  Blundell,  MSLD  Dejongh,  PTJ  Hendriks,  J  Singleton,  M Doporto, FL Pratt, A House, JAAJ Perenboom, W Hayes, M Kurmoo,  and  P  Day.  High-field  magnetoresistance  oscillations  on  alpha-  [bis(ethylenedithio)tetrathiafulvalene]2khg(scn)(4) - the effects of magnetic breakdown, exchange interactions, and fermi-surface reordering. Physical Review B, 51 (13):8325–8336, Apr 1 1995. ISSN 1098-0121. doi:10.1103/PhysRevB.51.8325. [15] J. Chang, R. Daou, Cyril Proust, David LeBoeuf, Nicolas Doiron-Leyraud, Francis Lalibert´e, B. Pingault, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, H. Takagi, A. B. Antunes, I. Sheikin, K. Behnia, and Louis Taillefer. Nernst and seebeck coefficients of the cuprate superconductor yba2 cu3 o6.67 : A study of fermi surface reconstruction. Phys. Rev. Lett., 104:057005, Feb 2010. doi:10.1103/PhysRevLett. 104.057005. URL http://link.aps.org/doi/10.1103/PhysRevLett.104.057005. [16] AV Chubukov and DK Morr. Electronic structure of underdoped cuprates. Physics Reports-Review Section Of Physics Letters, 288(1-6):355–387, SEP 1997. ISSN 03701573. doi:10.1016/S0370-1573(97)00033-1. [17] P. Coleman. Introduction to Many Body Physics. Cambridge University Press, 2008. ISBN 9780521864886. URL http://books.google.ca/books?id=PDg1HQAACAAJ. 162  Bibliography [18] A. Damascelli, D. H. Lu, K. M. Shen, N. P. Armitage, F. Ronning, D. L. Feng, C. Kim, Z.-X. Shen, T. Kimura, Y. Tokura, Z. Q. Mao, and Y. Maeno. Fermi surface, surface states, and surface reconstruction in sr2 ruo4 . Phys. Rev. Lett., 85:5194–5197, Dec 2000. doi:10.1103/PhysRevLett.85.5194. URL http://link.aps.org/doi/10. 1103/PhysRevLett.85.5194. [19] Andrea Damascelli, Zahid Hussain, and Zhi-Xun Shen. Angle-resolved photoemission studies of the cuprate superconductors. 2003.  doi:10.1103/RevModPhys.75.473.  Rev. Mod. Phys., 75:473–541, Apr  URL http://link.aps.org/doi/10.  1103/RevModPhys.75.473. [20] R. Daou, J. Chang, David LeBoeuf, Olivier Cyr-Choiniere, Francis Laliberte, Nicolas Doiron-Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis Taillefer. Broken rotational symmetry in the pseudogap phase of a high-t(c) superconductor. NATURE, 463(7280):519–522, JAN 28 2010. ISSN 0028-0836. doi: 10.1038/nature08716. [21] R. B. Dingle. Some magnetic properties of metals. ii. the influence of collisions on the magnetic behaviour of large systems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 211(1107):pp. 517–525, 1952. ISSN 00804630. URL http://www.jstor.org/stable/98939. [22] Nicolas Doiron-Leyraud, Cyril Proust, David LeBoeuf, Julien Levallois, Jean-Baptiste Bonnemaison, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis Taillefer. Quantum oscillations and the fermi surface in an underdoped high-tc superconductor. NATURE, 447(7144):565–568, May 31 2007. ISSN 0028-0836. doi:10.1038/nature05872. [23] I. S. Elfimov, G. A. Sawatzky, and A. Damascelli. Theory of fermi-surface pockets and correlation effects in underdoped Yba2 cu3 o6.5 . Phys. Rev. B, 77:060504, Feb 2008. doi:10.1103/PhysRevB.77.060504. URL http://link.aps.org/doi/ 10.1103/PhysRevB.77.060504. [24] S Engelsberg and G Simpson. Influence of electron-phonon interactions on de haasvan alphen effects. Physical Review B, 2(6):1657–&, 1970. doi:10.1103/PhysRevB. 2.1657. [25] B. Fauqu´e, Y. Sidis, V. Hinkov, S. Pailh`es, C. T. Lin, X. Chaud, and P. Bourges. Magnetic order in the pseudogap phase of high-TC superconductors. Phys. Rev. Lett., 96:197001, May 2006. doi:10.1103/PhysRevLett.96.197001. URL http://link. aps.org/doi/10.1103/PhysRevLett.96.197001. [26] Daniel S. Fisher, Matthew P. A. Fisher, and David A. Huse. Thermal fluctuations, quenched disorder, phase transitions, and transport in type-ii superconductors. Phys. 163  Bibliography Rev. B, 43:130–159, Jan 1991. doi:10.1103/PhysRevB.43.130. URL http://link. aps.org/doi/10.1103/PhysRevB.43.130. [27] D. Fournier, G. Levy, Y. Pennec, J. L. McChesney, A. Bostwick, E. Rotenberg, R. Liang, W. N. Hardy, D. A. Bonn, I. S. Elfimov, and A. Damascelli. Loss of nodal quasiparticle integrity in underdoped yba(2)cu(3)o(6+x). NATURE PHYSICS, 6(11): 905–911, NOV 2010. ISSN 1745-2473. doi:10.1038/NPHYS1763. [28] David Garcia-Aldea and Sudip Chakravarty.  Singlet versus triplet particle-hole  condensates in quantum oscillations in cuprates.  Phys. Rev. B, 82:184526, Nov  2010. doi:10.1103/PhysRevB.82.184526. URL http://link.aps.org/doi/10. 1103/PhysRevB.82.184526. [29] D.S. Ginley, C.I.H. Ashby, T.A. Plut, D. Urea, M.P. Siegal, and J.S. Martens. Di- and tricarboxylic-acid-based etches for processing high temperature superconducting thin films. 00036951.  Applied Physics Letters, 63(17):2429, 1993.  ISSN  URL http://search.ebscohost.com/login.aspx?direct=true&db=  aph&AN=4289515&site=ehost-live. [30] I.S. Gradshteyn and I.M. Ryzhik. Table of integrals, series, and products. Academic Press, 1980. URL http://books.google.ca/books?id=sIecpwAACAAJ. [31] PD Grigoriev. Theory of the shubnikov-de haas effect in quasi-two-dimensional metals. Physical Review B, 67(14), Apr 11 2003. ISSN 1098-0121. doi:10.1103/PhysRevB. 67.144401. [32] WN Hardy, DA Bonn, DC Morgan, RX Liang, and K Zhang. Precision-measurements of the temperature-dependence of lambda in yba2cu3o6.95 - strong evidence for nodes in the gap function. Physical Review Letters, 70(25):3999–4002, Jun 21 1993. ISSN 0031-9007. doi:10.1103/PhysRevLett.70.3999. [33] FJ Harris. Use of windows for harmonic-analysis with discrete fourier-transform. Proceedings Of The IEEE, 66(1):51–83, 1978. ISSN 0018-9219. doi:10.1109/PROC. 1978.10837. [34] R. Harris, A. Hosseini, Saeid Kamal, P. Dosanjh, Ruixing Liang, W. N. Hardy, and D. A. Bonn. Microwave spectroscopy of quasiparticle transport in the bˆ direction of yba2 cu3 o6.993 . Phys. Rev. B, 64:064509, Jul 2001. doi:10.1103/PhysRevB.64. 064509. URL http://link.aps.org/doi/10.1103/PhysRevB.64.064509. [35] N. Harrison, S. M. Hayden, P. Meeson, M. Springford, P. J. van der Wel, and A. A. Menovsky. de haas˘van alphen effect in the vortex state of nb3 sn. Phys. Rev. B, 50: 4208–4211, Aug 1994. doi:10.1103/PhysRevB.50.4208. URL http://link.aps. org/doi/10.1103/PhysRevB.50.4208. 164  Bibliography [36] T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, and R. Gross. Evolution of the fermi surface of the electron-doped hightemperature superconductor nd2−x cex cuo4 revealed by shubnikov˘de haas oscillations. Phys. Rev. Lett., 103:157002, Oct 2009. doi:10.1103/PhysRevLett.103.157002. URL http://link.aps.org/doi/10.1103/PhysRevLett.103.157002. [37] F.E. Hoare. Experimental Cryophysics. Edited by F.E. Hoare, L.C. Jackson (And¿ N. Kurti. Butterworth & Co., 1961. URL http://books.google.ca/books?id= pu4scgAACAAJ. [38] C. C. Homes, T. Timusk, R. Liang, D. A. Bonn, and W. N. Hardy.  Optical  conductivity of c axis oriented yba2 cu3 o6.70 : Evidence for a pseudogap.  Phys.  Rev. Lett., 71:1645–1648, Sep 1993.  URL  doi:10.1103/PhysRevLett.71.1645.  http://link.aps.org/doi/10.1103/PhysRevLett.71.1645. [39] C.C. Homes, T. Timusk, D.A. Bonn, R. Liang, and W.N. Hardy. Optical properties along the c-axis of yba2cu3o6+x, for x = 0.50 0.95 evolution of the pseudogap. Physica C: Superconductivity, 254:265 – 280, 1995. ISSN 0921-4534. doi:10. 1016/0921-4534(95)00579-X.  URL http://www.sciencedirect.com/science/  article/pii/092145349500579X. [40] M. A. Hossain, J. D. F. Mottershead, D. Fournier, A. Bostwick, J. L. McChesney, E. Rotenberg, R. Liang, W. N. Hardy, G. A. Sawatzky, I. S. Elfimov, D. A. Bonn, and A. Damascelli. In situ doping control of the surface of high-temperature superconductors. NATURE PHYSICS, 4(7):527–531, JUL 2008. ISSN 1745-2473. doi:10.1038/nphys998. [41] J Hubbard. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London Series A-Mathematical and Physical Sciences, 276(1364):238+, 1963. ISSN 0080-4630. doi:10.1098/rspa.1963.0204. [42] N. E. Hussey. Phenomenology of the normal state in-plane transport properties of high-T(c) cuprates. Journal of Physics-Condensed Matter, 20(12), MAR 26 2008. ISSN 0953-8984. doi:10.1088/0953-8984/20/12/123201. [43] N. E. Hussey, H. Takagi, Y. Iye, S. Tajima, A. I. Rykov, and K. Yoshida. Charge confinement on the cuo2 planes in slightly overdoped yba2 cu3 o7−δ and the role of metallic chains. Phys. Rev. B, 61:R6475–R6478, Mar 2000. doi:10.1103/PhysRevB. 61.R6475. URL http://link.aps.org/doi/10.1103/PhysRevB.61.R6475. [44] NE Hussey, M Abdel-Jawad, A Carrington, AP Mackenzie, and L Balicas. A coherent three-dimensional fermi surface in a high-transition-temperature superconductor.  NATURE, 425(6960):814–817, OCT 23 2003.  ISSN 0028-0836.  doi:  10.1038/nature01981. 165  Bibliography [45] Fast Track Team Inc.  Microsoft dreamspark, Feb 2012.  URL https://www.  dreamspark.com/. [46] Intel. Intel math kernel library reference manual, 2011. URL http://software. intel.com/sites/products/documentation/hpc/mkl/mklman/mklman.pdf. [47] T. J. B. M. Janssen, C. Haworth, S. M. Hayden, P. Meeson, M. Springford, and A. Wasserman.  Quantitative investigation of the de haas-van alphen effect in  the superconducting state.  Phys. Rev. B, 57:11698–11712, May 1998.  1103/PhysRevB.57.11698.  URL http://link.aps.org/doi/10.1103/PhysRevB.  doi:10.  57.11698. [48] H.-H. Klauss, W. Wagener, M. Hillberg, W. Kopmann, H. Walf, F. J. Litterst, M. H¨ ucker, and B. B¨ uchner. From antiferromagnetic order to static magnetic stripes: The phase diagram of (la, eu)2−x srx cuo4 . Phys. Rev. Lett., 85:4590–4593, Nov 2000. doi:10.1103/PhysRevLett.85.4590.  URL http://link.aps.org/doi/10.1103/  PhysRevLett.85.4590. [49] Walter Kohn. Cyclotron resonance and de haas-van alphen oscillations of an interacting electron gas. Phys. Rev., 123:1242–1244, Aug 1961. doi:10.1103/PhysRev.123. 1242. URL http://link.aps.org/doi/10.1103/PhysRev.123.1242. [50] AM Kosevich and IM Lifshitz. The dehaas-vanalphen effect in thin metal layers. Soviet Physics JETP-USSR, 2(4):646–649, 1956. ISSN 0038-5646. [51] Y Kurihara. A microscopic calculation of the angular-dependent oscillatory magnetoresistance in quasi-2-dimensional systems. Journal of the Physical Society of Japan, 61(3):975–982, MAR 1992. ISSN 0031-9015. doi:10.1143/JPSJ.61.975. [52] F. Laliberte, J. Chang, N. Doiron-Leyraud, E. Hassinger, R. Daou, M. Rondeau, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy, S. Pyon, T. Takayama, H. Takagi, I. Sheikin, L. Malone, C. Proust, K. Behnia, and Louis Taillefer. Fermi-surface reconstruction by stripe order in cuprate superconductors. NATURE COMMUNICATIONS, 2, AUG 2011. ISSN 2041-1723. doi:10.1038/ncomms1440. [53] David LeBoeuf, Nicolas Doiron-Leyraud, Julien Levallois, R. Daou, J.-B. Bonnemaison, N. E. Hussey, L. Balicas, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, S. Adachi, Cyril Proust, and Louis Taillefer. Electron pockets in the fermi surface of hole-doped high-t-c superconductors. Nature, 450(7169):533–536, Nov 22 2007. ISSN 0028-0836. doi:10.1038/nature06332. [54] David LeBoeuf, Nicolas Doiron-Leyraud, B. Vignolle, Mike Sutherland, B. J. Ramshaw, J. Levallois, R. Daou, Francis Lalibert´e, Olivier Cyr-Choini`ere, Johan 166  Bibliography Chang, Y. J. Jo, L. Balicas, Ruixing Liang, D. A. Bonn, W. N. Hardy, Cyril Proust, and Louis Taillefer. Lifshitz critical point in the cuprate superconductor yba2 cu3 oy from high-field hall effect measurements. Phys. Rev. B, 83:054506, Feb 2011. doi:10.1103/PhysRevB.83.054506. URL http://link.aps.org/doi/10. 1103/PhysRevB.83.054506. [55] R Liang, DA Bonn, WN Hardy, JC Wynn, KA Moler, L Lu, S Larochelle, L Zhou, M Greven, L Lurio, and SGJ Mochrie. Preparation and characterization of homogeneous ybco single crystals with doping level near the sc-afm boundary. Physica C-Superconductivity and its Applications, 383(1-2):1–7, Dec 1 2002. ISSN 0921-4534. doi:10.1016/S0921-4534(02)01269-8. [56] Ruixing Liang, D.A. Bonn, and W.N. Hardy. Growth of high quality ybco single crystals using bazro3 crucibles. Physica C: Superconductivity, 304:105 – 111, 1998. ISSN 0921-4534. doi:10.1016/S0921-4534(98)00275-5. URL http://www. sciencedirect.com/science/article/pii/S0921453498002755. [57] Ruixing Liang, D.A Bonn, and Walter N Hardy.  Preparation and x-ray char-  acterization of highly ordered ortho-ii phase yba2cu3o6.50 single crystals. Physica C: Superconductivity, 336:57 – 62, 2000. 1016/S0921-4534(00)00091-5.  ISSN 0921-4534.  doi:10.  URL http://www.sciencedirect.com/science/  article/pii/S0921453400000915. [58] Ruixing Liang, D. A. Bonn, W. N. Hardy, and David Broun. Lower critical field and superfluid density of highly underdoped yba2 cu3 o6+x single crystals. Phys. Rev. Lett., 94:117001, Mar 2005. doi:10.1103/PhysRevLett.94.117001. URL http: //link.aps.org/doi/10.1103/PhysRevLett.94.117001. [59] Ruixing Liang, D. A. Bonn, and W. N. Hardy. Evaluation of cuo2 plane hole doping in yba2 cu3 o6+x single crystals. Phys. Rev. B, 73:180505, May 2006. doi:10. 1103/PhysRevB.73.180505. URL http://link.aps.org/doi/10.1103/PhysRevB. 73.180505. [60] Ruixing Liang, Douglas A. Bonn, and Walter N. Hardy. Growth of ybco single crystals by the self-flux technique. Philosophical Magazine, 92(19-21):2563–2581, 2012. doi:10.1080/14786435.2012.669065.  URL http://www.tandfonline.com/doi/  abs/10.1080/14786435.2012.669065. [61] J. M. Luttinger. Fermi surface and some simple equilibrium properties of a system of interacting fermions. Phys. Rev., 119:1153–1163, Aug 1960. doi:10.1103/PhysRev. 119.1153. URL http://link.aps.org/doi/10.1103/PhysRev.119.1153.  167  Bibliography [62] J. M. Luttinger. Theory of the de haas-van alphen effect for a system of interacting fermions. Phys. Rev., 121:1251–1258, Mar 1961. doi:10.1103/PhysRev.121.1251. URL http://link.aps.org/doi/10.1103/PhysRev.121.1251. [63] G.D. Mahan. Many-particle physics. Physics of solids and liquids. Kluwer Academic/Plenum Publishers, 2000. ISBN 9780306463389. [64] JF Mercure. The de Haas van Alphen Effect Near a Quantum Critical End Point in Sr3Ru2O7. PhD thesis, University of St. Andrews, 2008. [65] Microsoft. Queryperformancecounter function, 2011. URL http://msdn.microsoft. com/en-us/library/windows/desktop/ms644904(v=vs.85).aspx. [66] NF Mott. The basis of the electron theory of metals, with special reference to the transition metals. Proceedings of the Royal Society of London Section A, 62(355): 416–422, 1949. doi:10.1088/0370-1298/62/7/303. [67] Peter Nagel, Volker Pasler, Christoph Meingast, Alexandre I. Rykov, and Setsuko Tajima. Anomalously large oxygen-ordering contribution to the thermal expansion of untwinned yba2 cu3 O6.95 single crystals: A glasslike transition near room temperature. Phys. Rev. Lett., 85:2376–2379, Sep 2000. doi:10.1103/PhysRevLett.85.2376. URL http://link.aps.org/doi/10.1103/PhysRevLett.85.2376. [68] M. R. Norman and Jie Lin. Spin zeros and the origin of fermi-surface reconstruction in the cuprates. Phys. Rev. B, 82:060509, Aug 2010. doi:10.1103/PhysRevB.82. 060509. URL http://link.aps.org/doi/10.1103/PhysRevB.82.060509. [69] M. R. Norman, Jie Lin, and A. J. Millis. Lifshitz transition in underdoped cuprates. Phys. Rev. B, 81:180513, May 2010. doi:10.1103/PhysRevB.81.180513. URL http: //link.aps.org/doi/10.1103/PhysRevB.81.180513. [70] H Ohkuni, Y Inada, Y Tokiwa, K Sakurai, R Settai, T Honma, Y Haga, E Yamamoto, Y Onuki, H Yamagami, S Takahashi, and T Yanagisawa. Fermi surface properties and de haas-van alphen oscillation in both the normal and superconducting mixed states of uru2si2. Philosophical Magazine B-Physics of Condensed Matter Statistical Mechanics Electronic Optical and Magnetic Properties, 79(7):1045–1077, Jul 1999. ISSN 0141-8637. doi:10.1080/014186399256916. [71] L Onsager. Interpretation of the de haas-van alphen effect. Philosophical Magazine, 43(344):1006–1008, 1952. ISSN 0031-8086. [72] J. Orenstein and A. J. Millis. Advances in the physics of high-temperature superconductivity. Science, 288(5465):468–474, 2000. doi:10.1126/science.288.5465.468. URL http://www.sciencemag.org/content/288/5465/468.abstract. 168  Bibliography [73] D. C. Peets. Why be Normal? Single Crystal Growth and X-ray Spectroscopy Reveal the Startlingly Unremarkable Electronic Structure of Tl-2201. PhD thesis, University of British Columbia, 2008. URL http://hdl.handle.net/2429/2365. [74] D. C. Peets, J. D. F. Mottershead, B. Wu, I. S. Elfimov, Ruixing Liang, W. N. Hardy, D. A. Bonn, M. Raudsepp, N. J. C. Ingle, and A. Damascelli. Tl2ba2cuo6+delta brings spectroscopic probes deep into the overdoped regime of the high-t-c cuprates. New Journal of Physics, 9, FEB 14 2007. ISSN 1367-2630. doi:10.1088/1367-2630/ 9/2/028. [75] T. Pereg-Barnea, P. J. Turner, R. Harris, G. K. Mullins, J. S. Bobowski, M. Raudsepp, Ruixing Liang, D. A. Bonn, and W. N. Hardy. Absolute values of the london penetration depth in yba2 cu3 o6+y measured by zero field esr spectroscopy on gd doped single crystals. Phys. Rev. B, 69:184513, May 2004. doi:10.1103/PhysRevB.69.184513. URL http://link.aps.org/doi/10.1103/PhysRevB.69.184513. [76] T. Pereg-Barnea, H. Weber, G. Refael, and M. Franz. lations from fermi arcs. 17452473.  Nature Physics,  6(1):44 – 49,  Quantum oscil2010.  ISSN  URL http://search.ebscohost.com/login.aspx?direct=true&db=  a9h&AN=47127015&site=ehost-live. [77] Warren E. Pickett. Electronic structure of the high-temperature oxide superconductors. Rev. Mod. Phys., 61:433–512, Apr 1989. doi:10.1103/RevModPhys.61.433. URL http://link.aps.org/doi/10.1103/RevModPhys.61.433. [78] M. Plat´e, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, Ruixing Liang, D. A. Bonn, W. N. Hardy, S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli. Fermi surface and quasiparticle excitations of overdoped tl2 ba2 cuo6+δ . Phys. Rev. Lett., 95:077001, Aug 2005. doi:10.1103/PhysRevLett.95.077001. URL http: //link.aps.org/doi/10.1103/PhysRevLett.95.077001. [79] B. J. Ramshaw, Baptiste Vignolle, James Day, Ruixing Liang, W. N. Hardy, Cyril Proust, and D. A. Bonn.  Angle dependence of quantum oscillations in  yba(2)cu(3)o(6.59) shows free-spin behaviour of quasiparticles. Nature Physics, 7 (3):234–238, Mar 2011. ISSN 1745-2473. doi:10.1038/NPHYS1873. [80] B.J. Ramshaw, P. Dosanjh, R. Liang, D.A. Bonn, and W.N. Hardy. Antiferromagnetic Order in Underdoped YBa2 Cu3 O6.x Probed by cˆ-axis Resistivity. Unpublished, 2007. [81] Scott C. Riggs, O. Vafek, J. B. Kemper, J. B. Betts, A. Migliori, F. F. Balakirev, W. N. Hardy, Ruixing Liang, D. A. Bonn, and G. S. Boebinger. Heat capacity through the magnetic-field-induced resistive transition in an underdoped high-temperature  169  Bibliography superconductor. NATURE PHYSICS, 7(4):332–335, APR 2011. ISSN 1745-2473. doi:10.1038/NPHYS1921. [82] P. M. C. Rourke, A. F. Bangura, T. M. Benseman, M. Matusiak, J. R. Cooper, A. Carrington, and N. E. Hussey. A detailed de haas-van alphen effect study of the overdoped cuprate tl(2)ba(2)cuo(6+delta). NEW JOURNAL OF PHYSICS, 12, OCT 29 2010. ISSN 1367-2630. doi:10.1088/1367-2630/12/10/105009. [83] J.J. Sakurai and S.F. Tuan. Modern Quantum Mechanics. Addison-Wesley Publishing Company, 1994. ISBN 9780201539295. [84] J.R. Schrieffer and J.S. Brooks. Handbook of High-Temperature Superconductivity: Theory And Experiment. Springer, 2007. ISBN 9780387350714. [85] L Schubnikow and WJ de Haas. New occurrences in the resistance changes of bismuth crystals in magnetic fields at the temperature of liquid hydrogen. PROCEEDINGS OF THE KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM, 33(1/5):363–388, 1930. [86] Suchitra E. Sebastian, N. Harrison, C. H Mielke, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich. Spin-order driven fermi surface reconstruction revealed by quantum oscillations in an underdoped high Tc superconductor. Phys. Rev. Lett., 103:256405, Dec 2009. doi:10.1103/PhysRevLett.103.256405. URL http://link. aps.org/doi/10.1103/PhysRevLett.103.256405. [87] Suchitra E. Sebastian, N. Harrison, M. M. Altarawneh, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich. Fermi-liquid behavior in an underdoped high-Tc superconductor. Phys. Rev. B, 81:140505, Apr 2010. doi:10.1103/PhysRevB.81. 140505. URL http://link.aps.org/doi/10.1103/PhysRevB.81.140505. [88] Suchitra E. Sebastian, N. Harrison, M. M. Altarawneh, C. H. Mielke, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich. Metal-insulator quantum critical point beneath the high t(c) superconducting dome. Proceedings of the National Academy of Sciences of the United States of America, 107(14):6175–6179, APR 6 2010. ISSN 0027-8424. [89] Suchitra E. Sebastian, N. Harrison, P. A. Goddard, M. M. Altarawneh, C. H. Mielke, Ruixing Liang, D. A. Bonn, W. N. Hardy, O. K. Andersen, and G. G. Lonzarich. Compensated electron and hole pockets in an underdoped high-t(c) superconductor. Physical Review B, 81(21), Jun 28 2010. ISSN 1098-0121. doi:10.1103/PhysRevB. 81.214524.  170  Bibliography [90] Suchitra E. Sebastian, N. Harrison, M. M. Altarawneh, F. F. Balakirevand, C. H Mielke, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich. Direct observation of multiple spin zeroes in the underdoped high temperature superconductor yba2cu3o6+x. arxiv.org, 2011. URL http://arxiv.org/abs/1103.4178v1. [91] r Shankar. Renormalization-group approach to interacting fermions. Reviews of Modern Physics, 66(1):129–192, Jan 1994. ISSN 0034-6861. doi:10.1103/revmodphys. 66.129. [92] D. Shoenberg. Magnetic oscillations in metals. Cambridge monographs on physics. Cambridge University Press, 1984. ISBN 9780521224802. [93] J Singleton. Studies of quasi-two-dimensional organic conductors based on bedt-ttf using high magnetic fields. Reports on Progress in Physics, 63(8):1111, 2000. URL http://stacks.iop.org/0034-4885/63/i=8/a=201. [94] John Singleton, Clarina de la Cruz, R. D. McDonald, Shiliang Li, Moaz Altarawneh, Paul Goddard, Isabel Franke, Dwight Rickel, C. H. Mielke, Xin Yao, and Pengcheng Dai. Magnetic quantum oscillations in yba2 cu3 o6.61 and yba2 cu3 o6.69 in fields of up to 85 t: Patching the hole in the roof of the superconducting dome. Phys. Rev. Lett., 104:086403, Feb 2010. doi:10.1103/PhysRevLett.104.086403. URL http: //link.aps.org/doi/10.1103/PhysRevLett.104.086403. [95] J.C. Slater.  Quantum theory of matter.  plied physics. McGraw-Hill, 1968.  International series in pure and ap-  URL http://books.google.ca/books?id=  VOREAAAAIAAJ. [96] J. E. Sonier, J. H. Brewer, R. F. Kiefl, D. A. Bonn, S. R. Dunsiger, W. N. Hardy, Ruixing Liang, W. A. MacFarlane, R. I. Miller, T. M. Riseman, D. R. Noakes, C. E. Stronach, and M. F. White. Measurement of the fundamental length scales in the vortex state of yba2 cu3 O6.60 .  Phys. Rev. Lett., 79:2875–2878, Oct 1997.  doi:10.1103/PhysRevLett.79.2875.  URL http://link.aps.org/doi/10.1103/  PhysRevLett.79.2875. [97] C. Stock, W. J. L. Buyers, R. Liang, D. Peets, Z. Tun, D. Bonn, W. N. Hardy, and R. J. Birgeneau. Dynamic stripes and resonance in the superconducting and normal phases of yba2 cu3 o6.5 ortho-ii superconductor. Phys. Rev. B, 69:014502, Jan 2004. doi:10. 1103/PhysRevB.69.014502. URL http://link.aps.org/doi/10.1103/PhysRevB. 69.014502. [98] Rainer Storn and Kenneth Price.  Differential evolution: A simple and efficient  heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11:341–359, 1997. ISSN 0925-5001. URL http://dx.doi.org/10.1023/A: 1008202821328. 10.1023/A:1008202821328. 171  Bibliography [99] L. Thompson and P. C. E. Stamp. temperature superconductors.  de Haas˘van Alphen oscillations in high-  Phys. Rev. B, 81:100514, Mar 2010.  doi:10.  1103/PhysRevB.81.100514. URL http://link.aps.org/doi/10.1103/PhysRevB. 81.100514. [100] T Timusk and B Statt. The pseudogap in high-temperature superconductors: an experimental survey. Reports on Progress in Physics, 62(1):61–122, JAN 1999. ISSN 0034-4885. doi:10.1088/0034-4885/62/1/002. [101] M. Tinkham. Introduction to Superconductivity. Dover books on physics and chemistry. Dover Publications, 2004. ISBN 9780486435039. [102] JM Tranquada. Antiferromagnetism in yba2cu3o6+x. Journal of Applied Physics, 64 (10, Part 2):6071–6074, Nov 15 1988. ISSN 0021-8979. doi:10.1063/1.342125. [103] B. Vignolle, A. Carrington, R. A. Cooper, M. M. J. French, A. P. Mackenzie, C. Jaudet, D. Vignolles, Cyril Proust, and N. E. Hussey. Quantum oscillations in an overdoped high-t(c) superconductor. NATURE, 455(7215):952–955, OCT 16 2008. ISSN 0028-0836. doi:10.1038/nature07323. [104] B. Vignolle, B. J. Ramshaw, J. Day, D. LeBoeuf, S. Lepault, R. Liang, W.N. Hardy, D.A. Bonn, L. Taillefer, and C. Proust. Coherent c-axis transport in the underdoped cuprate superconductor ybco. arxiv.org, 2011. URL http://arxiv.org/abs/1107. 5422. [105] M Von Zimmermann, JR Schneider, T Frello, NH Andersen, J Madsen, M Kall, HF Poulsen, R Liang, P Dosanjh, and WN Hardy. Oxygen-ordering superstructures in underdoped yba(2)cu(3)o(6+x) studied by hard x-ray diffraction. Physical Review B, 68(10), Sep 1 2003. ISSN 1098-0121. doi:10.1103/PhysRevB.68.104515. [106] RE Walstedt, Rf bell, LF Schneemeyer, JV Waszczak, and GP Espinosa. Diamagnetism in the normal state of yba2cu3o7. Physical Review B, 45(14):8074–8084, Apr 1 1992. ISSN 0163-1829. doi:10.1103/PhysRevB.45.8074. [107] Steven Watanabe and Jens Maurer. Mersenne twister.hpp, Feb 2001. URL http: //www.boost.org/doc/libs/1_48_0/boost/random/mersenne_twister.hpp. [108] MK Wu, JR Ashburn, CJ Torng, PH Hor, RL Meng, L Gao, ZJ Huang, YQ Wang, and CW Chu. Superconductivity at 93-k in a new mixed-phase y-ba-cu-o compound system at ambient pressure. Physical Review Letters, 58(9):908–910, Mar 2 1987. ISSN 0031-9007. doi:10.1103/PhysRevLett.58.908. [109] Jing Xia, Elizabeth Schemm, G. Deutscher, S. A. Kivelson, D. A. Bonn, W. N. Hardy, R. Liang, W. Siemons, G. Koster, M. M. Fejer, and A. Kapitulnik. Polar kerr-effect 172  measurements of the high-temperature yba(2)cu(3)o(6+x) superconductor: Evidence for broken symmetry near the pseudogap temperature. Physical Review Letters, 100 (12), Mar 28 2008. ISSN 0031-9007. doi:10.1103/PhysRevLett.100.127002. [110] K Yamaji. On the angle dependence of the magnetoresistance in quasi-2-dimensional organic superconductors. Journal of the Physical Society of Japan, 58(5):1520–1523, May 1989. ISSN 0031-9015. doi:10.1143/JPSJ.58.1520. [111] E. A. Yelland, J. Singleton, C. H. Mielke, N. Harrison, F. F. Balakirev, B. Dabrowski, and J. R. Cooper. Quantum oscillations in the underdoped cuprate yba2 cu4 o8 . Phys. Rev. Lett., 100:047003, Feb 2008. doi:10.1103/PhysRevLett.100.047003. URL http://link.aps.org/doi/10.1103/PhysRevLett.100.047003. [112] J. Zaanen, G. A. Sawatzky, and J. W. Allen. Band gaps and electronic structure of transition-metal compounds. Phys. Rev. Lett., 55:418–421, Jul 1985. doi:10.1103/ PhysRevLett.55.418. URL http://link.aps.org/doi/10.1103/PhysRevLett.55. 418.  173  Appendix A  Genetic Algorithm Code This appendix contains some of the key functions for the genetic algorithm code. Comments are in italics. The phase of the oscillations in this program is defined to run from 0 to 1, whereas in Equation 7.2 it was defined such that it ran from 0 to 2π. These two definitions are equivalent.  A.1  DataExtractor.cpp  This code extracts the background-subtracted oscillatory data from a *.dat file. There is some basic error checking to make sure that the file is in the correct format, and the data is stored in a 2D array of doubles.  #i n c l u d e ” StdAfx . h” #i n c l u d e ” D a t a E x t r a c t o r . h” #i n c l u d e ” A r r a y A l l o c a t o r . cpp ” d o u b l e ∗∗ dataArray ; int number of lines ; D a t a E x t r a c t o r : : D a t a E x t r a c t o r ( s t d : : s t r i n g name ) // c o n s t r u c t o r { s t d : : v e c t o r <s t d : : s t r i n g > b u f f e r 1 ; // s t r i n g b u f f e r n u m b e r o f l i n e s = dataReader ( name , &b u f f e r 1 ) ; // d a t a r e a d e r p u t s t h e ” name” d f i l e i n t o t h e b u f f e r and r e t u r n s i t s l e n t h i n l i n e s i f ( n u m b e r o f l i n e s == 0 ) throw ” F i l e Not Found” ;  // empty f i l e  s t d : : s t r i n g f t y p e = ” . dat ” ; // s h o u l d b e a . d a t f i l e c h a r ∗ p1 = ( c h a r ∗ ) name . c s t r ( ) ; // c o n v e r t s name o f f i l e t o a c s t r i n g w h i l e ( ∗ p1!=NULL) { p1++; // g o e s t o end o f s t r i n g } p1 = p1 −4; // g o e s b a c k 4 s p a c e s i f ( f t y p e . compare ( p1 ) !=0) throw ”Wrong F i l e Type” ; // c h e c k s i f t h e l a s t 4 c h a r a c t e r s a r e . d a t  174  A.1. DataExtractor.cpp  dataArray = A r r a y A l l o c a t o r : : AllocateDynamicArray<d o u b l e >( n u m b e r o f l i n e s , 4 ) ; // a l l o c a t e s an a r r a y o f t h e a p p r o p r i a t e s i z e , r i g h t now 3 w i d e MAY NEED ADJUSTMENT b o o l y = d a t a C o n v e r t e r (& b u f f e r 1 , dataArray ) ; } D a t a E x t r a c t o r : : ˜ D a t a E x t r a c t o r ( ) // d e s t r u c t o r { A r r a y A l l o c a t o r : : FreeDynamicArray<d o u b l e >(dataArray , n u m b e r o f l i n e s ) ; } i n t D a t a E x t r a c t o r : : dataReader ( s t d : : s t r i n g name , s t d : : v e c t o r <s t d : : s t r i n g >∗ b u f f e r ) // Reads t h e d a t a f i l e i n t o a v e c t o r i n s t r i n g f o r m a t . { i n t x = 0 ; // r e t u r n v a l u e s t d : : s t r i n g a ; // b u f f e r s t d : : i f s t r e a m i n ; // f i l e t o s t r e a m . i n . open ( name ) ; // o p e n s f i l e i f ( i n . i s o p e n ( ) ) // c h e c k s f o r opened f i l e { w h i l e ( g e t l i n e ( in , a ) ) // s u c c e s s i v l y g e t s l i n e s from f i l e ( Each l i n e h a s an x and a y d a t a v a l u e ) { ( ∗ b u f f e r ) . push back ( a ) ; // s t o r e s i n t h e v e c t o r b u f f e r } x=(∗ b u f f e r ) . s i z e ( ) ; // number o f l i n e s } in . close () ; r e t u r n x ; // r e t u r n s i f s u c c e s s f u l } b o o l D a t a E x t r a c t o r : : d a t a C o n v e r t e r ( s t d : : v e c t o r <s t d : : s t r i n g >∗ s t r i n g D a t , d o u b l e ∗∗ numDat ) // C o n v e r t s a s t r i n g f o r m a t v e c t o r o f numbers i n t o a 2D a r r a y . { s t d : : s t r i n g s t r e a m s b u f f 1 , s b u f f 2 ; // b u f f e r s f o r s t r i n g s t r e a m s s t d : : s t r i n g b u f f 1 ; // s t r i n g n b u f f e r s t d : : v e c t o r <s t d : : s t r i n g > v b u f f 1 ; // v e c t o r b u f f e r . f o r ( i n t i =0; i < ( i n t ) ( ∗ s t r i n g D a t ) . s i z e ( ) ; i ++) // f o r e a c h l i n e i n t h e b u f f e r ( each data f i l e l i n e ) . { s b u f f 1 << ( ∗ s t r i n g D a t ) [ i ] ; // p u t i t h e l e m e n t i n t o s t r i n b b u f f e r w h i l e ( g e t l i n e ( s b u f f 1 , b u f f 1 , ’ \ t ’ ) ) // g e t e l e m e n t s o f e a c h l i n e ( x and y ) { v b u f f 1 . push back ( b u f f 1 ) ; // p u t e l e m e n t s i n t o b u f f e r  175  A.2. Parameters.h } f o r ( i n t j =0; j < 4 ; j ++) // f o r x and y { numDat [ i ] [ j ] = a t o f ( v b u f f 1 [ j ] . c s t r ( ) ) ; // c o n v e r t s t r i n g t o f l o a t and stores it } v b u f f 1 . c l e a r ( ) ; // c l e a r s t h e b u f f e r s sbuff1 . clear () ; sbuff2 . clear () ; buff1 . clear () ; } return true ; } d o u b l e ∗∗ D a t a E x t r a c t o r : : getDataArray ( ) // r e t u r n s t h e d a t a a r r a y { r e t u r n dataArray ; } i n t D a t a E x t r a c t o r : : getNumberOfLines ( ) { return number of lines ; }  A.2  Parameters.h  The struct fitParameters contains the fit parameters. Higher order warping parameters, like parameter number 14 dF12, can be added to this list if needed. The struct arrayBounds stores information related to multi-threading.  #pragma once namespace Parameters { d e c l s p e c ( a l i g n ( 128) ) s t r u c t fi tP ar am ete rs { d o u b l e A1 ; // 0 d o u b l e A2 ; // 1 d o u b l e F1 ; // 2 d o u b l e F2 ; // 3 d o u b l e dF1 ; // 4 d o u b l e dF2 ; // 5 d o u b l e Td1 ; // 6 d o u b l e Td2 ; // 7 d o u b l e p h i 1 ; // 8  176  A.3. GeneticAlgorithm.cpp double double double double double double double double  p h i 2 ; // 9 ms1 ; // 10 ms2 ; // 11 m1 ; // 12 m2 ; // 13 dF12 ; // 14 c h i S q ; // 15 T ; // 16  }; s t r u c t arrayBounds { d o u b l e s t a r t ; // s t a r t and end d e s c r i b e w h i c h s e c t i o n o f t h e p o p u l a t i o n t h e t h r e a d t h a t owns t h i s p a r t i c u l a r copy o f arrayBounds i s responsible for d o u b l e end ; HANDLE h a n d l e ; // t h i s i s a f l a g t h a t i s s e t t o TRUE when t h e t h r e a d h a s finished execution d o u b l e time ; // s t o r a g e f o r t h e t o t a l e x e c u t i o n t i m e i n t threadID ; // i d e n t i f i e r f o r t h e t h r e a d t h a t owns t h i s copy o f arrayBounds }; }  A.3  GeneticAlgorithm.cpp  All of the following functions are contained in the class GeneticAlgorithm.cpp.  A.3.1  initializeParameters  This is the function that initializes the fit parameters to within the bounds specified.  v o i d G e n e t i c A l g o r i t h m 2 : : i n i t i a l i z e P a r a m e t e r s ( d o u b l e ∗∗ dataSet , i n t dataSetLength , i n t nPopulation , d o u b l e s c a l e F a c t o r , d o u b l e crossingProbability ){ scaleFactor = scaleFactor ; crossingProbability = crossingProbability ; dataSetLength = dataSetLength ; dataSet = dataSet ; r e s i d u a l A r r a y = new d o u b l e ∗ [ nThreads ] ; paramArray = new d o u b l e ∗ [ nThreads ] ;  177  A.3. GeneticAlgorithm.cpp f o r ( i n t i = 0 ; i < nThreads ; i ++){ r e s i d u a l A r r a y [ i ] = new d o u b l e [ d a t a S e t L e n g t h ] ; paramArray [ i ] = new d o u b l e [ nParams ] ; } p o p u l a t i o n P a r a m e t e r s O l d = ( Parameters : : f i t P a r a m e t e r s ∗ ) m k l m a l l o c ( s i z e o f ( Parameters : : f i t P a r a m e t e r s ) ∗ nPopulation , 1 6 ) ; p o p u l a t i o n P a r a m e t e r s N e w = ( Parameters : : f i t P a r a m e t e r s ∗ ) m k l m a l l o c ( s i z e o f ( Parameters : : f i t P a r a m e t e r s ) ∗ nPopulation , 1 6 ) ; f o r ( i n t i = 0 ; i < n P o p u l a t i o n ; i ++){ p o p u l a t i o n P a r a m e t e r s O l d [ i ] . A1 = randomDouble ( 0 . 0 , 5 . 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . A2 = randomDouble ( 0 . 0 , 5 . 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . F1 = randomDouble ( 4 6 0 , 4 8 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . F2 = randomDouble ( 5 0 0 , 5 5 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . dF1 = randomDouble ( 0 , 5 0 . 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . dF2 = randomDouble ( 0 , 2 0 . 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . p h i 1 = randomDouble ( 0 . 0 , 1 . 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . p h i 2 = randomDouble ( 0 . 0 , 1 . 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . Td1 = randomDouble ( 1 , 1 0 ) ; // The f i t program u s e d d i n g l e t e m p e r a t u r e i n s t e a d o f q u a s i p a r t i c l e l i f e t i m e b e c a u s e Td i s much c l o s e r t o u n i t y t h a n t a u . p o p u l a t i o n P a r a m e t e r s O l d [ i ] . Td2 = randomDouble ( 1 , 1 0 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . ms1 = randomDouble ( 0 , 3 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . ms2 = randomDouble ( 0 , 3 ) ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . m1 = 1 . 7 ; // t h e c y c l o t r o n mass was f i x e d f o r t h e a n g l e d e pe nd en c e , and g i v e n a r a n g e b e t w e e n 1 and 3 f o r t h e temperature dependence . p o p u l a t i o n P a r a m e t e r s O l d [ i ] . m2 = 1 . 7 ; p o p u l a t i o n P a r a m e t e r s O l d [ i ] . T = 4 . 2 ; // t h e a n g l e d e p e n d e n c e was c a r r i e d out at f i x e d temperature p o p u l a t i o n P a r a m e t e r s O l d [ i ] . c h i S q = c a l c u l a t e R e s i d u a l 2 (& populationParametersOld [ i ] , 0 ) ; } f o r ( i n t i = 0 ; i < n P o p u l a t i o n ; i ++){ populationParametersNew [ i ] . T = 0 ; p o p u l a t i o n P a r a m e t e r s N e w [ i ] . m1 = 0 ; p o p u l a t i o n P a r a m e t e r s N e w [ i ] . m2 = 0 ; } minimumParameters . A1 = 1 ; minimumParameters . A2 = 1 ; minimumParameters . F1 = 0 ; minimumParameters . F2 = 0 ; minimumParameters . dF1 = 0 ; minimumParameters . dF2 = 0 ; minimumParameters . p h i 1 = 0 ; minimumParameters . p h i 2 = 0 ;  178  A.3. GeneticAlgorithm.cpp minimumParameters . Td1 = 0 ; minimumParameters . Td2 = 0 ; minimumParameters . ms1 = 0 ; minimumParameters . ms2 = 0 ; minimumParameters . m1 = 0 ; minimumParameters . m2 = 0 ; minimumParameters . T = 0 ; minimumParameters . c h i S q = INFINITE ; }  A.3.2  calculateMinimum  This function simply iterates through each population member and finds the one with the lowest residual, or chiSq, value. The minimum parameters are updated to the new minimum. v o i d G e n e t i c A l g o r i t h m 2 : : calculateMinimum ( ) { f o r ( i n t i = 0 ; i < n P o p u l a t i o n ; i ++){ i f ( p o p u l a t i o n P a r a m e t e r s O l d [ i ] . c h i S q < minimumParameters . c h i S q ) { minimumParameters . A1 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . A1 ; minimumParameters . A2 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . A2 ; minimumParameters . F1 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . F1 ; minimumParameters . F2 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . F2 ; minimumParameters . dF1 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . dF1 ; minimumParameters . dF2 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . dF2 ; minimumParameters . p h i 1 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . p h i 1 ; minimumParameters . p h i 2 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . p h i 2 ; minimumParameters . Td1 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . Td1 ; minimumParameters . Td2 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . Td2 ; minimumParameters . ms1 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . ms1 ; minimumParameters . ms2 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . ms2 ; minimumParameters . m1 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . m1 ; minimumParameters . m2 = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . m2 ; minimumParameters . c h i S q = p o p u l a t i o n P a r a m e t e r s O l d [ i ] . c h i S q ; } } }  A.3.3  calculateNewGenerations  This function executes the creation of the mutant generation, the crossing with the parent generation, and the creating of the new generation. It also calculates the residual for  179  A.3. GeneticAlgorithm.cpp each population member. It does this by creating 4 threads (or nThreads, the global variable which can be changed) and farming out 1/4 of the parameter sets to each of them. It reports on the total time taken, as the sum of the time taken by each thread, after execution. void GeneticAlgorithm2 : : calculateNewGenerations ( i n t nGenerations ) { d o u b l e averageTime = 0 ; f o r ( i n t i = 0 ; i < n G e n e r a t i o n s ; i ++){  viRngUniform ( VSL RNG METHOD UNIFORM STD, stream , nPopulation ) ; viRngUniform ( VSL RNG METHOD UNIFORM STD, stream , nPopulation ) ; viRngUniform ( VSL RNG METHOD UNIFORM STD, stream , nPopulation ) ; for ( int // i n t // i n t // i n t  j g1 g2 g3  double double double double double  = = = =  ∗ ∗ ∗ ∗ ∗  0; (∗ (∗ (∗  nPopulation , ints1 , 0 , nPopulation , ints2 , 0 , nPopulation , ints3 , 0 ,  j < n P o p u l a t i o n ; j ++){ i n t e g e r D i s t r i b u t i o n ) ( ∗ randomNumberGenerator ) ; i n t e g e r D i s t r i b u t i o n ) ( ∗ randomNumberGenerator ) ; i n t e g e r D i s t r i b u t i o n ) ( ∗ randomNumberGenerator ) ;  p o i n t e r T o O l d V a r i a b l e = & p o p u l a t i o n P a r a m e t e r s O l d [ j ] . A1 ; po int erT oNe wVa ria ble = & p o p u l a t i o n P a r a m e t e r s N e w [ j ] . A1 ; p o i n t e r T o g 1 V a r i a b l e = & p o p u l a t i o n P a r a m e t e r s O l d [ i n t s 1 [ j ] ] . A1 ; p o i n t e r T o g 2 V a r i a b l e = & p o p u l a t i o n P a r a m e t e r s O l d [ i n t s 2 [ j ] ] . A1 ; p o i n t e r T o g 3 V a r i a b l e = & p o p u l a t i o n P a r a m e t e r s O l d [ i n t s 3 [ j ] ] . A1 ;  // c o u l d b e o p t i m z e d f o r v e c t o r a r i t h m e t i c f o r ( i n t k = 0 ; k < nVars ; k++){ d o u b l e p = randomDouble ( 0 , 1 ) ; i f (p > crossingProbability ){ ∗ po int erT oNe wVa ria ble = ∗ p o i n t e r T o O l d V a r i a b l e ; } else{ ∗ po int erT oNe wVa ria ble = ∗ p o i n t e r T o g 1 V a r i a b l e + s c a l e F a c t o r ∗ ( ∗ pointerTog2Variable − ∗ pointerTog3Variable ) ; i f ( ( k == 8 ) | | ( k == 9 ) ) i f ( ∗ po int erT oNe wVa ria ble > 1 . 0 ) ∗ po int erT oNe wVa ria ble = ∗ po int erT oNe wVa ria ble − 1 . 0 ; i f ( ∗ po int erT oNe wVa ria ble < −1.0) ∗ po int erT oNe wVa ria ble = ∗ po int erT oNe wVa ria ble + 1 . 0 ; } po int erT oNe wVa ria ble++; p o i n t e r T o O l d V a r i a b l e ++; p o i n t e r T o g 1 V a r i a b l e ++; p o i n t e r T o g 2 V a r i a b l e ++;  180  A.3. GeneticAlgorithm.cpp p o i n t e r T o g 3 V a r i a b l e ++; } } totalTime = 0 ; HANDLE t h r e a d E v e n t s [ nThreads ] ; Parameters : : arrayBounds threadBounds [ nThreads ] ; t h r e a d C o n t e n t s t h r e a d C o n t e n t s [ nThreads ] ; f o r ( i n t m = 0 ; m<nThreads ; m++){ t h r e a d E v e n t s [m] = CreateEvent (NULL, FALSE, FALSE, NULL) ; i n t nPopulationPerThread = n P o p u l a t i o n / nThreads ; i f (m != ( nThreads −1) ) { threadBounds [m ] . s t a r t = m∗ nPopulationPerThread ; threadBounds [m ] . end = (m+1)∗ nPopulationPerThread − 1 ; } else{ threadBounds [m ] . s t a r t = m∗ nPopulationPerThread ; threadBounds [m ] . end = (m+1)∗ nPopulationPerThread − 1 + n P o p u l a t i o n% nThreads ; } threadBounds [m ] . h a n d l e = t h r e a d E v e n t s [m] ; threadBounds [m ] . time = 0 ; threadBounds [m ] . threadID = m; t h r e a d C o n t e n t s [m ] . arrayBounds = threadBounds [m ] ; t h r e a d C o n t e n t s [m ] . pThis = t h i s ; AfxBeginThread ( s t a r t R e s i d u a l T h r e a d , (LPVOID) &t h r e a d C o n t e n t s [m] ) ; } // // //  CrtMemCheckpoint ( &s 2 ) ; C r t M e m D u m p S t a t i s t i c s ( &s 2 ) ; s t d : : c o u t <<s 2 . l T o t a l C o u n t <<s t d : : e n d l ;  W a i t F o r M u l t i p l e O b j e c t s ( nThreads , t h r ea d Ev e n ts ,TRUE, INFINITE ) ;  t o t a l T i m e = t h r e a d C o n t e n t s [ 0 ] . arrayBounds . time+t h r e a d C o n t e n t s [ 1 ] . arrayBounds . time+t h r e a d C o n t e n t s [ 2 ] . arrayBounds . time+t h r e a d C o n t e n t s [ 3 ] . arrayBounds . time ; averageTime +=t o t a l T i m e ; // c a l c u l a t e M i n i m u m ( ) ; // e x p o r t C h i S q ( ) ; } s t d : : cout<<” Average time p e r g e n e r a t i o n p e r t h r e a d i s : ”<<averageTime / ( 4 ∗ n G e n e r a t i o n s )<<”ms”<<s t d : : endl <<s t d : : e n d l ;  181  A.3. GeneticAlgorithm.cpp  }  A.3.4  startResidualThread  This function is called by the calculateNewGenerations function to start the threads. This function is required because of the way threading works in Windows.  UINT G e n e t i c A l g o r i t h m 2 : : s t a r t R e s i d u a l T h r e a d (LPVOID param ) { t h r e a d C o n t e n t s ∗ c o n t e n t s = ( t h r e a d C o n t e n t s ∗ ) param ; c o n t e n t s −>pThis−>r e s i d u a l C a l c u l a t i n g T h r e a d (&( c o n t e n t s −>arrayBounds ) ) ; return 0; }  This is a function called by startResidualThread that calls the residual calculating function. It also records the time taken to calculate the residual.  A.3.5  residualCalculatingThread  v o i d G e n e t i c A l g o r i t h m 2 : : r e s i d u a l C a l c u l a t i n g T h r e a d ( Parameters : : arrayBounds ∗ arrayBounds ) { LARGE INTEGER time1 , time2 ; #pragma i v d e p f o r ( i n t i = arrayBounds−>s t a r t ; i <= arrayBounds−>end ; i ++){ QueryPerformanceCounter(& time1 ) ; p o p u l a t i o n P a r a m e t e r s N e w [ i ] . c h i S q = c a l c u l a t e R e s i d u a l 2 (& p o p u l a t i o n P a r a m e t e r s N e w [ i ] , arrayBounds−>threadID ) ; QueryPerformanceCounter(& time2 ) ; arrayBounds−>time += 1 0 0 0 ∗ ( d o u b l e ) ( time2 . QuadPart−time1 . QuadPart ) / ( f r e q . QuadPart ) ; i f ( populationParametersNew [ i ] . chiSq < populationParametersOld [ i ] . chiSq ) { populationParametersOld [ i ] = populationParametersNew [ i ] ; } } SetEvent ( arrayBounds−>h a n d l e ) ; return ; }  182  A.3. GeneticAlgorithm.cpp  A.3.6  calculateResidual  This is the function that actually calculates the residual. It loads the memory location of the first fit parameter into the pointer paramPointer, and passes this to the function that will calculate the model value at the specific data point. It then iterates over all data points.  d o u b l e G e n e t i c A l g o r i t h m 2 : : c a l c u l a t e R e s i d u a l 2 ( Parameters : : f i t P a r a m e t e r s ∗ parameters , i n t threadID ) { double t o t a l = 0; d o u b l e ∗ paramPointer = &( parameters −>A1) ; f o r ( i n t i = 0 ; i < nParams ; i ++){ paramArray [ threadID ] [ i ] = ∗ paramPointer ; paramPointer++; } #pragma i v d e p for ( int i = 0; i <  d a t a S e t L e n g t h ; i ++){  r e s i d u a l A r r a y [ threadID ] [ i ] = i n t e g r a t e L e g e n d r e ( d a t a S e t [ i ] [ 0 ] , d a t a S e t [ i ] [ 2 ] , d a t a S e t [ i ] [ 3 ] , paramArray [ threadID ] ) − d a t a S e t [ i ] [ 1 ] ; } #pragma i v d e p f o r ( i n t i = 0 ; i < d a t a S e t L e n g t h ; i ++){ t o t a l += r e s i d u a l A r r a y [ threadID ] [ i ] ∗ r e s i d u a l A r r a y [ threadID ] [ i ] ; } return total ; }  A.3.7  getYValue  getYValue calculates the model value if there is only one warping parameter.  d o u b l e G e n e t i c A l g o r i t h m 2 : : getYValue ( d o u b l e H, i n t t h e t a , d o u b l e T, d o u b l e c o n s t ∗ parameters ) { d o u b l e k f s q F 1 t a n = fToKfConstant ∗ s q r t ( p a r a m e t e r s [ 2 ] ) ∗ t a n A n g l e s [ t h e t a ]; d o u b l e k f s q F 2 t a n = fToKfConstant ∗ s q r t ( p a r a m e t e r s [ 3 ] ) ∗ t a n A n g l e s [ t h e t a ];  183  A.3. GeneticAlgorithm.cpp d o u b l e i n v H c o s = 1 . 0 / (H∗ c o s A n g l e s [ t h e t a ] ) ; d o u b l e s u r f 1 = p a r a m e t e r s [ 0 ] ∗ c o s (TWOPI∗ ( p a r a m e t e r s [ 1 0 ] + (H/ 3 0 . ) ∗ p a r a m e t e r s [ 1 5 ] ) / ( 2 . 0 ∗ c o s A n g l e s [ t h e t a ] ) ) ∗ exp(−OSC CONST∗ ( p a r a m e t e r s [ 6 ] ) ∗ ( p a r a m e t e r s [ 1 2 ] ) ∗ i n v H c o s ) ∗ (OSC CONST∗T∗ ( p a r a m e t e r s [ 1 2 ] ) ∗ i n v H c o s ) / s i n h (OSC CONST∗T ∗ ( p a r a m e t e r s [ 1 2 ] ) ∗ i n v H c o s ) ∗ c o s (TWOPI∗ ( p a r a m e t e r s [ 2 ] ∗ i n v H c o s+p a r a m e t e r s [ 8 ] ) ) ∗ j 0 (TWOPI∗ p a r a m e t e r s [ 4 ] ∗ i n v H c o s ∗ j 0 ( k f s q F 1 t a n ) ) ; d o u b l e s u r f 2 = p a r a m e t e r s [ 1 ] ∗ c o s (TWOPI∗ ( p a r a m e t e r s [ 1 1 ] + (H/ 3 0 . ) ∗ p a r a m e t e r s [ 1 6 ] ) / ( 2 . 0 ∗ c o s A n g l e s [ t h e t a ] ) ) ∗ exp(−OSC CONST∗ ( p a r a m e t e r s [ 7 ] ) ∗ ( p a r a m e t e r s [ 1 3 ] ) ∗ i n v H c o s ) ∗ (OSC CONST∗T∗ ( p a r a m e t e r s [ 1 3 ] ) ∗ i n v H c o s ) / s i n h (OSC CONST∗T ∗ ( p a r a m e t e r s [ 1 3 ] ) ∗ i n v H c o s ) ∗ c o s (TWOPI∗ ( p a r a m e t e r s [ 3 ] ∗ i n v H c o s+p a r a m e t e r s [ 9 ] ) ) ∗ j 0 (TWOPI∗ p a r a m e t e r s [ 5 ] ∗ i n v H c o s ∗ j 0 ( k f s q F 2 t a n ) ) ; r e t u r n ( s u r f 1+s u r f 2 ) ; }  A.3.8  integrateLegendre  integrateLegendre calculates the model value when there is more than one warping parameter, or if the other parameters are given a kz dependence.  d o u b l e G e n e t i c A l g o r i t h m 2 : : i n t e g r a t e L e g e n d r e ( d o u b l e H, i n t t h e t a , d o u b l e T, double ∗ parameters ) { double t o t a l 1 = 0; double t o t a l 2 = 0; d o u b l e k f s q F 1 t a n = fToKfConstant ∗ s q r t ( p a r a m e t e r s [ 2 ] ) ∗ t a n A n g l e s [ t h e t a ]; d o u b l e k f s q F 1 t a n 2 = 2 . 0 ∗ fToKfConstant ∗ s q r t ( p a r a m e t e r s [ 2 ] ) ∗ t a n A n g l e s [ theta ] ; d o u b l e k f s q F 2 t a n = fToKfConstant ∗ s q r t ( p a r a m e t e r s [ 3 ] ) ∗ t a n A n g l e s [ t h e t a ]; d o u b l e i n v H c o s = 1 . 0 / (H∗ c o s A n g l e s [ t h e t a ] ) ; ; #pragma i v d e p f o r ( i n t i = n I n t e g r a t i o n N o d e s P 5 ; i < n I n t e g r a t i o n N o d e s ; i ++){ t o t a l 1 += TWOPI ∗ i n t e g r a t i o n W e i g h t s [ i ] ∗ c o s (TWOPI∗ ( ( p a r a m e t e r s [ 2 ] + ( p a r a m e t e r s [ 4 ] ) ∗ c o s V a l s [ i ] ∗ j 0 ( k f s q F 1 t a n ) ) ∗ i n v H c o s+p a r a m e t e r s [ 8 ] ) ) ; t o t a l 2 += TWOPI ∗ i n t e g r a t i o n W e i g h t s [ i ] ∗ c o s (TWOPI∗ ( ( p a r a m e t e r s [ 3 ] + ( p a r a m e t e r s [ 5 ] ) ∗ c o s V a l s [ i ] ∗ j 0 ( k f s q F 2 t a n ) ) ∗ i n v H c o s+p a r a m e t e r s [ 9 ] ) ) ; } r e t u r n p a r a m e t e r s [ 0 ] ∗ c o s (TWOPI∗ ( p a r a m e t e r s [ 1 0 ] ) / ( 2 . 0 ∗ c o s A n g l e s [ t h e t a ] ) ) ∗ exp (−OSC CONST∗ ( p a r a m e t e r s [ 6 ] ) ∗ ( p a r a m e t e r s [ 1 2 ] ) ∗ i n v H c o s ) ∗ (OSC CONST∗T∗ ( p a r a m e t e r s [ 1 2 ] ) ∗ i n v H c o s ) / s i n h (OSC CONST∗T∗ ( p a r a m e t e r s [ 1 2 ] ) ∗ i n v H c o s ) ∗  184  A.4. FittingProgram.cpp t o t a l 1 + p a r a m e t e r s [ 1 ] ∗ c o s (TWOPI∗ ( p a r a m e t e r s [ 1 1 ] ) / ( 2 . 0 ∗ c o s A n g l e s [ t h e t a ] ) ) ∗ exp(−OSC CONST∗ ( p a r a m e t e r s [ 7 ] ) ∗ ( p a r a m e t e r s [ 1 3 ] ) ∗ i n v H c o s ) ∗ (OSC CONST∗ T∗ ( p a r a m e t e r s [ 1 3 ] ) ∗ i n v H c o s ) / s i n h (OSC CONST∗T∗ ( p a r a m e t e r s [ 1 3 ] ) ∗ inv H cos )∗ total2 ; }  A.4  FittingProgram.cpp  FittingProgram.cpp imports the data, takes the user-specified inputs for the crossing rate, the crossing probability, the number of generations, and creates the GeneticAlgorithm class object. After the number of iterations specified have executed, the current minimum parameter set is output to the screen (and to a file), and the user is prompted to input the next number of generations to execute, if needed.  #i n c l u d e ” s t d a f x . h” #i n c l u d e ” D a t a E x t r a c t o r . h” #i n c l u d e ” G e n e t i c A l g o r i t h m 2 . h” u s i n g namespace s t d ; i n t tm ai n ( i n t argc , TCHAR∗ argv [ ] ) { LARGE INTEGER time1 , time2 , f r e q ; // s t o r e s t i m e s and CPU f r e q u e n c y f o r profiling QueryPerformanceFrequency (& f r e q ) ; // g e t s CPU f r e q u e n c y d o u b l e s c a l e F a c t o r , c r o s s i n g P r o b a b i l i t y ; // how f a r t h e a l g o r i t h m ” l o o k s ” when m u t a t i n g , and c h a n c e o f m u t a t i o n i n t n G e n e r a t i o n s ; // number o f g e n e r a t i o n s , w h i c h i s b a s i c a l l y number o f iterations try { D a t a E x t r a c t o r e x t r a c t o r ( ”C: / U s e r s / r o o t / Dropbox / T h e s i s /C++ data a n a l y s i s / f u l l D a t a 6 7 . dat ” ) ; // g e t t h e da ta , w h i c h i s f i e l d / a m p l i t u d e / a n g l e number . maybe bad f o r m a t ? s h o u l d b e c o n s i s t e n t w i t h m at he m at ic a maybe . d o u b l e ∗∗ data = e x t r a c t o r . getDataArray ( ) ; // p u t s d a t a i n 2 x2 a r r a y o f doubles . i n t n L i n e s = e x t r a c t o r . getNumberOfLines ( ) ; // number o f l i n e s i n t h e d a t a set cout<<” S c a l e F a c t o r : ” ; c i n >>s c a l e F a c t o r ; // s e t s c a l e f a c t o r . what i s a good v a l u e ? . 1 ? i forget cout<<” C r o s s i n g P r o b a b i l i t y : ” ;  185  A.4. FittingProgram.cpp c i n >>c r o s s i n g P r o b a b i l i t y ; // c r o s s i n g p r o b . p r o b a b l y around . 7 i s good ? G e n e t i c A l g o r i t h m 2 g e n e t i c A l g o r i t h m 2 ( data , nLines , 2 0 0 0 , s c a l e F a c t o r , c r o s s i n g P r o b a b i l i t y ) ; // c o n s t r u c t s t h e a l g o r i t h m o b j e c t . d a t a s e t / lines / population size / scale / crossing . while ( true ) // p o p u l a t i o n i s s e t f o r e v e r . maybe c h a n g e t h i s ? { cout<<”Number o f g e n e r a t i o n s : ” ; c i n >>n G e n e r a t i o n s ; // number o f g e n e r a t i o n s n i e i t e r a t i o n s . QueryPerformanceCounter(& time1 ) ; g e n e t i c A l g o r i t h m 2 . calculateMinimum ( ) ; g e n e t i c A l g o r i t h m 2 . printMinimumParameters ( ) ; geneticAlgorithm2 . calculateNewGenerations ( nGenerations ) ; g e n e t i c A l g o r i t h m 2 . calculateMinimum ( ) ; g e n e t i c A l g o r i t h m 2 . printMinimumParameters ( ) ;  QueryPerformanceCounter(& time2 ) ; cout<<”Time f o r ”<<” g e n e t i c a l g o r i t h m ”<<” i s : ” <<1000∗( d o u b l e ) ( time2 . QuadPart−time1 . QuadPart ) / ( f r e q . QuadPart )<<”ms”<<endl <<e n d l ; } } catch ( const char ∗ s t r ) { cout<<s t r <<e n d l ; } return 0; }  186  Appendix B  Background Subtraction The following code produces an interface for removing background magnetoresistance from oscillatory data. The variable “p00” is a plot of the data that the inteface is built over top of. The variable “backSubDat” stores the oscillatory part of the data after background subtraction. Manipulate[{Show[Graphics[{Red, BSplineCurve[pts, SplineDegree → d]}, PlotRange → {{27, 71}, {0, 1.0}}], p00, Axes → False, AspectRatio → .6, TicksStyle → Directive[Black, 18], Frame → True, FrameStyle → Directive[Black, Thick], FrameTicksStyle → Directive[Black, 25], ImageSize → 800], ListPlot[Module[{spline = BSplineFunction[pts]}, backSubDat = Table[{spline[i][[1]], dataFunction[spline[i][[1]]]/spline[i][[2]] − 1}, {i, .05, .9942, .0005}]], ImageSize → 300, AspectRatio → .6, PlotRange → All]}, {{d, 3}, 1, Length[pts], 1}, {{pts, {{25.69, 0.064}, {29.48, 0.215}, {34.54, 0.291}, {40.1, 0.374}, {44.18, 0.47}, {49.12, 0.544}, {58.97, 0.645}, {70.33, 0.772}}}, {25, −.1}, {73, 1.0}, Locator, LocatorAutoCreate → 2}]  187  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0072925/manifest

Comment

Related Items