"Science, Faculty of"@en . "Physics and Astronomy, Department of"@en . "DSpace"@en . "UBCV"@en . "Veenstra, Christian Neil"@en . "2013-04-10T09:09:53Z"@en . "2013"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "This thesis represents two bodies of work: a detailed look at what angle- resolved photoemission spectroscopy (ARPES) measures, as well as ARPES and circularly polarized photon spin- and angle-resolved photoemission spectroscopy (CPS-ARPES) measurements on the unconventional superconductor Sr\u00E2\u0082\u0082RuO\u00E2\u0082\u0084.\n\nIn the first part I present a study of both established methods of ARPES analysis and some new variations on model spectral functions. This modelling was done in a realistic regime, yet far from the limits often assumed. Away from these limits I show that any \"effective coupling\" inferred from quasiparticle renormalizations differs drastically and unpredictably from the true coupling. Conversely, I show that perturbation theory retains good predictive power where expected, that the momentum dependence of the self-energy can be revealed via the relationship between velocity renormalization and quasiparticle strength, and that it is often possible to infer the self-energy and bare electronic structure through lineshape analysis.\n\nIn the second part I present experimental ARPES and CPS-ARPES data on Sr\u00E2\u0082\u0082RuO\u00E2\u0082\u0084. Newly discovered and unexplained ARPES features are characterized and compared with a variety of different possible structural distortions through bulk and slab local-density approximation (LDA) band structure calculations. I thereby rule out phases driven by electronic interaction, such as Dirac- or Rashba-type surface states, and instead find that there exists a progressive structural modulation whereby both the surface and (at a minimum) sub-surface layers exhibit a (\u00E2\u0088\u009A2 x \u00E2\u0088\u009A2)R45\u00C2\u00B0 reconstruction.\n\nThrough CPS-ARPES on Sr\u00E2\u0082\u0082RuO\u00E2\u0082\u0084 I also directly demonstrate that the effects of spin-orbit (SO) coupling are not limited to a modification of the band structure, but fundamentally entangle the spin and spatial parts of the wave-function. This must drive the superconducting state in Sr\u00E2\u0082\u0082RuO\u00E2\u0082\u0084 to be even more unconventional than is generally assumed, with mixing between singlet and triplet states that varies around the Fermi surface, and thereby offers a possible resolution to a number of experiments that clash with the categorization of Sr\u00E2\u0082\u0082RuO\u00E2\u0082\u0084 as a hallmark spin-triplet chiral p-wave superconductor."@en . "https://circle.library.ubc.ca/rest/handle/2429/44166?expand=metadata"@en . "Spin- and angle-resolved photoemission spectroscopy on unconventional superconductor strontium ruthenate by Christian Neil Veenstra B.Sc., McMaster University, 2005 M.Sc., 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) April 2013 c\u00C2\u00A9 Christian Neil Veenstra 2013 Abstract This thesis represents two bodies of work: a detailed look at what angle- resolved photoemission spectroscopy (ARPES) measures, as well as ARPES and circularly polarized photon spin- and angle-resolved photoemission spec- troscopy (CPS-ARPES) measurements on the unconventional superconduc- tor Sr2RuO4. In the first part I present a study of both established methods of ARPES analysis and some new variations on model spectral functions. This mod- elling was done in a realistic regime, yet far from the limits often assumed. Away from these limits I show that any \u00E2\u0080\u009Ceffective coupling\u00E2\u0080\u009D inferred from quasiparticle renormalizations differs drastically and unpredictably from the true coupling. Conversely, I show that perturbation theory retains good predictive power where expected, that the momentum dependence of the self-energy can be revealed via the relationship between velocity renormal- ization and quasiparticle strength, and that it is often possible to infer the self-energy and bare electronic structure through lineshape analysis. In the second part I present experimental ARPES and CPS-ARPES data on Sr2RuO4. Newly discovered and unexplained ARPES features are characterized and compared with a variety of different possible structural distortions through bulk and slab local-density approximation (LDA) band structure calculations. I thereby rule out phases driven by electronic inter- action, such as Dirac- or Rashba-type surface states, and instead find that there exists a progressive structural modulation whereby both the surface and (at a minimum) sub-surface layers exhibit a ( \u00E2\u0088\u009A 2\u00C3\u0097\u00E2\u0088\u009A2)R45\u00E2\u0097\u00A6 reconstruc- tion. Through CPS-ARPES on Sr2RuO4 I also directly demonstrate that the effects of spin\u00E2\u0080\u0093orbit (SO) coupling are not limited to a modification of the ii Abstract band structure, but fundamentally entangle the spin and spatial parts of the wave-function. This must drive the superconducting state in Sr2RuO4 to be even more unconventional than is generally assumed, with mixing between singlet and triplet states that varies around the Fermi surface, and thereby offers a possible resolution to a number of experiments that clash with the categorization of Sr2RuO4 as a hallmark spin-triplet chiral p-wave superconductor. iii Preface I have published, as first author, almost all of the work presented here. That which is not published was either used to support these publications or is in preparation for publication. That is not to say that I produced this work alone; here I will detail the relative contributions of my co-authors to all chapters and manuscripts. Chapter 2 \u00E2\u0080\u0093 Spectral function analysis Portions of Chapter 2 and Appendices A and D have been published in Refs. 1 and 2. In these publications, as in these chapters, I was responsible for writ- ing and preparing the manuscript, figures, and derivations, as well as man- aging collaboration and feedback from the co-authors. I was also responsible for all data analysis, including writing all software to perform this analysis. G. L. Goodvin, a fellow student and co-author, was primarily responsible for creating a code library that applied the momentum average approximation to the two model systems studied here, which he used in his own publica- tions [3, 4]. I wrote a software interface to this library to generate the large quantity of simulated data required for analysis. M. Berciu, as Goodvin\u00E2\u0080\u0099s supervisor and a member of my advisory committee, and A. Damascelli, as my supervisor, provided direction and oversight to guide the project. A. Damascelli provided guidance particularly on experimental aspects and how people have performed similar analysis in the past, while M. Berciu provided guidance on theoretical aspects in addition to insight on the mo- mentum average approximation which she devised [5]. iv Preface Chapter 3 \u00E2\u0080\u0093 Chamber characterization Chapter 3 is based on analysis and experiments done to characterize the in- house angle-resolved photoemission spectroscopy (ARPES) system at UBC, as well as specifically to characterize the data found in Chapter 4. Although many people have assisted in the characterization of the system, all work presented in this chapter is my own, with three exceptions: the electron gun data presented in \u00C2\u00A7 3.1.2 was taken by G. Levy; the gold Fermi edge data presented in Fig. 3.8 panel (a) was taken by B. Ludbrook, as it corresponds with data on Sr2RuO4 used in Chapter 4, as described below; and the gas cell photoemission data presented in Fig. 3.9 was taken by SPECS, the man- ufacturer of our analyzer. I performed all the analysis presented on these data sets and wrote most of software to do so (see last section of this preface: Software for ARPES data preparation). Chapter 4 \u00E2\u0080\u0093 ARPES on Sr2RuO4 Chapter 4 and Appendix B are based on work and experiments conducted with the ARPES group at UBC using our in-house spectroscopy system, a system I have played a large role in helping to build and characterize over the past seven years. Portions of Chapter 4 and Appendix B have been submitted for publication [6]. The long co-author list represents two dis- tinct groups: C. N. Veenstra, Z.-H. Zhu, B. Ludbrook, G. Levy, A. Nicolaou, J. A. Rosen, and R. Comin (in addition to others) have assisted with the con- struction and/or ongoing maintenance and characterization of the system used under the direction of A. Damascelli, while S. Kittaka and Y. Maeno grew the samples. Z.-H. Zhu, under the direction of I. S. Elfimov, was re- sponsible for running the local-density approximation (LDA) calculations. In this publication, as in this chapter, I was responsible writing and prepar- ing the manuscript and figures, as well as managing collaborations with the co-authors. Additionally, I performed all the data analysis and wrote much of the software required to do so (see last section of this preface: Soft- ware for ARPES data preparation). Most, but not all, of the data presented in this chapter was also physically taken by me using samples I v Preface had prepared \u00E2\u0080\u0093 the exception being the data presented in Fig. 4.3 panels (d,e) and Fig. 4.4 panels (b,c,d), which are based on the same data set that was taken by B. Ludbrook using a sample I had polished and prepared to be free of ruthenium metal inclusions. B. Ludbrook and Z.-H. Zhu also took additional data on Sr2RuO4, including some as a function of potassium dop- ing. This additional data, while helpful to piece together an overall picture of what was going on, was not presented in [6], nor is it discussed in this thesis. M. Capsoni, as a summer student working with our group under my direction, also assisted me in taking some of the data presented here and additional data not presented. Chapter 5 \u00E2\u0080\u0093 CPS-ARPES on Sr2RuO4 Chapter 5 and Appendix C are based on work performed at the Swiss Light Source Complete Photoemission Experiment (COPHEE) endstation. Por- tions of Chapter 5 appear in a manuscript under preparation for submission. C. N. Veenstra, Z.-H. Zhu, B. Ludbrook, and A. Nicolaou were the group that travelled together from UBC to COPHEE to perform the measurements dur- ing one week of beamtime. Owing to the long acquisition times necessary (data used to make Fig. 5.6, for example, took 16 hours) and close group en- vironment of a short beamtime, it is only fair to say that everybody was re- sponsible for all data taken. Although I led this group during the beamtime, everybody also took part in discussions regarding best courses of action, and in this regard Z.-H. Zhu was particularly helpful. M. Raichle, a postdoctoral fellow working with our group, had been in charge of the project idea before I took over and even led some measurements on Sr2RuO4 at COPHEE with a similar goal. Although the project did not succeed at that time, review- ing Raichle\u00E2\u0080\u0099s data and the avenues previously explored was very helpful in forming a new plan. B. Slomski and G. Landolt were graduate students who, under the direction J. H. Dil, were ultimately responsible for the well-being of the COPHEE system and directed us on physically running experiments there, in addition to providing valuable insight into how the system works. S. Kittaka and Y. Maeno were responsible for sample growth. I. S. Elfimov, vi Preface M. W. Haverkort, and A. Damascelli all served in a supervisory role on this project and have provided much valuable insight. M. W. Haverkort, in par- ticular, has provided an exceptional amount of guidance on the theoretical side of the work and also wrote the software libraries I used that execute the LMTO/NMTO+SO method on Sr2RuO4, provide the quantum operators on the correct basis, and allow for the illustration of orbital character. For this manuscript and chapter I have prepared most of the text and all of the figures; \u00C2\u00A7 5.3 was originally drafted by M. W. Haverkort. I performed all of the analysis on the circularly polarized photon spin- and angle-resolved photoemission spectroscopy (CPS-ARPES) data and also wrote the software to do so. Software for ARPES data preparation I wrote all of the software to perform the data analysis presented in this the- sis; however, (as mentioned above for Chapters 3 and 4) some aspects of the software required for ARPES data preparation were co-authored with other members of the ARPES group at UBC. In order to analyze ARPES data, our group wrote a full object-oriented software suite in the Interactive Data Language (IDL), with many command-driven tools but with a graphical user interface (GUI) for most steps. This effort was undertaken largely by G. Levy, J. A. Rosen, and me. Each of us developed most of our own code regarding data management, workflow, and analysis, with many core rou- tines (or pieces of them) for data preparation shared and often co-authored. In total this suite involved 138 000 lines of code1, which could be roughly divided as 20k managed by Rosen, 86k managed by Levy2, 2k produced by summer students, and 30k that I managed \u00E2\u0080\u0093 although there has been a lot of intermingling, particularly between Levy and me. This total naturally also includes many routines not directly used for this thesis, including soft- ware for low-energy electron diffraction (LEED) and x-ray photoelectron spectroscopy (XPS) analysis. 1As counted by CLOC 1.56; this does not include blank lines or comments. 2Levy\u00E2\u0080\u0099s code base also includes many libraries for quickly building GUI interfaces, modified to suit our own purposes. vii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Abbreviations and Symbols . . . . . . . . . . . . . . . . . xvii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 A quantum many-body problem . . . . . . . . . . . . . . . . 1 1.2 ARPES and spectral function analysis . . . . . . . . . . . . . 1 1.3 ARPES on Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 CPS-ARPES, Spin\u00E2\u0080\u0093orbit coupling, and superconductivity in Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Spectral function analysis . . . . . . . . . . . . . . . . . . . . 6 2.1 The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . 11 2.1.2 An example spectral function . . . . . . . . . . . . . 12 2.2 Quasiparticle analysis . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Quasiparticle dispersion, velocity, mass, strength, and their renormalizations . . . . . . . . . . . . . . . . . . 14 viii Table of Contents 2.2.2 Renormalization as a function of coupling strength . 19 2.2.3 Renormalizations for different phonon energies . . . . 21 2.2.4 Renormalizations in different dimensions . . . . . . . 24 2.2.5 Perturbation theory predictions near \u00CE\u00BB = 0 . . . . . . 24 2.2.6 Verifying momentum independence of the self-energy 29 2.3 Self-energy analysis . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.1 Momentum-independent self-energies . . . . . . . . . 31 2.3.2 Momentum-dependent self-energies . . . . . . . . . . 37 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Chamber characterization . . . . . . . . . . . . . . . . . . . . 41 3.1 Angular resolution . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.2 Electron gun tests . . . . . . . . . . . . . . . . . . . . 43 3.1.3 Sample tests . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 Event counting . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 Functional forms and a comparison of different measures . . 60 4 ARPES on Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1 Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands . . . 64 4.1.1 Additional features found . . . . . . . . . . . . . . . . 68 4.1.2 Exploring the 3K phase . . . . . . . . . . . . . . . . . 72 4.1.3 Time-dependent surface degradation and dispersion analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1.4 Polarization dependence . . . . . . . . . . . . . . . . 85 4.1.5 A comparison with folded features in the cuprates . . 88 4.2 LDA calculations and structural distortions . . . . . . . . . . 88 4.2.1 Preliminary bulk calculations . . . . . . . . . . . . . . 90 4.2.2 Slab choice . . . . . . . . . . . . . . . . . . . . . . . . 94 4.2.3 Slab results . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ix Table of Contents 5 CPS-ARPES on Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . 99 5.1 Single-particle predictions \u00E2\u0080\u0093 entanglement at the Fermi en- ergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2 Measuring spin\u00E2\u0080\u0093orbital entanglement . . . . . . . . . . . . . 104 5.2.1 States measured . . . . . . . . . . . . . . . . . . . . . 104 5.2.2 ARPES results . . . . . . . . . . . . . . . . . . . . . . 106 5.2.3 CPS-ARPES results . . . . . . . . . . . . . . . . . . . 106 5.3 Implications for pairing . . . . . . . . . . . . . . . . . . . . . 113 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Appendices A Spectral function analysis . . . . . . . . . . . . . . . . . . . . 139 A.1 Calculating renormalizations from perturbation theory . . . 139 A.2 Kramers\u00E2\u0080\u0093Kronig bare-band fitting . . . . . . . . . . . . . . . 140 B ARPES data preparation . . . . . . . . . . . . . . . . . . . . . 145 B.1 Angle (slit) correction . . . . . . . . . . . . . . . . . . . . . . 148 B.2 Energy (gold) correction . . . . . . . . . . . . . . . . . . . . 148 B.3 Angle to k-space mapping . . . . . . . . . . . . . . . . . . . . 149 C CPS-ARPES data preparation . . . . . . . . . . . . . . . . . 152 C.1 How polarization is measured . . . . . . . . . . . . . . . . . . 152 C.2 Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? . . . . . 157 D Software routines . . . . . . . . . . . . . . . . . . . . . . . . . . 166 D.1 Monte Carlo integration using the GNU VEGAS algorithm . 166 D.1.1 shared.h . . . . . . . . . . . . . . . . . . . . . . . . . 167 D.1.2 Holstein.cpp . . . . . . . . . . . . . . . . . . . . . . . 167 D.1.3 Breathing.cpp . . . . . . . . . . . . . . . . . . . . . . 168 x Table of Contents D.1.4 Integrator.cpp . . . . . . . . . . . . . . . . . . . . . . 169 D.1.5 makefile . . . . . . . . . . . . . . . . . . . . . . . . . 184 D.1.6 do them all . . . . . . . . . . . . . . . . . . . . . . . . 186 xi List of Tables 4.1 A comparison between Fermi surface volumes, velocities, and their renormalizations as measured by ARPES, LDA+SO, and dHvA for Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . 81 xii List of Figures 2.1 The spectral function and self-energies of the Holstein po- laron, for different microscopic couplings . . . . . . . . . . . . 9 2.2 A(k, \u00CF\u0089) calculated for the momentum-independent Holstein self-energy in 1D; the quasiparticle dispersion, perturbation theory prediction, and bare-band are also shown . . . . . . . 17 2.3 A(k, \u00CF\u0089) calculated for a momentum-dependent self-energy from coupling to a single breathing mode in 1D; the quasiparticle dispersion, perturbation theory prediction, and bare-band are also shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Effective coupling plotted vs. the true dimensionless coupling 20 2.5 Effective coupling plotted vs. the true dimensionless coupling, for a range of phonon energies . . . . . . . . . . . . . . . . . . 22 2.6 Effective coupling plotted vs. the true dimensionless coupling, for different dimensionalities . . . . . . . . . . . . . . . . . . . 23 2.7 \u00E2\u0088\u0082\u00CE\u00BBeff/\u00E2\u0088\u0082\u00CE\u00BB \u00E2\u0088\u00A3\u00E2\u0088\u00A3 k=0,\u00CE\u00BB=0 as a function of phonon energy, for the Hol- stein model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 \u00E2\u0088\u0082\u00CE\u00BBeff/\u00E2\u0088\u0082\u00CE\u00BB \u00E2\u0088\u00A3\u00E2\u0088\u00A3 k=0,\u00CE\u00BB=0 as a function of phonon energy, for coupling to a breathing mode . . . . . . . . . . . . . . . . . . . . . . . 27 2.9 Diagrammatic description of MDC self-energy analysis . . . . 33 2.10 MDC self-energy analysis performed on the Holstein model at low coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.11 Deviation between extracted and known self-energies . . . . . 35 2.12 MDC self-energy analysis performed on the breathing mode model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1 Illustration of slit array and slit array data . . . . . . . . . . 42 xiii List of Figures 3.2 Angular resolution tests in WAM, using an electron gun . . . 44 3.3 Angular resolution tests in LAD, using an electron gun . . . . 45 3.4 Angular resolution tests in WAM, on Sr2RuO4 . . . . . . . . 47 3.5 Angular resolution tests in LAD, on Sr2RuO4 . . . . . . . . . 48 3.6 Illustration of energy discrimination through the hemispher- ical analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.7 Energy resolution test in LAD, on gold . . . . . . . . . . . . . 53 3.8 Energy resolution test in WAM, on gold . . . . . . . . . . . . 54 3.9 Gas cell photoemission demonstrating the form of single elec- tron events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.10 A zoom-in on a group of events before and after the applica- tion of a single-event filter . . . . . . . . . . . . . . . . . . . . 58 3.11 A comparison of Gaussian and Lorentzian functions . . . . . 61 4.1 The near-FS bulk band structure of Sr2RuO4 from LDA+SO 65 4.2 Laue diffraction pattern for sample alignment on Sr2RuO4 . . 66 4.3 Expected vs. measured Fermi surface of Sr2RuO4 . . . . . . . 69 4.4 MDC cuts on the Fermi surface of Sr2RuO4 . . . . . . . . . . 70 4.5 Orientation of the Sr2RuO4 single crystal that was polished to produces pieces with and without the presence of the 3K phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Polarized light microscopy image of the 3K phase . . . . . . . 75 4.7 FS map of Sr2RuO4 as measured by ARPES on fresh samples both without and with the presence of the 3K phase . . . . . 76 4.8 Time evolution of a 45\u00E2\u0097\u00A6 MDC cut at EF . . . . . . . . . . . . 79 4.9 Band maps for all data in Fig. 4.8 . . . . . . . . . . . . . . . . 80 4.10 Polarization dependence of the ARPES FS of Sr2RuO4 . . . . 83 4.11 Polarization dependence of selected FS features of Sr2RuO4 . 84 4.12 LDA+SO bulk band structure of Sr2RuO4, with incremental octahedra rotations . . . . . . . . . . . . . . . . . . . . . . . . 91 4.13 Bulk band structure of Sr2RuO4, with buckling in oxygen octahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.14 Bulk band structure of Sr2RuO4, with electronic interaction U 93 xiv List of Figures 4.15 Atomic positions in the five-layer slab used for LDA+SO cal- culations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.16 Results of LDA+SO slab calculations . . . . . . . . . . . . . . 97 5.1 Band structure of Sr2RuO4 and expectation values of ~l \u00C2\u00B7 ~s . . 101 5.2 Orbital character as well as expectation value of ~l \u00C2\u00B7 ~s, and its components around the Fermi surface sheets of Sr2RuO4 . . . 102 5.3 Illustration of the states measured by CPS-ARPES . . . . . . 105 5.4 Spin-integrated ARPES on 5.2 K Sr2RuO4 at the \u00CE\u0093 point, performed at UBC . . . . . . . . . . . . . . . . . . . . . . . . 107 5.5 CPS-ARPES on Sr2RuO4 at the \u00CE\u0093 point, performed at COPHEE with 24 eV photon energy . . . . . . . . . . . . . . . . . . . . 109 5.6 CPS-ARPES on Sr2RuO4 at the \u00CE\u0093 point, performed at COPHEE with 56 eV photon energy . . . . . . . . . . . . . . . . . . . . 110 5.7 CPS-ARPES on Sr2RuO4 at the \u00CE\u0093 point, performed at COPHEE with 56 eV photon energy over an extended energy range . . . 111 5.8 Orbital and spin character around the Fermi surface sheets of Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.9 Orbitals illustrated in greater detail, for selected k-points in Sr2RuO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.10 \u00E3\u0080\u0088~sk \u00C2\u00B7 ~s\u00E2\u0088\u0092k \u00E3\u0080\u0089 around the Fermi surface sheets in Sr2RuO4 . . . . 117 A.1 Tail extrapolation for use in KK transformations . . . . . . . 142 B.1 An illustration of ARPES data preparation workflow . . . . . 146 B.2 A screen shot of IDL while defining peak positions in prepa- ration for correcting angular distortions . . . . . . . . . . . . 147 B.3 A screen shot of the GUI from which offset angles are chosen; this GUI drives the angle-to\u00E2\u0080\u0093momentum space transformation 150 C.1 Illustration of the Mott detector setup at COPHEE . . . . . 153 C.2 Raw detector data from CPS-ARPES measurements . . . . . 155 C.3 Spin-polarized photoemission from a model SO-split system . 158 C.4 Peak splitting simulation example, varying the splitting . . . 159 xv List of Figures C.5 Peak splitting simulation example, varying the peak width . . 160 C.6 Peak splitting simulation example, varying the background . 161 C.7 Peak splitting simulation example, varying the splitting with background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 C.8 Peak splitting simulation example, varying the width with background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 xvi List of Abbreviations and Symbols Acronyms ARPES angle-resolved photoemission spectroscopy a.u. arbitrary units BCT body-centred tetragonal BZ Brillouin zone COPHEE Complete Photoemission Experiment CPS-ARPES circularly polarized photon spin- and angle-resolved photoemission spectroscopy dHvA de Haas\u00E2\u0080\u0093van Alphen EDC energy distribution curve FS Fermi surface FWHM full width at half maximum GUI graphical user interface HWHM half width at half maximum IDL Interactive Data Language KK Kramers\u00E2\u0080\u0093Kronig xvii Frequently used symbols KKBF Kramers\u00E2\u0080\u0093Kronig bare-band fitting LAD low-angular-dispersion mode LDA local-density approximation LEED low-energy electron diffraction LMTO linear muffin-tin orbital LSDA local-spin-density approximation MA momentum average approximation; MA(n) refers to the n terms kept exactly [4, 5], while MA(v,n) includes variational considerations [3]. MDC momentum distribution curve NMTO order-N muffin-tin orbital SARPES spin- and angle-resolved photoemission spectroscopy SO spin\u00E2\u0080\u0093orbit SPECS SPECS Surface Nano Analysis GmbH TB tight-binding UBC University of British Columbia WAM wide-angle mode XPS x-ray photoelectron spectroscopy Frequently used symbols \u00E2\u008A\u0095 (\t) is used to designate circular plus (minus) polarization of light. \u00E2\u0080\u00B2 primes are used to denote real/imaginary parts in Chapter 2, folded fea- tures in Chapter 4, and the detector space coordinate system in Ap- pendix C. xviii Frequently used symbols \u00CE\u00B1 a FS-crossing band in Sr2RuO4; see Fig. 4.1. \u00CE\u00B2 a FS-crossing band in Sr2RuO4; see Fig. 4.1. \u00CE\u00B3 a FS-crossing band in Sr2RuO4; see Fig. 4.1. \u00CE\u0093 the origin in the momentum space BZ; see Fig. 4.1. \u00CE\u00B4 a band that can sometimes be predicted to cross the FS in Sr2RuO4 for certain lattice distortions but that is not observed to do so; see Fig. 4.1. \u00CE\u00B5bk the \u00E2\u0080\u009Cbare-band\u00E2\u0080\u009D single-particle electronic dispersion, as would be found with no interactions, as a function of momentum k; see Eq. 2.1 in general or Fig. 2.2 and Eq. 2.3 for the models used in Chapter 2. \u00CE\u00B5qk the quasiparticle dispersion, identified as the EDC peak location in the spectral function as a function of momentum k; see Fig. 2.2. \u00CE\u00B5pertk the quasiparticle dispersion, as predicted by perturbation theory; see Fig. 2.2 and Eqs. 2.8 and 2.9. \u00CE\u00A3(k, \u00CF\u0089) the self-energy, or its real (\u00CE\u00A3\u00E2\u0080\u00B2) or imaginary (\u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2) part; see Eqs. 2.1 and 2.10. \u00CE\u00BB the dimensionless coupling, \u00CE\u00BB=\u00E3\u0080\u0088|g|2\u00E3\u0080\u0089/2Dt\u00E2\u0084\u00A6; see Eq. 2.6. \u00CE\u00BBeff the \u00E2\u0080\u009Ceffective coupling\u00E2\u0080\u009D, as would be interpreted in the Migdal limit based on quasiparticle renormalization. \u00CF\u0089 energy. \u00E2\u0084\u00A6 phonon energy. a lattice constant; for Chapter 2 see Eq. 2.7, and for Sr2RuO4 see Fig. 4.15. A(k, \u00CF\u0089) the spectral function; see Eqs. 2.1. b a subscript to indicate a bulk-like band; see Fig. 4.3. D dimension. xix Frequently used symbols EF the Fermi energy. Ek kinetic energy; also an analyzer setting for the kinetic energy (as pho- toemitted) an electron requires to be mapped to the centre of the detector. Epass pass energy \u00E2\u0080\u0093 an analyzer setting for the kinetic energy (inside the analyzer) an electron requires to pass through the centre of the hemi- spheres. This also controls the energy resolution of the analyzer. g the coupling term between phonons and electrons in the Hamiltonian; see Eqs. 2.2, 2.4, and 2.5. k momentum. km(\u00CF\u0089) momentum location of MDC peak maxima as a function of energy \u00CF\u0089; related to, but not the same as, \u00CE\u00B5k. mk dispersion mass, defined as 1/mk=\u00E2\u0088\u0082 2\u00CE\u00B5k/\u00E2\u0088\u0082k 2; see Fig. 2.2. M a location in the momentum space BZ; see Fig. 4.1. s a subscript to indicate a surface-like band; see Fig. 4.3. t tight-binding hopping; see Eq. 2.7. Tc critical temperature of a phase transition, usually between supercon- ducting and normal. vk dispersion velocity, defined as vk=\u00E2\u0088\u0082\u00CE\u00B5k/\u00E2\u0088\u0082k; see Fig. 2.2. X a location in the momentum space BZ; see Fig. 4.1. Z a location in the momentum space BZ; see Fig. 4.1. Zqk quasiparticle strength, where Z q k = \u00E2\u0088\u00AB q A(k, \u00CF\u0089)d\u00CF\u0089 is the integral over the coherent, quasiparticle part of the spectral function only. xx Acknowledgements Many, many people have helped me along my journey to complete this work, but some deserve special mention. Thank you to my fellow lab mates: Giorgio Levy, Jeff Mottershead, Thomas Roth, David Fournier, Ryan Wicks, Jason Zhu, Bart Ludbrook, Nik Ingle and Riccardo Comin. As well, thanks to Doug Wong and Pinder Dosanjh for technical assistance. After all those years we did it. We built a world-class ARPES system, and it works. Jason, you deserve a second thank-you for your additional support as we wrote our respective theses and publications together up in AMPEL 346. Thank you as well to those theorists who\u00E2\u0080\u0099ve taken the time to help me, an experimentalist, understand - Maurits Haverkort, Mona Berciu, Ilya Elfimov, and Glen Goodvin. Thank you to all the personelle who keep the COPHEE beamline at the Swiss Light Source going, and to Hugo Dil, Bartosz Slomski, and Gabriel Landolt for showing us how to use it. Thanks especially to Hugo, who met with me to discuss the experiment both before and after the beamtime, thereby ensuring its success. Thank you to Yoshi Maeno and Shunichiro Kittaka; although we\u00E2\u0080\u0099ve never met, without your samples I would have nothing to measure. Thank you to Iva Cheung for editing this thesis, and showing me the difference between \u00E2\u0080\u009Cwhich\u00E2\u0080\u009D and \u00E2\u0080\u009Cthat\u00E2\u0080\u009D. Finally I would like to thank my supervisor, Andrea Damascelli. Thank you for not only coming up with ideas, but also the funding and guidance to see them through. xxi Chapter 1 Introduction 1.1 A quantum many-body problem Although the foundations of modern quantum mechanics were laid at the beginning of the 20th century, the \u00E2\u0080\u009Ccomplications\u00E2\u0080\u009D that arise from the na- ture of the many-body problem in many materials have survived decades of active research and remain a playground for condensed matter physi- cists. In materials where one can no longer consider each electron to feel only the average effect from a \u00E2\u0080\u009Csea\u00E2\u0080\u009D of its neighbours the complex network of microscopic particle interactions can foster a diverse variety of observ- able phenomena. From the first discovery of Bardeen\u00E2\u0080\u0093Cooper\u00E2\u0080\u0093Schrieffer superconductivity over a century ago to Mott\u00E2\u0080\u0093Hubbard insulation, colossal magnetoresistance, and unconventional superconductivity, these materials continue to provide fascinating manifestations of quantum mechanics and the many-body problem for theorists and experimentalists alike. 1.2 ARPES and spectral function analysis By measuring both the angle and energy of photoemitted electrons, angle- resolved photoemission spectroscopy (ARPES) gives access to information about the energy and momentum of the electrons in a material. This makes it a powerful tool in the study of materials where microscopic interactions cannot be ignored [7]. In Chapter 2 we present a detailed study on the analysis of ARPES data, in particular focusing on quasiparticle renormal- izations and self-energy analysis. In the quasiparticle picture, a simplified model of the complex many- body problem inherent in such materials is constructed through the intro- 1 1.2. ARPES and spectral function analysis duction of quasiparticles, which themselves behave in a more understandable manner. In this picture the quasiparticle \u00E2\u0080\u0093 a collection of fundamental par- ticles, excitations, and the interactions between them \u00E2\u0080\u0093 will have its own energy\u00E2\u0080\u0093momentum dispersion relation as well as a self-energy. The quasi- particle renormalizations are the ratios between various properties of the quasiparticle dispersion and those of the \u00E2\u0080\u009Cbare\u00E2\u0080\u009D-particle (as they would be were the particle not \u00E2\u0080\u009Cdressed\u00E2\u0080\u009D by its association with all the interactions included in the collection). The self-energy, a complex quantity, represents the difference in energy between the quasiparticle and bare-particle through the real part, and the inverse lifetime of the quasiparticle through the imag- inary part. In materials well-suited to this picture the quasiparticle is not only an abstract theoretical concept, but something whose properties \u00E2\u0080\u0093 the quasiparticle dispersion and lineshape \u00E2\u0080\u0093 can actually be observed through ARPES and related to the renormalization and self-energy. These are the relationships studied in Chapter 2, where we present a study of quasiparti- cle renormalization and self-energy analysis. Both established methods of ARPES analysis as well as some new variations are applied to simulated single-electron removal spectral functions, which are generated using two different models of electron-phonon coupling in a filled band system. The quasiparticle renormalizations found in these simulations are ini- tially surprising when compared to the microscopic coupling found in the Hamiltonian. While these simulations were done in a realistic regime, the filled-band nature of the model is far from the limits often assumed, i.e., where the phonon energy is very small compared with the Fermi energy in a parabolic band and Migdal\u00E2\u0080\u0099s theorem [8], which predicts quasiparticle renormalizations that scale linearly with the microscopic coupling, is valid. Away from these limits we find that the \u00E2\u0080\u009Ceffective coupling\u00E2\u0080\u009D, as would be inferred from quasiparticle renormalizations directly, will differ drastically and unpredictably from the true coupling. This work helps explain the exceedingly large quasiparticle renormalizations observed in some systems [9\u00E2\u0080\u009312], which seem implausible when understood using the de-facto stan- dard ARPES interpretation where Migdal\u00E2\u0080\u0099s theorem is assumed to hold. In contrast with Migdal\u00E2\u0080\u0099s theorem, we show that the perturbation theory 2 1.3. ARPES on Sr2RuO4 predictions for quasiparticle renormalization remain good where expected. Additionally, we find that the momentum dependence of the self-energy can be revealed via the relationship between velocity renormalization and quasi- particle strength. Through lineshape self-energy analysis on these simulations, we show that (although not strictly valid) it is often possible to infer the self-energy and bare electronic structure through an iterative fitting process \u00E2\u0080\u0093 even if the self-energy is momentum dependent, a condition that is generally neglected. We also show that through lineshape alone, when that lineshape is Lorentzian, it is possible to reliably extract the shape of the imaginary part of a momentum-dependent self-energy without reference to the bare- band. Finally, through simulations with a momentum-dependent self-energy, I explore exactly which self-energy is extracted by these methods, and find that the self-energy extracted is that which follows a path through energy\u00E2\u0080\u0093 momentum space along the observed spectral maximum. 1.3 ARPES on Sr2RuO4 In addition to this theoretical background work on typical ARPES analy- sis, I have helped to construct a new state-of-the-art ARPES system here at UBC. Results pertaining to the characterization of this system are pre- sented in Chapter 3. A logical extension of the spectral function analysis work presented in Chapter 2 could have been to apply these methods, par- ticularly the detailed self-energy analysis schemes, to a real material using this ARPES system. This was attempted with the unconventional super- conductor Sr2RuO4 [13], believed to also be a model Fermi-liquid [14, 15] \u00E2\u0080\u0093 a trait that should simplify such analysis. However, this did not turn out to be feasible even with the high resolution of the new ARPES system at UBC. This failure was due to some previously undetected spectral features \u00E2\u0080\u0093 an apparent splitting of certain bands \u00E2\u0080\u0093 found to exist in Sr2RuO4, which confounded detailed lineshape analysis 3. 3In the meantime (although not discussed in this thesis) another member of our group, B. Ludbrook, has begun measurements on MgB2, on which it seems likely that such anal- 3 1.4. CPS-ARPES, Spin\u00E2\u0080\u0093orbit coupling, and superconductivity in Sr2RuO4 These features themselves proved to be quite interesting, and their char- acterization and the ultimate resolution of their origin forms the entirety of Chapter 4. These features were not unnoticed by the rest of the ARPES community, with another group publishing results offering no definitive ex- planation but suspecting a surface state driven by electronic interactions [16] even as our own publication [6] was under review. However, through a detailed study of the characteristics of these features along with their degra- dation over time, their polarization dependence, a topological comparison to folding in the Bi-cuprates, and a comparison to local-density approx- imation (LDA) slab calculations we rule out phases driven by electronic interaction. Instead we find that a structural crystal-lattice distortion al- ready known to be present in the surface must extend to (at a minimum) the sub-surface layer, with an associated progression of electronic states. The different electronic structure from these two layers is thus responsible for the apparent splitting observed. This experience with Sr2RuO4 led to a further opportunity to explore more interesting aspects of its electronic structure \u00E2\u0080\u0093 directly verifying the effects of spin\u00E2\u0080\u0093orbit coupling, which has the potential to profoundly change the unsolved description of superconductivity in this material. 1.4 CPS-ARPES, Spin\u00E2\u0080\u0093orbit coupling, and superconductivity in Sr2RuO4 Spin\u00E2\u0080\u0093orbit (SO) coupling is the well-characterized interaction between an electron\u00E2\u0080\u0099s spin and orbital motion owing to relativity and the electric field of the nuclei. Because of this it is well-known to be of importance in the energy of core levels and materials incorporating the 3d transition metals. However, in Sr2RuO4 (with electronic structure based on the 4d transition metal ruthenium), it is only due to degeneracies and the overall proximity in energy of the three Fermi-surface crossing bands (always of a similar energy to the coupling) that it becomes important near the Fermi energy. ysis will succeed. 4 1.4. CPS-ARPES, Spin\u00E2\u0080\u0093orbit coupling, and superconductivity in Sr2RuO4 By breaking these degeneracies SO coupling modifies the band structure, an effect that has already been observed [17]. A much more important effect, however, is due to the physics that causes these degeneracies to be broken in the first place. By entangling the spin and spatial descriptions, SO coupling invalidates the standard method of writing the electronic wave-function as a product of independent spin and spatial descriptions. Instead the two are fundamen- tally intertwined. It is this entanglement that we, for the first time, directly measure in Chapter 5. We accomplish this by using circularly polarized light on carefully selected initial states that, owing to selection rules and the predictions of their SO entanglement, will generate spin-polarized photoe- mission. These experiments were performed at the Complete Photoemission Experiment (COPHEE) beamline at the Swiss Light Source, which can per- form spin- and angle-resolved photoemission spectroscopy (SARPES) with circularly polarized light (CPS-ARPES). We also, by calculating the wave-functions on the minimal basis of ruthe- nium d and oxygen p orbitals and adding SO coupling as a local term in the Hamilton, explicitly calculate some of the effects of SO coupling for both single electrons and possible pairings. We show this entanglement must necessarily drive the superconducting state in Sr2RuO4 to be even more un- conventional than is generally assumed, by allowing mixing between singlet, triplet, and other spin states. This mixing varies around the Fermi surface, and offers a possible resolution to a small number of experiments that clash with the broader categorization of Sr2RuO4 as a hallmark spin-triplet chiral p-wave superconductor. 5 Chapter 2 Spectral function analysis The many-body problem allows relatively simple interactions to transform into a wide range of exciting yet often complicated phenomena. The quasi- particle picture simplifies these complications by grouping fundamental par- ticles and excitations together into quasiparticles, which themselves behave in a more understandable manner. In this picture the real part of the self- energy represents the energy difference from the bare-particle energy, and the imaginary part the inverse lifetime of the combined excitation. Angle- resolved photoemission spectroscopy (ARPES) is a well-established tool for the investigation of such electronic systems, as it provides access to the electron-removal part of the momentum-resolved spectral function A(k, \u00CF\u0089) [7], which is generally written in the following form: A(k, \u00CF\u0089) = \u00E2\u0088\u0092 1 pi \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(k, \u00CF\u0089) [\u00CF\u0089 \u00E2\u0088\u0092 \u00CE\u00B5bk \u00E2\u0088\u0092 \u00CE\u00A3\u00E2\u0080\u00B2(k, \u00CF\u0089)]2 + [\u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(k, \u00CF\u0089)]2 . (2.1) The analysis of this extremely rich data source can be both difficult and rewarding, as it depends on both the interaction self-energy \u00CE\u00A3(k, \u00CF\u0089) = \u00CE\u00A3\u00E2\u0080\u00B2(k, \u00CF\u0089)+ i\u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(k, \u00CF\u0089), as well as the single-particle electronic dispersion \u00CE\u00B5bk (the so-called bare-band). A variety of approaches to analyzing this spec- tral function have been utilized and often focus on analysis of either the path of quasiparticle dispersions through (k, \u00CF\u0089) space, or lineshape and its implications for the structure of the self-energy. Both methods generally cut the spectral function into curves constant in either momentum [gener- ating a series of energy distribution curves (EDCs)] or energy [for a series of momentum distribution curves (MDCs)]. In this chapter, using simula- tions that have no experimental limitations, we will perform quasiparticle analysis on EDCs (which allows the identification of a quasiparticle peak in 6 Chapter 2. Spectral function analysis each k slice, thereby forming a quasiparticle dispersion, \u00CE\u00B5qk) and self-energy analysis on MDCs (as self-energies often show weak momentum and strong energy dependence, allowing more straightforward approximation for slices of constant energy). In quasiparticle analysis one can estimate properties such as the disper- sion\u00E2\u0080\u0099s velocity vqk = \u00E2\u0088\u0082\u00CE\u00B5 q k/\u00E2\u0088\u0082k; effective mass m q k, where 1/m q k = \u00E2\u0088\u0082 2\u00CE\u00B5qk/\u00E2\u0088\u0082k 2; and quasiparticle strength Zqk, where Z q k = \u00E2\u0088\u00AB q A(k, \u00CF\u0089)d\u00CF\u0089 is the integral over the coherent part of the spectral function (this is the quasiparticle weight only, which in somewhat loose terminology is often referred to as quasi- particle coherence [7]). If the bare-band dispersion \u00CE\u00B5bk is known, the renor- malization of these properties can also be calculated. This concept has been used to generate an \u00E2\u0080\u009Ceffective coupling\u00E2\u0080\u009D (which we will denote \u00CE\u00BBeff but which is often denoted simply as \u00CE\u00BB in ARPES literature) in the analysis of many complex systems, often through the so-called mass enhancement fac- tor mbk/m q k = v q k/v b k = Z q k = 1/(1+\u00CE\u00BBeff) 4. This factor has become a de facto standard in ARPES analysis [7, 15, 18\u00E2\u0080\u009320], since, in the Migdal/Eliashberg limit after few approximations, it is equivalent to the true dimensionless microscopic coupling found in the Hamiltonian (denoted by \u00CE\u00BB here and in theoretical literature) and is expected to manifest itself in a variety of differ- ent measurements [21, 22]. However, the large values sometimes measured for these renormalizations and effective couplings via ARPES (see, for exam- ple, Refs. 9\u00E2\u0080\u009312), should raise the question of this scheme\u00E2\u0080\u0099s universal utility [23\u00E2\u0080\u009328] and the applicability of the limits implied by such analysis to the systems being measured [1, 29\u00E2\u0080\u009331]. Another common goal of spectral function analysis is to extract the self- energy. In most circumstances, under the assumption of k-independence of the self-energy, MDCs cuts through Eq. 2.1 reduce to a simple Lorentzian form, thus allowing a measurement of \u00CE\u00A3\u00E2\u0080\u00B2(\u00CF\u0089) and \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(\u00CF\u0089) through ARPES [19, 23, 32\u00E2\u0080\u009336]. However, not only do these methods hinge on some assumptions and/or approximations for the bare-band \u00CE\u00B5bk, but more fundamentally the 4Here we use the equals sign between different renormalization quantities, as is often assumed in ARPES literature, but, as we will show, these quantities are in general not equivalent. 7 Chapter 2. Spectral function analysis problem of how momentum dependence in \u00CE\u00A3\u00E2\u0080\u00B2(k, \u00CF\u0089) and \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(k, \u00CF\u0089) affects this analysis is unaddressed \u00E2\u0080\u0093 even though it is known that a Lorentzian lineshape does not guarantee a momentum-independent self-energy [37]. In this chapter we present a careful study of established methods and present some new variations using one of the most studied interactions \u00E2\u0080\u0093 that of electrons and phonons. This work resulted in two publications [1, 2]. We generate self-energies and spectral functions, where the inclusion of mo- mentum dependence and all energy scales are controlled using the least complicated electron\u00E2\u0080\u0093phonon interaction models possible. We do this in a filled-band system \u00E2\u0080\u0093 far from the asymptotic limit often assumed, i.e., where the phonon energy is very small compared with the Fermi energy in a parabolic band and Migdal\u00E2\u0080\u0099s theorem [8] predicting (1 + \u00CE\u00BB) quasiparticle renormalizations is valid. These spectral functions are examined over a wide range of parameter space through techniques often used in ARPES. Ana- lyzing over 1200 simulations we consider variations of the microscopic cou- pling strength (\u00C2\u00A7 2.2.2), phonon energy (\u00C2\u00A7 2.2.3), and dimensionality (\u00C2\u00A7 2.2.4) for two models: a momentum-independent Holstein model and momentum- dependent coupling to a breathing mode phonon. In this limit we find that any \u00E2\u0080\u009Ceffective coupling\u00E2\u0080\u009D, \u00CE\u00BBeff, inferred from the quasiparticle renormal- izations differs from the microscopic dimensionless coupling characterizing these Hamiltonians, \u00CE\u00BB, and could drastically either over- or underestimate it, depending on the particular parameters and model. Conversely, we show that perturbation theory retains good predictive power for low coupling and small momenta (\u00C2\u00A7 2.2.5) and that the momentum dependence of the self- energy can be revealed via the relationship between velocity renormaliza- tion and quasiparticle strength (\u00C2\u00A7 2.2.6). Additionally in \u00C2\u00A7 2.3 we show that (although not strictly valid) it is often possible to infer the self-energy and bare electronic structure through self-consistent Kramers\u00E2\u0080\u0093Kronig bare-band fitting (KKBF) and also that through lineshape alone, when the lineshape is Lorentzian, it is possible to reliably extract the shape of the imaginary part of a momentum-dependent self-energy without reference to the bare-band. 8 2.1. The models M om en tu m (1 /\u00C3\u0085 ) 0 1 2 3 0 100 2000 100 200 0 100 200 Binding Energy (meV) \u00CE\u00B5 \u00CE\u00B5 km(\u00CF\u0089) A(k,\u00CF\u0089) \u00CE\u00A3' \u00CE\u00A3'' MA(1) Holstein \u00CE\u00BB=0.1, \u00CE\u00A9=50 meV Se lf- En er gy (m eV ) -1500 -1000 -500 0 500 -750 -500 -250 0 250 -50 0 MA(1) Holstein \u00CE\u00BB=0.5, \u00CE\u00A9=50 meV MA(1) Holstein \u00CE\u00BB=1.0, \u00CE\u00A9=50 meV (a) (b) (c) (d) (e) (f) b k q k Figure 2.1: The spectral function (a\u00E2\u0080\u0093c) and self-energies (d\u00E2\u0080\u0093f) of the 1D momentum-independent Holstein polaron model calculated with MA(1) for \u00E2\u0084\u00A6 = 50 meV, for different microscopic couplings, with the bare-band \u00CE\u00B5bk, quasiparticle band \u00CE\u00B5qk, and km(\u00CF\u0089) path shown. 2.1 The models In order to model photoemission from a completely filled system we use single-electron addition to an empty band; this is done at 0 Kelvin. Note that this is exact through particle\u00E2\u0080\u0093hole symmetry, which essentially amounts to the replacement \u00CF\u0089 \u00E2\u0086\u0092 \u00E2\u0088\u0092\u00CF\u0089. This is an ideal test case, as it provides the sim- plest possible description of electron\u00E2\u0080\u0093phonon coupling and is uncomplicated by further interactions such as strong correlations between electrons (as in, for example, Ref. 38), or even a Fermi sea, which would add yet another energy scale to the problem (as in, for example, Refs. 39, 40). The chemical potential in our treatment is then the top of the first electron removal state, labelled as 0 binding energy on all plots. For momentum-independent study we will use spectral functions and self-energies generated with the momen- 9 2.1. The models tum average approximation MA(1)[4, 5]5. Since MA(1) has been shown to be accurate everywhere in parameter space [4], it will enable us to study A(k, \u00CF\u0089) and \u00CE\u00A3(\u00CF\u0089) over a broad range of electron\u00E2\u0080\u0093phonon coupling and phonon en- ergies. For momentum-dependent study we use an extension of this ap- proximation with variational considerations, denoted MA(v,n) [3]. Although generally accurate everywhere in parameter space, for reasons specific to this approximation, details studied through EDC quasiparticle analysis are best realized through MA(v,1), and MDC-based self-energy analysis is best realized through MA(v,0). In all cases the spectral function remains entirely self-consistent with the associated self-energy6. Our test case for a momentum-independent self-energy is the simplest possible in momentum space \u00E2\u0080\u0093 namely, the Holstein polaron [42]: momentum- independent coupling between a single dispersionless phonon mode and tight-binding electrons. In reality, however, even for the Holstein model, the self-energy is weakly dependent on momentum, which can be seen at the MA(2) level of approximation [4]. We overcome this complication by choos- ing the momentum-independent self-energy from the MA(1) level in order to see how well these methods work for a truly momentum-independent self- energy. For strongly momentum-dependent self-energy study we will model coupling to a single optical mode, where the phonons live on half-integer lat- tice sites in between the electron sites and modify the on-site energy of their neighbours. In 2D this describes lattice vibrations in a CuO2-like plane, where the motion of the O ions is the most important vibrational degree of freedom; this has been the topic of many ARPES studies [11, 32\u00E2\u0080\u009334]. Throughout this chapter we will refer to this as the breathing mode model. 5Here the superscript on MA(n) refers to the n terms kept exactly [4, 5], while MA(v,n) (which appears later) includes variational considerations [3]. 6With more terms kept exactly, MA(v,1) should show overall improvement over MA(v,0); however, for reasons that are not understood, it does so only in the quasiparticle regime \u00E2\u0080\u0093 conversely the continuum of A(k, \u00CF\u0089) below the quasiparticle band is worsened; toward the Brillouin zone edge it is pushed further down in energy than exact diagonalization results indicate it should be [3, 41]. We use the lower-order MA(v,0) for the MDC-based self-energy analysis, as this analysis uses A(k, \u00CF\u0089) in both the continuum and quasiparticle regime. 10 2.1. The models 2.1.1 The Hamiltonian We may write both these models in the following form in momentum space: H = \u00E2\u0088\u0091 k \u00CE\u00B5bkc \u00E2\u0080\u00A0 kck+\u00E2\u0084\u00A6 \u00E2\u0088\u0091 Q b\u00E2\u0080\u00A0QbQ+ \u00E2\u0088\u0091 k,Q gQ\u00E2\u0088\u009A N c\u00E2\u0080\u00A0k\u00E2\u0088\u0092Qck(b \u00E2\u0080\u00A0 Q+b\u00E2\u0088\u0092Q). (2.2) The terms describe, in order, an electron with dispersion \u00CE\u00B5bk = \u00E2\u0088\u00922t D\u00E2\u0088\u0091 i=1 cos(kia) (2.3) in D dimensions, an optical phonon with energy \u00E2\u0084\u00A6 and momentum Q, and the on-site momentum-dependent electron\u00E2\u0080\u0093phonon coupling gQ [for N sites with periodic boundary conditions; c\u00E2\u0080\u00A0k (ck) and b \u00E2\u0080\u00A0 k (bk) are the usual electron and phonon creation (annihilation) operators]. For the Holstein case gQ\u00E2\u0089\u00A1g (2.4) is a constant, while for coupling to the breathing mode gQ\u00E2\u0089\u00A1 \u00E2\u0088\u0092i \u00E2\u0088\u009A 2g D\u00E2\u0088\u0091 i=1 sin(Qia/2), (2.5) which has an average value of \u00E3\u0080\u0088|gQ|2\u00E3\u0080\u0089 = g2 across the Brillouin zone7. This allows us to define the same dimensionless coupling for both models: \u00CE\u00BB= \u00E3\u0080\u0088|g|2\u00E3\u0080\u0089 2Dt\u00E2\u0084\u00A6 , (2.6) 7In most implementations, the coupling is found via a scattering integral around the Fermi surface [21], which for the breathing mode model would be zero at all coupling strengths. We take the Brillouin zone as a sensible alternative in this case (the choice is irrelevant for the momentum-independent Holstein coupling). 11 2.1. The models the ratio between lattice deformation energy\u00E2\u0088\u0092g2/\u00E2\u0084\u00A6 and free-electron ground- state energy \u00E2\u0088\u00922Dt. For this thesis we set a = ~ = 1, t = 50 meV, (2.7) such that the 1D bandwidth is 200 meV and the Brillouin zone is 2piA\u00CC\u008A \u00E2\u0088\u00921 wide. Also note that an additional constant 1 meV full width at half maxi- mum (FWHM) Lorentzian broadening is used, similar to an impurity scat- tering, to allow the numerical resolution of the sharpest features in A(k, \u00CF\u0089). 2.1.2 An example spectral function The spectral function calculated with MA(1) for the Holstein problem in 1D, with \u00E2\u0084\u00A6 = 50 meV and \u00CE\u00BB= 0.1, 0.5, 1.0, is presented as a false-colour plot in Fig. 2.1(a,b,c), along with the path of peak maxima measured through MDCs [km(\u00CF\u0089)] and EDCs (\u00CE\u00B5 q k) compared with the bare-band dispersion \u00CE\u00B5 b k. In the bottom panels (d,e,f) of Fig. 2.1 we present the real, \u00CE\u00A3\u00E2\u0080\u00B2(\u00CF\u0089), and imaginary, \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(\u00CF\u0089), parts of the self-energy for this momentum-independent model. Here each \u00CE\u00B5qk is a true (and the lowest) pole of the Green\u00E2\u0080\u0099s function (it has zero width, hence an infinite lifetime) and is only resolved numerically owing to the impurity scattering inserted in the energy direction. One can see from Eq. 2.1 that the peak width should go roughly like \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(k, \u00CF\u0089), and it is reassuring to see in Fig. 2.1(d,e,f) that the imaginary part of the self-energy is indeed zero near \u00CE\u00B5qk. The pole structure \u00CE\u00B5 q k is also distinct from that of km, the path of peak maxima observed during MDC analysis; not only are they fundamentally different (as one is a function of \u00CF\u0089 and the other of k), but the path of peak maxima observed when cutting A(k, \u00CF\u0089) in energy vs. momentum will not necessarily overlap, as has already been noted in the literature [15, 43]. For small couplings [Fig. 2.1 panel (a)] most of the spectral weight re- mains along \u00CE\u00B5bk, with only a small feature formed at energy \u00E2\u0084\u00A6 below the top of the band. With experimental resolution such a feature might appear only as a \u00E2\u0080\u009Ckink\u00E2\u0080\u009D in a quasiparticle dispersion; however, from looking at the 12 2.2. Quasiparticle analysis self-energy [panel (d)] one can see that a distinction between \u00CE\u00B5bk and km(\u00CF\u0089) should be made at this feature. The lowest pole, where \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(\u00CF\u0089) \u00E2\u0089\u0088 0 and which we will identify as the quasiparticle, exists only between the top of the band and \u00E2\u0084\u00A6. This pole forms a narrow dispersion, \u00CE\u00B5qk, of bandwidth approximately \u00E2\u0084\u00A6, although for k near the zone edge the electron spectral weight is very weak owing to its significant phonon character. The km(\u00CF\u0089) path of MDC peak maxima, however, does not follow this quasiparticle dis- persion but instead carries on close to the original bare-band \u00CE\u00B5bk into what we will identify as the continuum, owing to its broader structure and finite \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2. As the coupling is increased [Fig. 2.1 panels (b,c)], this distinction be- comes increasingly evident; the quasiparticle band gains spectral weight to- ward the zone boundary and becomes better defined. Also, its bandwidth narrows, becoming less than \u00E2\u0084\u00A6, as the quasiparticle mass increases and the quasiparticle velocity decreases. At the same time the spectral weight in the continuum becomes more spread out at deep energies, and new quasiparticle- like features begin to appear at the top of the continuum. At very large coupling (not shown) these additional features and the quasiparticle will eventually form a ladder of states with flat dispersions, although this cou- pling regime is well beyond the scope of this thesis. 2.2 Quasiparticle analysis As can be seen in Fig. 2.1, quasiparticle renormalizations do increase as the microscopic coupling increases. This monotonicity has led to widespread acceptance of measuring coupling through the quasiparticle mass, velocity, or strength renormalizations observed with ARPES, often without refer- ence as to whether or not the system should be expected to fall in the Migdal/Eliashberg framework. In this section we use our simple models to demonstrate that this scheme is not universal and to make other observa- tions by performing quasiparticle analysis, as is typically done with ARPES data (Figs. 2.2 and 2.3), on \u00E2\u0088\u00BC1200 generated spectral functions. These allow us to explore a range of couplings (Fig. 2.4), parameters (Fig. 2.5), and di- 13 2.2. Quasiparticle analysis mensionality (Fig. 2.6) on models that provide both momentum-dependent and momentum-independent self-energies. Following a discussion of these results we will follow the mass renormalization behaviour as \u00CE\u00BB\u00E2\u0086\u0092 0 for k \u00E2\u0088\u00BC 0 in detail through perturbation theory in \u00C2\u00A7 2.2.5, the predictions for which are also plotted in Figs. 2.2, 2.3, and 2.4 for later comparison. Finally, in \u00C2\u00A7 2.2.6, we will show that it is possible to determine whether or not the self-energy is momentum dependent from quasiparticle analysis. 2.2.1 Quasiparticle dispersion, velocity, mass, strength, and their renormalizations In order to perform quasiparticle analysis we generate an entire spectral function for each combination of the following parameters: model, dimen- sionless coupling \u00CE\u00BB=\u00E3\u0080\u0088|g|2\u00E3\u0080\u0089/2Dt\u00E2\u0084\u00A6, phonon energy scale \u00E2\u0084\u00A6/2t, dimensionality D, and (for 2 and 3D) the desired cut through momentum space. For all simulations the form of the bare-band is not changed, and the hopping is set to a constant of t = 50 meV to give physically familiar values, a bandwidth of 200 meV in the 1D case. (To consider other bandwidths, one should simply scale the bandwidth, phonon energy, and coupling together, as seen in the Hamiltonian, Eq. 2.2). On each of the \u00E2\u0088\u00BC1200 generated spectral functions the quasiparticle dispersion is found by fitting a Lorentzian peak with linear background to each EDC within the quasiparticle regime. The inclusion of a linear background allows any spectral weight from the continuum that has bled in to be excluded (a problem especially at low couplings and high dimensions). We illustrate this analysis in Figs. 2.2 and 2.3, where for the Holstein and coupling to breathing mode models (respectively), we present the spectral function A(k, \u00CF\u0089) for a mid-range coupling \u00CE\u00BB = 0.5 and phonon energy \u00E2\u0084\u00A6 = 50 meV, as well as dispersions found from the Lorentzian fits \u00CE\u00B5qk, perturbation theory prediction \u00CE\u00B5 pert k , and the bare electronic structure \u00CE\u00B5bk [panel (a)]. Also shown are the velocities [v q k, v pert k , and v b k in panel (b)], inverse masses [1/mqk, 1/m pert k , and 1/m b k in panel (c)], the corresponding renormalization ratios (vbk/v q k, m q k/m b k, and their perturbation theory pre- dictions), the inverse quasiparticle strength 1/Zqk , and bandwidth renormal- 14 2.2. Quasiparticle analysis ization \u00E2\u0084\u00A6/W q [panel (d); see caption for definitions]. In Fig. 2.2 the results are for the Holstein model with a momentum-independent self-energy, while Fig. 2.3 is for coupling to a breathing mode, which yields a momentum- dependent self-energy. Figs. 2.2 and 2.3(d) show that the velocity, mass, and spectral weight renormalizations are all functions of momentum, which raises concerns if one wishes to compare them with the bandwidth renormalization \u00E2\u0084\u00A6/W q or an \u00E2\u0080\u009Cexpected\u00E2\u0080\u009D renormalization factor of (1 + \u00CE\u00BB), which are both constant. Although they do cross at certain values of k, this is merely accidental, and none match at the top of the band \u00E2\u0080\u0093 our \u00E2\u0080\u009CFermi surface\u00E2\u0080\u009D. More problematic is that the mass renormalization must necessarily contain a divergence if the inflection point of \u00CE\u00B5bk is different from \u00CE\u00B5 q k, where 1/m q k vanishes (empha- sized by the horizontal dashed line). Similarly, in the case of momentum- dependent coupling [panel (e)], it can be seen that the quasiparticle disper- sion \u00CE\u00B5qk is not even monotonic, causing another divergence when v q k vanishes in the middle of the dispersion. (This non-monotonic dispersion is a direct consequence of the structure of the polaronic cloud, which causes a larger second-nearest-neighbour hopping and is discussed at length in Ref. 41.) Be- cause of this momentum dependence, any estimation of \u00CE\u00BB drawn from vbk/v q k, mqk/m b k, or 1/Z q k would depend heavily on the momentum chosen; if either of vbk/v q k or m q k/m b k were used close to their divergences, the estimated value could be off by an unlimited amount. Even \u00E2\u0084\u00A6/W q, although constant in k, does not match the value of (1+\u00CE\u00BB) for either the momentum-independent or momentum-dependent case. From Figs. 2.2 and 2.3 we draw the conclusion that none of the renormalization quantities gives a good direct estimate of the dimensionless coupling \u00CE\u00BB= \u00E3\u0080\u0088|g2|\u00E3\u0080\u0089/2t\u00E2\u0084\u00A6. Further, we find that, with the exception of the quasiparticle strength and velocity renormalization in the Holstein model only (which we will return to in \u00C2\u00A7 2.2.6), the renormaliza- tions do not even match each other \u00E2\u0080\u0093 even though the models were kept as similar and simple as possible. This indicates that making even qualitative comparisons of \u00E2\u0080\u009Ccoupling\u00E2\u0080\u009D from experiments on different materials (or even different experiments on the same material) through these renormalization parameters may not be meaningful. However, modelling of the parameters 15 2.2. Quasiparticle analysis in question from the original Hamiltonian via perturbation theory might be a start, as these results show much closer agreement near k = 0 despite the relatively high (for perturbation theory) coupling. (We will return to discuss perturbation theory later in \u00C2\u00A7 2.2.5.) 16 2.2. Q u asip article an aly sis M o m e n t u m ( 1 / \u00C3\u0085 ) 1 2 3 0 50 100 150 200 25 50 75 -50 0 50 Binding Energy (meV) Velocity (meV \u00C3\u0085) Inverse Mass (meV \u00C3\u00852) Renormalization 10 20 30 40 (c)(a) (d)MA(1) Holstein \u00CE\u00BB=0.5, \u00CE\u00A9=50 meV 0 50 1.0 1.5 0.0 0.8 A(k,\u00CF\u0089) \u00CE\u00B5 \u00CE\u00B5 \u00CE\u00B5 q k pert k b k v v v q k pert k b k 1/m 1/m 1/m q k pert k b k 1/Z v /v m /m q k q k b k b k q k v /v , m /mpertk pertk bkbk \u00CE\u00A9/W q 1+\u00CE\u00BB (b) Figure 2.2: (a) A(k, \u00CF\u0089) calculated for the momentum-independent Holstein self-energy in 1D with MA(1) for \u00E2\u0084\u00A6 = 50 meV and \u00CE\u00BB= 0.5; the quasiparticle dispersion \u00CE\u00B5qk, perturbation theory (about k = 0) prediction \u00CE\u00B5 pert k , and bare-band \u00CE\u00B5bk are also shown. (b) Quasiparticle (v q k), perturbation theory (v pert k ), and bare-band (v b k) velocities, as well as (c) corresponding inverse masses, 1/mqk, 1/m pert k , and 1/m b k, according to the definitions vk = \u00E2\u0088\u0082\u00CE\u00B5k/\u00E2\u0088\u0082k and 1/mk =\u00E2\u0088\u0082 2\u00CE\u00B5k/\u00E2\u0088\u0082k 2. (d) Momentum-dependent quasiparticle renormalization as obtained from vbk/v q k, m q k/m b k, as well as the inverse quasiparticle strength 1/Zqk , where Z q k = \u00E2\u0088\u00AB q A(k, \u00CF\u0089)d\u00CF\u0089 is the quasiparticle-only integrated spectral weight; in the inset, these quantities are compared near k=0 with the renormalization factors \u00E2\u0084\u00A6/W q and (1+\u00CE\u00BB), obtained from quasiparticle bandwidth W q (defined as the energy difference between top and bottom of the quasiparticle band) and dimensionless coupling \u00CE\u00BB=g2/2t\u00E2\u0084\u00A6, as well as the perturbation theory prediction for mass and velocity renormalizations (shown with the same line style but which can be distinguished by their proximity to the quasiparticle curves). In subsequent figures (2.4, 2.5, 2.6) the quantity plotted is the effective coupling, \u00CE\u00BBeff that would be implied by these renormalizations in the Migdal limit; this simply amounts to subtracting 1 from the renormalization. The noise is due to the finite simulation grid and subsequent lineshape fitting; slight variations in peak position are exaggerated by taking the derivative numerically, and therefore they are most visible in mqk. 17 2.2. Q u asip article an aly sis MA(v,1) Breathing \u00CE\u00BB=0.5, \u00CE\u00A9=50 meV 1.0 1.5 0.0 0.8 M o m e n t u m ( 1 / \u00C3\u0085 ) 1 2 3 0 50 100 150 200 25 50 75 -50 0 50 Binding Energy (meV) Velocity (meV \u00C3\u0085) Inverse Mass (meV \u00C3\u00852) Renormalization 10 20 30 40 (c)(a) (d) 0 50 A(k,\u00CF\u0089) \u00CE\u00B5 \u00CE\u00B5 \u00CE\u00B5 q k pert k b k v v v q k pert k b k 1/m 1/m 1/m q k pert k b k 1/Z v /v m /m q k q k b k b k q k v /v , m /mpertk pertk bkbk \u00CE\u00A9/W q 1+\u00CE\u00BB (b) Figure 2.3: (a) A(k, \u00CF\u0089) calculated for a momentum-dependent self-energy from coupling to a single breathing mode in 1D with MA(v,1) for \u00E2\u0084\u00A6=50 meV and \u00CE\u00BB=0.5; the quasiparticle dispersion, perturbation theory prediction, and bare-band are also shown. (b) Quasiparticle, perturbation theory, and bare-band velocities, as well as (c) corresponding inverse masses. (d) Momentum-dependent quasiparticle renormalization for these quantities, as well as the inverse quasiparticle strength, quasiparticle bandwidth, dimensionless coupling, and perturbation theory predictions; these are shown in greater detail near k = 0 in the inset. All definitions are identical to those in Fig. 2.2, where \u00CE\u00BB=\u00E3\u0080\u0088|g|2\u00E3\u0080\u0089/2t\u00E2\u0084\u00A6 is the average of the coupling across the Brillouin zone. 18 2.2. Quasiparticle analysis 2.2.2 Renormalization as a function of coupling strength Despite their differences from each other and their momentum dependence, however, these renormalizations do monotonically increase as the micro- scopic coupling increases (as previously observed in Fig. 2.1) which natu- rally leads one to wonder how, precisely, these quantities scale with \u00CE\u00BB as well as \u00E2\u0084\u00A6 in our different models, so that one might be able to capture the trend if the material and measured quantity were held constant \u00E2\u0080\u0093 for example, in an experiment performed as a function of doping, if that dop- ing does not cause structural distortions. In Fig. 2.4 we will follow the \u00E2\u0080\u009Ceffective coupling\u00E2\u0080\u009D, \u00CE\u00BBeff (which is simply the observed renormalization mi- nus 1), that each of these renormalization quantities would predict using the Migdal/Eliashberg framework as a function of \u00CE\u00BB, as well as renormaliza- tions found using the perturbation theory results around k = 0 (Eqs. 2.8 and 2.9), for both momentum-independent [panel (a)] and momentum-dependent [panel (b)] self-energies. In Fig. 2.5 we plot a selection of these quantities in the same fashion, for a range of phonon energies. For the momentum- dependent quantities we must choose a k value: we plot vb0/v q 0, m q 0/m b 0, and 1/Zq0 at k = 0 (our \u00E2\u0080\u009CFermi surface\u00E2\u0080\u009D), as well as v b ki /vqki and 1/Z q ki at the inflection point k=ki of the quasiparticle band \u00CE\u00B5 q k, where m q ki /mbki diverges. In Fig. 2.4 we find that the predictions from all quantities scale monotoni- cally with the microscopic coupling and are concave up. In the low-coupling regime (below about \u00CE\u00BB = 0.3) the perturbation theory results match the simulations \u00E2\u0080\u0093 however, nowhere does \u00CE\u00BBeff match \u00CE\u00BB from the Hamiltonian. At small coupling values, using this model, many renormalization quantities would drastically underestimate the true microscopic coupling, by a factor ranging from infinite (\u00E2\u0084\u00A6/W near \u00CE\u00BB = 0, where it is not renormalized in the Holstein model) to \u00E2\u0088\u00BC6 (1/Zki near \u00CE\u00BB = 0.1, breathing mode). Conversely, at larger coupling values (\u00CE\u00BB \u00E2\u0088\u00BC 2) all quantities would overestimate the true mi- croscopic coupling, with factors ranging from \u00E2\u0088\u00BC 4 (\u00E2\u0084\u00A6/W , Holstein) to \u00E2\u0088\u00BC 22 (vbki/v q ki , breathing mode). We also find that, depending on the coupling or model, the relative renormalization strength of quantities changes \u00E2\u0080\u0093 for the momentum-independent model \u00E2\u0084\u00A6/W is renormalized the least, whereas 19 2.2. Quasiparticle analysis \u00CE\u00BB e ff = R en or m al iz at io n -1 \u00CE\u00BB=\u00E3\u0080\u0088|g|2\u00E3\u0080\u0089/2t\u00CE\u00A9 0.0 0.5 1.0 1.5 2.0 10 20 \u00CE\u00BB e ff = R en or m al iz at io n -1 10 0 (a) (b) MA(1) Holstein \u00CE\u00A9 = 50 meV MA(v,1) Breathing \u00CE\u00A9 = 50 meV 0.0 0.5 1.0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 \u00CE\u00BB m /m -1 v /v -1, m /m -1, 1/Z -1 (k=0) v /v -1, 1/Z -1 (k=ki) \u00CE\u00A9/W q-1 pert k b k b k b k q k q k q k b k q k q k \u00CE\u00BB m /m -1 v /v -1, m /m -1 (k=0) 1/Z -1 (k=0) v /v -1 (k=ki) 1/Z -1 (k=ki) \u00CE\u00A9/W q -1 pert k b k b k b k q k b k q k q k q k q k Figure 2.4: Effective coupling, \u00CE\u00BBeff (as would be interpreted in the Migdal limit from the renormalization parameters as defined in Fig. 2.2), plotted vs. the true dimensionless coupling \u00CE\u00BB= \u00E3\u0080\u0088|g|2\u00E3\u0080\u0089/2t\u00E2\u0084\u00A6; g is a constant for the Hol- stein model (a), and gQ\u00E2\u0089\u00A1 \u00E2\u0088\u0092i \u00E2\u0088\u009A 2g sin(Qa/2) for the breathing mode model (b). Also shown, in the insets only, are the predictions for observed effec- tive coupling found via the mass renormalizations in perturbation theory (Eqs. 2.8 and 2.9) for the low-coupling regime at k = 0. Note that the noise in v and 1/Z at k= ki originates from the numerical determination of the inflection point ki. 20 2.2. Quasiparticle analysis in the momentum-dependent model 1/Zq0 shows the least renormalization. This indicates, yet again, that comparing different materials via renormal- izations is not feasible, nor is comparing different renormalizations on the same material without a detailed model. We note again that, in the Holstein model only, quasiparticle strength and velocity renormalization are identical for all couplings at both k = 0 and k = ki (as previously seen in Fig. 2.2, where they are shown to be identical at all momenta, contrasted against Fig. 2.3, where they do not match), which will be explored further in \u00C2\u00A7 2.2.6. 2.2.3 Renormalizations for different phonon energies In Fig. 2.5 we follow the same quantities for a variety of phonon energies, allowing \u00E2\u0084\u00A6 to vary from 1/16 to 3/4 of the bare-band width for both models (although inverse quasiparticle weight is omitted from the breathing mode plots for clarity). First we note that there are some qualitative similarities, but just as many differences. In all these 1D cases the concavity increases as phonon energy decreases, so that by the mid-coupling regime (\u00CE\u00BB \u00E2\u0089\u0088 1) we recover the expected dependence \u00E2\u0080\u0093 phonons that are easier (require less energy) to excite renormalize the band more. However, in the low-coupling regime we do not find this dependence (later seen again in Figs. 2.7 and 2.8 and in agreement with Eqs. 2.8 and 2.9). For both models the band- width [panels (a, d)] shows the opposite behaviour for low coupling, with a transition near \u00CE\u00BB = 0.8. Still considering the low-coupling regime, mass and velocity renormalizations show little dependence on the phonon energy for the breathing mode model yet strong dependence in the Holstein case. Again we find that the renormalizations and their corresponding \u00E2\u0080\u009Ceffective couplings\u00E2\u0080\u009D vary widely from each other and depend on the model and pa- rameters chosen \u00E2\u0080\u0093 sometimes in counterintuitive ways (with the exception of quasiparticle strength and velocity renormalization, which again overlap for the Holstein model). 21 2.2. Quasiparticle analysis \u00CE\u00BB e ff (R en or m al iz at io n -1 ) \u00CE\u00BB=g2/2t \u00CE\u00A9 0.0 0.5 1.0 1.5 \u00CE\u00BB e ff (R en or m al iz at io n -1 ) 0.0 0.5 1.0 \u00CE\u00BB e ff (R en or m al iz at io n -1 ) 0.0 0.5 1.0 10.2 0.4 0.6 0.8 \u00CE\u00BB=g2/2t \u00CE\u00A9 0.20.0 0.4 0.6 0.8 (a) \u00CE\u00BB \u00CE\u00A9/W q-1 \u00CE\u00A9=12.5 \u00CE\u00A9=25 \u00CE\u00A9=150 \u00CE\u00A9=100 \u00CE\u00A9=50 MA(1) Holstein \u00CE\u00A9=12.5 \u00CE\u00A9=25 \u00CE\u00A9=150 \u00CE\u00A9=100 \u00CE\u00A9=50 (d) \u00CE\u00BB \u00CE\u00A9/W q-1 MA(v,1) Breathing \u00CE\u00A9=150 \u00CE\u00A9=100 (b) \u00CE\u00BB v /v -1, m /m -1, 1/Z -1 (k=0) \u00CE\u00A9=12.5 \u00CE\u00A9=25 \u00CE\u00A9=50 b k b k q k q k q k \u00CE\u00A9=25 \u00CE\u00A9=12.5 \u00CE\u00A9=150 \u00CE\u00A9=50 \u00CE\u00A9=100 (e) \u00CE\u00BB v /v -1, m /m -1 (k=0)bk bkqk qk (c) \u00CE\u00BB v /v -1, 1/Z -1 (k=ki) \u00CE\u00A9=12.5 \u00CE\u00A9=25 \u00CE\u00A9=150 \u00CE\u00A9=100 \u00CE\u00A9=50 b k q k q k \u00CE\u00A9=25 \u00CE\u00A9=12.5 \u00CE\u00A9=150 \u00CE\u00A9=50 \u00CE\u00A9=100 (f) \u00CE\u00BB v /v -1 (k=ki)bk q k Figure 2.5: Effective coupling, \u00CE\u00BBeff, (as would be interpreted in the Migdal limit from the renormalization parameters defined in Fig. 2.2) from the Hol- stein model (a\u00E2\u0080\u0093c) and breathing mode model (d\u00E2\u0080\u0093f) plotted vs. the true dimensionless coupling \u00CE\u00BB=g2/2t\u00E2\u0084\u00A6 for a range of phonon energies \u00E2\u0084\u00A6 labelled in meV. In panels (b, c) those from multiple renormalization parameters (which lie directly on top of each other) are all plotted, whereas in (e, f) those from inverse quasiparticle weight do not fall on any other curve and are therefore omitted for clarity. The slope at \u00CE\u00BB = 0 in panels (b, e) is the quantity plotted in Figs. 2.7 and 2.8. Note that the noise in panels (c, f) originates mostly from the numerical determination of the inflection point ki, while in (a, d) it stems mostly from variations in fitting the quasiparticle peak location at high momentum, where it is has less weight. 22 2.2. Quasiparticle analysis \u00CE\u00BB e ff = R en or m al iz at io n -1 0 1 2 v /v -1, m /m -1, 1/Z -1 (k=0) v /v -1, 1/Z -1 (k=ki) \u00CE\u00BB \u00CE\u00BB=g2/2Dt \u00CE\u00A9 0.25 0.5 0.75 (0,0)-(\u00CF\u0080,\u00CF\u0080) and (0,0)-(\u00CF\u0080,0) (0)-(\u00CF\u0080) (0,0)-(\u00CF\u0080,\u00CF\u0080) (0,0)-(\u00CF\u0080,0) 0.0 1.0 MA(1) Holstein \u00CE\u00A9 = 50 meV (0,0,0)-(\u00CF\u0080,0,0) (0,0,0)-(\u00CF\u0080,\u00CF\u0080,\u00CF\u0080) (0,0,0)-(\u00CF\u0080,\u00CF\u0080,\u00CF\u0080) and (0,0,0)-(\u00CF\u0080,\u00CF\u0080,\u00CF\u0080) (0)-(\u00CF\u0080) b k b k q k q k q k b k q k q k Figure 2.6: Effective coupling, \u00CE\u00BBeff, (as would be interpreted in the Migdal limit from the mass and velocity renormalization parameters defined in Fig. 2.2) from the Holstein model and plotted vs. the true dimensionless coupling \u00CE\u00BB=g2/2Dt\u00E2\u0084\u00A6 for different dimensionalities and high-symmetry cuts. Note that the noise in v and 1/Z at k = ki originates from the numerical determination of the inflection point ki. 23 2.2. Quasiparticle analysis 2.2.4 Renormalizations in different dimensions The final parameter to be varied is dimensionality, which we explore briefly with Fig. 2.6 for the Holstein Hamiltonian in the low-coupling regime only. Here we find that, for a fixed dimensionality and phonon energy where \u00E2\u0084\u00A6 \u00E2\u0088\u00BC t, the renormalizations as a function of \u00CE\u00BB look qualitatively similar. The vari- ous renormalized quantities increase monotonically yet remain distinct from the microscopic coupling as well as each other (with the exception of quasi- particle strength and velocity renormalization, which are again the same), with details that depend on phonon energy and dimensionality. We feel it is important to note, however, that at larger couplings not explored here other studies on the dynamics of the Holstein (and momentum-dependent Su\u00E2\u0080\u0093Schrieffer\u00E2\u0080\u0093Heeger) models have found more complicated behaviour in higher dimensions, where a critical coupling value marks a drastic change in quasiparticle properties, which is most prominent as \u00E2\u0084\u00A6 \u00E2\u0086\u0092 0 [44\u00E2\u0080\u009347]. Al- though interesting, this type of behaviour would not simplify quasiparticle renormalization analysis on such a system and is not investigated here. 2.2.5 Perturbation theory predictions near \u00CE\u00BB = 0 So far we have shown that, while the slope may not be 1, the renormal- ization curves could all still be reasonably well approximated as linear in \u00CE\u00BB in the very-low-coupling regime and that perturbation theory makes an excellent prediction for them near k = 0. This allows us to follow this slope continuously through parameter space with perturbation theory rather than by simulating even larger numbers of spectral functions. It is worth noting that in the classic implementation of the (1 + \u00CE\u00BB) effective coupling scheme (see Ref. 22), perturbation theory is discussed but dismissed as a possible avenue, because the resulting corrections are too large for perturbation the- ory to be valid. However, in that instance, some approximations are made to ease completion of the integrals \u00E2\u0080\u0093 these approximations eliminate the possibility of the arbitrarily small couplings we have used here. In our case the lowest non-zero order in the phonon\u00E2\u0080\u0093electron interaction term from Eq. 2.2 is the second (see Appendix \u00C2\u00A7A.1), and in 1D we find that for both 24 2.2. Quasiparticle analysis models it is possible to complete the integrals without further approxima- tion. In the Holstein case the energy dispersion should be modified from \u00CE\u00B5bk = \u00E2\u0088\u00922t cos (ka) to \u00CE\u00B5pertk \u00E2\u0089\u0088 \u00E2\u0088\u00922t \u00EF\u00A3\u00AB\u00EF\u00A3\u00ADcos(ka) + \u00CE\u00BB \u00E2\u0084\u00A62t\u00E2\u0088\u009A( cos(ka) + \u00E2\u0084\u00A62t )2 \u00E2\u0088\u0092 1 \u00EF\u00A3\u00B6\u00EF\u00A3\u00B8 , (2.8) which agrees with the results calculated for the quasiparticle residue at k = 0 in Ref. 48. For the breathing mode model we find that the dispersion becomes \u00CE\u00B5pertk \u00E2\u0089\u0088 \u00E2\u0088\u00922t ( cos(ka) + \u00CE\u00BB \u00E2\u0084\u00A6 2t F (\u00E2\u0084\u00A62t , k)) , F (\u00E2\u0084\u00A62t , k) \u00E2\u0089\u00A1 cos(ka) + sin2(ka)\u00E2\u0088\u0092 \u00E2\u0084\u00A62tcos(ka)\u00E2\u0088\u009A( cos(ka) + \u00E2\u0084\u00A62t )2 \u00E2\u0088\u0092 1 . (2.9) This demonstrates that, at the very least, we should expect the renor- malizations to depend not solely on the dimensionless coupling \u00CE\u00BB = g2/2t\u00E2\u0084\u00A6 but also on the other relevant energy scale in the problem \u00E2\u0080\u0093 the ratio of phonon energy to bandwidth. By taking derivatives of these dispersions we can also find the predicted mass and velocity renormalizations. In Figs. 2.2 and 2.3 we plot the predicted dispersion, derivatives, and renormalizations contrasted against the simulated spectral function and find close but not perfect agreement for both models near k = 0 (but failing at large mo- menta), despite the relatively high coupling (\u00CE\u00BB = 0.5). As seen in Fig. 2.4 (and later in Figs. 2.7 and 2.8), near k = 0 for vanishing \u00CE\u00BB there is perfect agreement within our measurement accuracy; perturbation theory begins to show signs of failure near \u00CE\u00BB \u00E2\u0089\u0088 0.25. In higher dimensions we did not complete the integrals exactly but instead used the VEGAS Monte Carlo integration algorithm (see Appendix \u00C2\u00A7D.1) to evaluate them numerically [49\u00E2\u0080\u009351]. Using this routine for all dimensionalities allowed us to validate the results by comparing them with these known solutions for 1D for both 25 2.2. Quasiparticle analysis (000-\u00CF\u008000)(000-\u00CF\u0080\u00CF\u0080\u00CF\u0080) (00-\u00CF\u00800)(00-\u00CF\u0080\u00CF\u0080) (0-\u00CF\u0080) 1 2 0 1 2 Phonon Energy (\u00CE\u00A9/2t) \u00CE\u00BB e ff / \u00CE\u00BB a t k =0 , \u00CE\u00BB =0 Holstein Model Pertubation Theory MA(1) Simulations Figure 2.7: \u00E2\u0088\u0082\u00CE\u00BBeff/\u00E2\u0088\u0082\u00CE\u00BB \u00E2\u0088\u00A3\u00E2\u0088\u00A3 k=0,\u00CE\u00BB=0 (rate of change in observed effective coupling, \u00CE\u00BBeff, per change in true dimensionless coupling, \u00CE\u00BB) as a function of phonon energy, for the Holstein model (where \u00CE\u00BBeff \u00E2\u0089\u00A1 mqk/mbk \u00E2\u0088\u0092 1 is as predicted by perturbation theory and measured from simulated spectral functions, and \u00CE\u00BB = \u00E3\u0080\u0088|g|2\u00E3\u0080\u0089/2Dt\u00E2\u0084\u00A6). If the Migdal limit holds, this derivative would be a constant, 1, for all values of \u00E2\u0084\u00A6/2t. For D > 1 cuts from the \u00CE\u0093 point to both the corner and face of the Brillouin zone were considered, as labelled. Vertical error in the results from simulated spectral functions is comparable to symbol size and results from approximating the slope at \u00CE\u00BB = 0 from the finite data points in Figs. 2.5 and 2.6, as well as similar simulation sets not shown. Uncertainty in the perturbation theory results stems from the nu- meric Monte Carlo integration used to determine the perturbation energies, taken from the distribution of independent subsets of total points evaluated, and is higher for non-diagonal cuts because of the narrower bandwidth in that direction. The curves predicted by Ref. 44 for the Holstein model in one and two dimensions are not shown, as they would be hidden by symbols, but fall exactly onto the perturbation theory results. 26 2.2. Quasiparticle analysis (000-\u00CF\u008000) (000-\u00CF\u0080\u00CF\u0080\u00CF\u0080) (00-\u00CF\u00800) (00-\u00CF\u0080\u00CF\u0080) (0-\u00CF\u0080) 0 1 2 0 1 2 Phonon Energy (\u00CE\u00A9/2t) Breathing Mode Pertubation Theory MA(v,1) Simulations \u00CE\u00BB e ff / \u00CE\u00BB a t k =0 , \u00CE\u00BB =0 Figure 2.8: \u00E2\u0088\u0082\u00CE\u00BBeff/\u00E2\u0088\u0082\u00CE\u00BB \u00E2\u0088\u00A3\u00E2\u0088\u00A3 k=0,\u00CE\u00BB=0 (as defined in Fig. 2.7) as a function of phonon energy, for coupling to a breathing mode. 27 2.2. Quasiparticle analysis models and the results found in Refs. 44, 48 for the Holstein model in one and two dimensions, where they show perfect agreement (once corrected for a slightly different definition of \u00CE\u00BB in 2D). In Figs. 2.7 (for Holstein) and 2.8 (for coupling to a breathing mode) we show how the renormalization with the dimensionless coupling parameter \u00CE\u00BB near k = 0, \u00CE\u00BB = 0 (and hence the observed effective coupling, \u00CE\u00BBeff) varies as a function of the other energy scale \u00E2\u0084\u00A6/2t for both models in one, two and three dimensions, and how this matches nearly perfectly against measure- ments of the same quantity on the simulations. Interestingly, despite spectral functions that have outwardly similar characteristics [as seen, for example, comparing panel (a) in Fig. 2.2 vs. Fig. 2.3 or Fig. 2.10 vs. Fig. 2.12], we find that neither model would be well approximated by a \u00CE\u00BBeff = \u00CE\u00BB scheme, which is shown as the dashed line along 1. Additionally, we find a drastic difference in how the actual renormalizations vary with phonon energy de- pending on the model. The 1D Holstein model shows a perhaps expected dependence: phonons that require very little energy to excite have a dra- matic effect on the electronic renormalization (blowing up as \u00E2\u0084\u00A6 \u00E2\u0086\u0092 0), but as the phonon energy increases, the mode has progressively less effect. In the 2D case, however, we find very limited dependence on phonon energy, with a curve that is almost flat and could therefore be rescaled to match if \u00CE\u00BB was chosen to be defined appropriately. In 3D we find the opposite of the 1D case, in that the renormalization vanishes as \u00E2\u0084\u00A6 \u00E2\u0086\u0092 0. These very different limits are directly related to the bare-electron density of states at the band-edge and its strong dependence on dimensionality [44]. In all di- mensions we find that the renormalization is isotropic (as one might expect from an isotropic coupling) and that it asymptotically approaches a simi- lar value for large phonon energies \u00E2\u0080\u0093 reminiscent of a renormalization that depends solely on \u00CE\u00BB, if only for \u00E2\u0084\u00A6/2t \u001D 1. In stark contrast, however, the more realistic breathing mode model shows entirely different behaviour. For all dimensionalities the overall functional form is similar for \u00E2\u0084\u00A6 \u00E2\u0086\u0092 0, where mass renormalizations vanish. This low-energy behaviour may be expected, as for weak coupling and energies close to k = 0 in the bare-band, the elec- tron couples mostly to q \u00E2\u0089\u0088 0 phonons \u00E2\u0080\u0093 and in this model such coupling 28 2.2. Quasiparticle analysis vanishes, g(q \u00E2\u0086\u0092 0) \u00E2\u0086\u0092 0. As the phonon energy increases, however, we dis- cover that the renormalization is anisotropic, with stronger renormalization along the diagonal cut (as may be expected for an anisotropic coupling), and we find a coupling that gets stronger as dimensionality increases (op- posite of the Holstein case). We also find that the renormalizations do not asymptotically approach any fixed value for large phonon energies, as they did for the Holstein case. Overall we find that there is much variability in quasiparticle analysis, to the point that one cannot make a general rule about renormalizations in this regime. There are, however, two common threads. The first is that, for both our models, perturbation theory works in the low-coupling regime: it correctly predicts the quasiparticle band structure near k = 0 for all combinations of parameters tested, although it fails at higher momenta (as seen in Figs. 2.2 and 2.3). The second, and perhaps more interesting, hints at something that may be quantitatively gained through quasiparticle analysis \u00E2\u0080\u0093 without even requiring a more detailed model on which to attempt perturbation theory or other tools \u00E2\u0080\u0093 and that we explore in the following section. 2.2.6 Verifying momentum independence of the self-energy In Figs. 2.2, 2.3, 2.4, 2.5, and 2.6 one observes that, in the momentum- independent case only, vbk/v q k and 1/Z q k lie precisely on top of each other for all values and match mqk/m b k at k = 0. Although the velocity and mass renormalizations at k = 0 are simply a consequence of derivatives following each other near an extremum, the relationship between velocity renormalization and quasiparticle strength has implications for the structure of the self-energy, as will be discussed in detail here. By definition, the Green\u00E2\u0080\u0099s function is: G(k, \u00CF\u0089) = 1 \u00CF\u0089 \u00E2\u0088\u0092 \u00CE\u00B5bk \u00E2\u0088\u0092 \u00CE\u00A3(k, \u00CF\u0089) + i\u00CE\u00B7 . (2.10) In the infinite-lifetime quasiparticle regime the self-energy should be purely 29 2.3. Self-energy analysis real, with any broadening coming solely from the small impurity scattering, \u00CE\u00B7. We may then identify the implicitly defined quasiparticle dispersion as \u00CE\u00B5qk = \u00CE\u00B5 b k + \u00CE\u00A3(k, \u00CF\u0089) \u00E2\u0088\u00A3\u00E2\u0088\u00A3 \u00CF\u0089=\u00CE\u00B5qk and, since we are interested in an EDC, expand the self-energy about \u00CF\u0089 = \u00CE\u00B5qk to first order in energy. Taking \u00E2\u0088\u00921/pi times the imaginary part yields the spectral function: A(k, \u00CF\u0089) \u00E2\u0089\u0088 1 pi \u00CE\u00B7 \u00CE\u00B72 + (\u00CF\u0089 \u00E2\u0088\u0092 \u00CE\u00B5qk)2(1\u00E2\u0088\u0092 \u00E2\u0088\u0082\u00CE\u00A3(k,\u00CF\u0089)\u00E2\u0088\u0082\u00CF\u0089 \u00E2\u0088\u00A3\u00E2\u0088\u00A3 \u00CF\u0089=\u00CE\u00B5qk )2 . (2.11) We can see that, cut along energy in the quasiparticle regime, the spectral function will be a Lorentzian at \u00CE\u00B5qk, with width given by \u00CE\u00B7 and with weight Zk = 1/(1 \u00E2\u0088\u0092 \u00E2\u0088\u0082\u00CE\u00A3(k,\u00CF\u0089)\u00E2\u0088\u0082\u00CF\u0089 \u00E2\u0088\u00A3\u00E2\u0088\u00A3 \u00CF\u0089=\u00CE\u00B5qk ). This relationship between quasiparticle weight and the energy derivative of the self-energy is often derived and is usually associated directly with the success of an effective coupling scheme [21, 22], but we do not make that association here. Velocity renormalization is simply the ratio of momentum derivatives of the bare, vbk, and quasiparticle v q k = vbk + \u00E2\u0088\u0082\u00CE\u00A3(k,\u00CF\u0089) \u00E2\u0088\u0082\u00CF\u0089 \u00E2\u0088\u00A3\u00E2\u0088\u00A3 \u00CF\u0089=\u00CE\u00B5qk vqk + \u00E2\u0088\u0082\u00CE\u00A3(k,\u00CE\u00B5qk) \u00E2\u0088\u0082k bands, which reduces to: vbk vqk = 1 Zk \u00E2\u0088\u0092 \u00E2\u0088\u0082\u00CE\u00A3(k, \u00CE\u00B5 q k) \u00E2\u0088\u0082k 1 vqk . (2.12) We see that, for momentum-independent self-energies, the velocity renor- malization must follow the inverse spectral weight. This means that the renormalization quantities can be used to conclusively check the momen- tum dependence of the self-energy, in the quasiparticle regime. Whether or not the self-energy is momentum dependent is of great importance to MDC self-energy analysis, on which we focus in the rest of the chapter. 2.3 Self-energy analysis Since quasiparticle analysis fails to reveal the true microscopic coupling through renormalizations, we look toward other options for spectral func- tion analysis. In this section we describe how it is possible to estimate the self-energy through the analysis of MDC lineshapes. We will begin, for sim- 30 2.3. Self-energy analysis plicity, with a description for momentum-independent self-energy in \u00C2\u00A7 2.3.1 and move on to describe the implications of momentum dependence on the procedure in \u00C2\u00A7 2.3.2. 2.3.1 Momentum-independent self-energies Cases of a momentum-independent self-energy can be verified through quasi- particle analysis; as seen in \u00C2\u00A7 2.2.6, Eq. 2.12, the overlap of vbk/vqk and 1/Zqk is possible only when the self-energy is momentum independent along the quasiparticle dispersion. Although the quasiparticle dispersion and the path of MDC peak maxima where MDC analysis is carried out may vary, in prac- tice they are often very close in the quasiparticle regime. One must always keep in mind that although a momentum-independent self-energy causes a Lorentzian MDC lineshape, Lorentzian lineshape alone is not sufficient to conclude that \u00CE\u00A3(k, \u00CF\u0089)=\u00CE\u00A3(\u00CF\u0089) [37]. In cases where the self-energy is momentum independent, we may ana- lyze A(k, \u00CF\u0089) in terms of MDCs at constant energy \u00CF\u0089 = \u00CF\u0089\u00CC\u0083, where the self- energy may then also be considered a constant. Under this condition, as long as \u00CE\u00B5bk can be linearized in the vicinity of the MDC peak maximum observed at k = km, the MDC lineshape will be Lorentzian. By Taylor expanding \u00CE\u00B5 b k about an MDC peak maximum at k = km, i.e., \u00CE\u00B5 b k=\u00CE\u00B5 b km +vbkm \u00C2\u00B7 (k\u00E2\u0088\u0092 km)+...; ignoring higher-order terms (which must be negligible if the curve appears Lorentzian); and noticing that \u00CE\u00B5bkm +\u00CE\u00A3 \u00E2\u0080\u00B2 \u00CF\u0089\u00CC\u0083 \u00E2\u0088\u0092 \u00CF\u0089\u00CC\u0083 = 0 will implicitly define the observed peak maximum, we can rewrite Eq. 2.1 as A\u00CF\u0089\u00CC\u0083(k)w A0 pi \u00E2\u0088\u0086km (k \u00E2\u0088\u0092 km)2 + (\u00E2\u0088\u0086km)2 , (2.13) with \u00E2\u0088\u0086km = \u00E2\u0088\u0092\u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2\u00CF\u0089\u00CC\u0083/vbkm = HWHM, A0 = 1/v b km = \u00E2\u0088\u00AB A\u00CF\u0089\u00CC\u0083(k)dk. (2.14) Here \u00E2\u0088\u0086km is the half width at half maximum (HWHM) of a Lorentzian 31 2.3. Self-energy analysis of weight A0. For each constant energy cut, \u00CF\u0089 = \u00CF\u0089\u00CC\u0083, the observed peak maximum is labelled km. The self-energies are then easily found as \u00CE\u00A3\u00E2\u0080\u00B2\u00CF\u0089\u00CC\u0083 = \u00CF\u0089\u00CC\u0083 \u00E2\u0088\u0092 \u00CE\u00B5bkm , \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2\u00CF\u0089\u00CC\u0083 = \u00E2\u0088\u0092\u00E2\u0088\u0086kmvbkm . (2.15) One must be careful visualizing the relationships; although a simple picture might be that the peak, whose width scales with the imaginary self-energy and band velocity, has been \u00E2\u0080\u009Cpushed up\u00E2\u0080\u009D by the real self-energy to its ob- served location at \u00CF\u0089\u00CC\u0083, one must remember that these quantities are defined implicitly and evaluated at different locations in the (k, \u00CF\u0089) plane: the self- energy is evaluated at (km, \u00CF\u0089\u00CC\u0083) and the bare-band at (km, \u00CE\u00B5 b km ). Of course, these implicit definitions are no trouble if you simply wish to observe A(k, \u00CF\u0089) and not calculate it based on this simple approximation. These relations are illustrated graphically in Fig. 2.9. These relationships work exactly where they are applicable \u00E2\u0080\u0093 namely, when the self-energy is momentum independent, when km is far from a band edge (where vbkm must vanish), where the peak shape is truly Lorentzian, and when the peak width is narrow enough that a first-order expansion of \u00CE\u00B5bk is appropriate. Since the convolution of two Lorentzians is another Lorentzian where the peak width is a simple sum of the widths of the original functions, the inserted impurity scattering shows up directly as an addition to the measured \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2\u00CF\u0089\u00CC\u0083. (For comparison purposes a constant \u00CE\u00B7 = 1 meV has therefore been subtracted from all plots of \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2\u00CF\u0089\u00CC\u0083.) However, these relationships still hinge on knowledge of the bare-band. If \u00CE\u00B5bk is unknown it is possible to fit it, to within an arbitrary energy offset, to any functional form that provides a value and derivative using a Kramers\u00E2\u0080\u0093Kronig bare-band fitting (KKBF) routine (see Appendix \u00C2\u00A7A.2). Alternatively, as previously noted in Ref. 1 and used in Ref. 52, the imaginary part of the self-energy requires knowledge of only vbkm , which can be obtained directly from A0 = 1/v b km , allowing us to write it as the MDC width/integral ratio: \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2ratio =\u00E2\u0088\u0092\u00E2\u0088\u0086km/A0 (2.16) 32 2.3. Self-energy analysis (b) 1.0 2.0 3.0 M om en tu m (1 /\u00C3\u0085 ) Binding Energy (meV) A\u00CF\u0089(k)~ 90 100 110 120 130 1.2 1.4 1.6 1.8 2.0 (a) \u00CE\u00A3\u00CF\u0089~' \u00CE\u0094km(\u00CF\u0089)~ \u00CF\u0089 km path km (\u00CF\u0089) \u00CE\u00B5 known A(k,\u00CF\u0089) ~ b k ~ 80 Figure 2.9: Diagrammatic description of MDC self-energy analysis (Eqs. 2.13 and 2.15) for a momentum independent self-energy. (a) is a false-colour plot with the bare-band (\u00CE\u00B5bk) and path of peak maxima (km path) shown in addition to an example MDC cut at \u00CF\u0089\u00CC\u0083 and the peak maximum location for that cut [km(\u00CF\u0089\u00CC\u0083)]. (b) the cut through momentum of A(k, \u00CF\u0089) at constant energy \u00CF\u0089\u00CC\u0083, observed to be a Lorentzian, with a peak maximum located at km(\u00CF\u0089\u00CC\u0083), a HWHM \u00E2\u0088\u0086km(\u00CF\u0089\u00CC\u0083), and an area A0(\u00CF\u0089\u00CC\u0083). These lineshape properties are related to the self-energy at \u00CF\u0089\u00CC\u0083 through the bare-band evaluated at km through Eq. 2.15. 33 2.3. Self-energy analysis (c) M om en tu m (1 /\u00C3\u0085 ) 1 2 3 -50 -25 0 0 50 100 150 200 -50 -25 0 (b) \u00CE\u00A3' ' ( m eV ) Binding Energy (meV) \u00CE\u00A3'MDC \u00CE\u00A3'KK \u00CE\u00A3'known (a) \u00CE\u00A3''MDC \u00CE\u00A3''KK \u00CE\u00A3''known \u00CE\u00A3''ratio \u00CE\u00A3' ( m eV ) km(\u00CF\u0089) \u00CE\u00B5 known \u00CE\u00B5 found A(k,\u00CF\u0089) MA(1) Holstein \u00CE\u00BB = 0.1, \u00CE\u00A9 = 50 meV b k Figure 2.10: MDC self-energy analysis performed on the Holstein model at low coupling. (a) A(k, \u00CF\u0089), calculated for the momentum-independent Holstein self-energy with MA(1) for \u00E2\u0084\u00A6 = 50 meV and \u00CE\u00BB = 0.1, as a false- colour plot; also shown are the km path of MDC maxima along which the analysis is performed, as well as the known bare-band and the third order polynomial approximation found through the KKBF analysis (the arbitrary energy offset introduced through KKBF has been shifted back by hand to allow comparison between the two). (b,c) Real and imaginary parts of the self-energy from the model (\u00CE\u00A3known), the bare-band and MDC fitting routine (\u00CE\u00A3MDC) as found via Eq. 2.15, and the KK transform of \u00CE\u00A3 \u00E2\u0080\u00B2\u00E2\u0080\u00B2 MDC (\u00CE\u00A3 \u00E2\u0080\u00B2 KK) and \u00CE\u00A3\u00E2\u0080\u00B2MDC (\u00CE\u00A3 \u00E2\u0080\u00B2\u00E2\u0080\u00B2 KK) used as an internal check in KKBF. In (c) the MDC ratio results, \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2ratio, as found via Eq. 2.16, are also shown. 34 2.3. Self-energy analysis -150 -100 -50 0 \u00CE\u00A3' '(\u00CF\u0089 ) (m eV ) 0 50 100 150 Binding Energy (meV) \u00CE\u00A3''MDC \u00CE\u00A3''KK \u00CE\u00A3''ratio \u00CE\u00A3''known 0 10 0.01 0.02 Av er ag e D ev ia tio n fro m K no w n ( eV 2 ) \u00CE\u00BB=g2/2t \u00CE\u00A9 MA(1) Holstein \u00CE\u00BB = 0.5, \u00CE\u00A9 = 50 meV (b)(a) Figure 2.11: (a) deviation (i.e., average of the squared difference at each km) between extracted and known self-energies vs. \u00CE\u00BB. (b) various estimates for the imaginary part of the self-energy, defined as in the caption of Fig. 2.10, for A(k, \u00CF\u0089) calculated within MA(1) for \u00E2\u0084\u00A6=50 meV and \u00CE\u00BB=0.5. This variation allows us to tackle the problem over a larger range of \u00CE\u00BB, as it does not rely on the KKBF routine to succeed over the entire range in order to ensure the fitness of the Kramers\u00E2\u0080\u0093Kronig transform and fit the bare- band. Eq. 2.16 is free to work over energies where the peak is Lorentzian (i.e., Eq. 2.13 holds) and to fail in others without allowing these failures to block the procedure. Experimentally, when using this ratio, one must be careful that the observed spectra are properly normalized \u00E2\u0080\u0093 otherwise, it will be off by a constant multiple \u00E2\u0080\u0093 but even if this is not possible the form of the imaginary self-energy should be nevertheless recoverable. It is also possible, in cases of momentum-independent self-energy for which vbk/v q k = 1/Z q k from Eq. 2.12, to find the same ratio using only quasiparticle 35 2.3. Self-energy analysis properties as \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2MDC =\u00E2\u0088\u0092vqkm\u00E2\u0088\u0086km/Z q km . The results of both the KKBF and the ratio method, performed as if the bare-band was not known on a momentum-independent self-energy in the low-coupling regime, are presented in Fig. 2.10. The internal self-consistency of the KKBF results is confirmed by the good match between \u00CE\u00A3MDC and \u00CE\u00A3KK, and the agreement of \u00CE\u00A3ratio adds further confidence. These \u00E2\u0080\u009Cmeasured\u00E2\u0080\u009D quantities show good agreement with their known counterparts everywhere Eq. 2.13 is applicable, demonstrating that these methods work well in the low-coupling regime; however, they become progressively less accurate as the coupling increases. In Fig. 2.11(a) we show the progressive failure of the method applied for couplings where \u00CE\u00BB ranges from 0 to 1, which demon- strates a rapid degeneration of the accuracy of the method outside of the low-coupling regime. Note, however, that the two methods fail in different ways. \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2MDC tends to fail more globally, whereas \u00CE\u00A3 \u00E2\u0080\u00B2\u00E2\u0080\u00B2 ratio often continues to work almost exactly in some energy regions while failing catastrophically in others. (These regions cause its average deviation, shown in Fig. 2.11, to indicate perhaps a higher degree of failure than deserved.) In Fig. 2.11(b) we demonstrate these differences by showing the results of the methods applied blindly at \u00CE\u00BB = 0.5, even though lineshapes show that there are areas where Eq. 2.13 does not apply. One can see how the internal KKBF check has begun to fail, as \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2MDC and \u00CE\u00A3 \u00E2\u0080\u00B2\u00E2\u0080\u00B2 KK do not match; there are structural differ- ences, and \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2KK has picked up different offsets in the various flatter parts of the spectrum, making setting its overall offset difficult. As the disagreement between \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2MDC and \u00CE\u00A3 \u00E2\u0080\u00B2\u00E2\u0080\u00B2 KK increases with coupling, it will eventually cause the KKBF routine to fail to capture the bare electronic structure. None of the methods reproduce \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2known near the sharp one-phonon structure; note that \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2MDC and \u00CE\u00A3 \u00E2\u0080\u00B2\u00E2\u0080\u00B2 ratio overestimate and underestimate it, respectively. Our experience with this model leads us to believe that this behaviour is typical: when the methods fail, they do not tend to fail in identical ways, so that in regions where they do match one can still have confidence that the methods are working. 36 2.3. Self-energy analysis 2.3.2 Momentum-dependent self-energies The methods previously described to extract the self-energy from the spec- tral function hinge on the momentum independence of the self-energy in two ways. For fitting lineshape, a momentum-independent self-energy guaran- tees a Lorentzian lineshape, but the inverse is not true \u00E2\u0080\u0093 it is still possible to have a momentum-dependent self-energy that generates a Lorentzian. Ad- ditionally, fitting the bare-band relies on the Kramers\u00E2\u0080\u0093Kronig transforms in energy, which are valid only for a fixed momentum. In cases where the momentum dependence is not too strong locally near km, however, we have found that it is still possible to gain insight using similar approaches, al- though even more care must be taken in the interpretation of the results. If, despite momentum dependence, the MDC appears Lorentzian in shape, cer- tain higher-order terms must be small when expanding each of \u00CE\u00B5bk, \u00CE\u00A3 \u00E2\u0080\u00B2(\u00CF\u0089, k), and \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2(\u00CF\u0089, k) about km. Under this condition we may drop terms of order (k \u00E2\u0088\u0092 km)3 and higher from the denominator as well as (k \u00E2\u0088\u0092 km) and higher from the numerator, which also implies we may drop \u00E2\u0088\u0082\u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2/\u00E2\u0088\u0082k and \u00E2\u0088\u00822\u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2/\u00E2\u0088\u0082k2 everywhere. We end up with an expression identical to Eq. 2.13, only with new interpretations for the HWHM and the spectral weight of the peak: \u00E2\u0088\u0086km = \u00E2\u0088\u0092 \u00CE\u00A3 \u00E2\u0080\u00B2\u00E2\u0080\u00B2(\u00CF\u0089\u00CC\u0083, km) vbkm + \u00E2\u0088\u0082\u00CE\u00A3 \u00E2\u0080\u00B2(\u00CF\u0089\u00CC\u0083, km)/\u00E2\u0088\u0082k , A0 = \u00E2\u0088\u00AB A\u00CF\u0089\u00CC\u0083(k)dk = 1 vbkm + \u00E2\u0088\u0082\u00CE\u00A3 \u00E2\u0080\u00B2(\u00CF\u0089\u00CC\u0083, km)/\u00E2\u0088\u0082k . (2.17) Now that the self-energies are momentum dependent, it becomes more important to remember that the self-energy extracted using this method will follow the km path through (\u00CF\u0089, k) space; this path is demonstrated as a false-colour plot in Fig. 2.12 (d, e, i, j). From Eq. 2.17 we find that, in this momentum-dependent case, the ratio check Eq. 2.16 proves to be in- valuable, as it removes the possible inaccuracies when strong momentum dependence near km in \u00CE\u00A3 \u00E2\u0080\u00B2 might provide a Lorentzian lineshape with a mis- leading width, when viewed from a momentum-independent perspective. 37 2.3. Self-energy analysis Binding Energy (meV) 0 100 200 \u00CE\u00A3' ' (m eV ) M om en tu m (1 /\u00C3\u0085 ) \u00CE\u00A3' (m eV ) 1 2 3 -75 -50 -25 0 -75 -100 -50 -25 0 M om en tu m (1 /\u00C3\u0085 ) 1 2 3 M om en tu m (1 /\u00C3\u0085 ) 1 2 3 1 2 3 1 2 3 1 2 3 -200 -100 0 -200 -100 0 Binding Energy (meV) 0 100 200 MA(v,0) Breathing \u00CE\u00BB=0.15, \u00CE\u00A9=50 meV \u00CE\u00A3'(k,\u00CF\u0089) \u00CE\u00A3'(k,\u00CF\u0089) \u00CE\u00A3''(k,\u00CF\u0089) \u00CE\u00A3'known @ km, 0, \u00CF\u0080 \u00CE\u00A3'MDC \u00CE\u00A3'KK , , MA(v,0) Breathing \u00CE\u00BB=0.45, \u00CE\u00A9=50 meV M os t N eg at iv e M os t P os iti ve Ze ro (d) (e) (f) (g) (h) (i) (j) (a) (b) (c) \u00CE\u00A3''(k,\u00CF\u0089) km(\u00CF\u0089) \u00CE\u00B5 known \u00CE\u00B5 found A(k,\u00CF\u0089) b k b k km(\u00CF\u0089) \u00CE\u00B5 known \u00CE\u00B5 found A(k,\u00CF\u0089) b k b k km(\u00CF\u0089) \u00CE\u00B5 knownbk km(\u00CF\u0089) \u00CE\u00B5 knownbk km(\u00CF\u0089) \u00CE\u00B5 knownbk km(\u00CF\u0089) \u00CE\u00B5 knownbk \u00CE\u00A3'known @ km, 0, \u00CF\u0080 \u00CE\u00A3'MDC \u00CE\u00A3'KK , , \u00CE\u00A3''known @ km, 0, \u00CF\u0080 \u00CE\u00A3''MDC \u00CE\u00A3''KK \u00CE\u00A3''ratio , , \u00CE\u00A3''Known @ km, 0, \u00CF\u0080 \u00CE\u00A3''MDC \u00CE\u00A3''KK \u00CE\u00A3''ratio , , Figure 2.12: MDC self-energy analysis performed on the breathing mode model. (a\u00E2\u0080\u0093c,f\u00E2\u0080\u0093h) A(k, \u00CF\u0089) and self-energies as defined in Fig. 2.10; here, as the self-energy is momentum dependent, the known self-energies are plotted along the path of peak maxima (\u00CE\u00A3Known@km) to show good agreement with the derived values, as well as along both edges of the Brillouin zone for com- parison (\u00CE\u00A3Known@k = 0 and \u00CE\u00A3Known@k = pi). Finally (d,e,i,j) show the full extent of the energy and momentum dependence of the real and imaginary self-energies as a false-colour plot, with the km path superimposed. 38 2.4. Conclusions In Fig. 2.12 (a\u00E2\u0080\u0093c, f\u00E2\u0080\u0093h) we present the results of both KKBF and the ratio method for a momentum-dependent coupling. From a comparison between the measured and known self-energies on paths through k-space along the zone boundaries and along km, we find agreement only along the km path, as expected. Further, especially in Fig. 2.12 (h) near the third phonon struc- ture close to \u00E2\u0088\u00BC125 meV, one can see how it is possible for \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2MDC and \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2KK to agree with each other and yet not correctly predict \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2known, despite the peak shape being reasonably Lorentzian, owing to sufficient local momen- tum dependence in the real self-energy [Fig. 2.12 (i)]. In this location we note, however, that \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2ratio still correctly predicts \u00CE\u00A3 \u00E2\u0080\u00B2\u00E2\u0080\u00B2 known as it is not affected by this local momentum dependence. Overall we find that, in a similar fash- ion to the momentum-independent case, there is generally good agreement between the found self-energies and the self-energy along the km path in the low-coupling regime, and the methods progressively fail as we move into the mid-coupling regime. The available modes of failure are increased: there are more locations where the lineshape is not Lorentzian because of strong local momentum dependence of the self-energy; there are places where it is still Lorentzian but with a misleading width; and the Kramers\u00E2\u0080\u0093Kronig relations are not valid along an arbitrary path through (\u00CF\u0089, k) space, which disrupts the fitness of the KKBF routine. While it is not surprising that the KKBF routine eventually fails for large couplings in the momentum- dependent case, it is interesting that it works at all, as the Kramers\u00E2\u0080\u0093Kronig relations in energy are formally valid only for a fixed momentum, but the measured self-energies follow the km path at all couplings. Despite this, as can be seen in Fig. 2.12 (b, c), the Kramers\u00E2\u0080\u0093Kronig relations appear to work relatively well along the km path in the low-coupling case. Nevertheless we find that, in this model, failures occur at qualitatively similar couplings when momentum dependence is added. 2.4 Conclusions The spectral function is an extremely rich data source. We have shown that, despite the allure of extracting the true microscopic coupling through quasi- 39 2.4. Conclusions particle renormalizations of ARPES data with an effective coupling scheme, away from the Migdal limit it is not always possible to do so \u00E2\u0080\u0093 and cer- tainly not for cases close to a dispersion maximum. Away from this limit \u00CE\u00BBeff 6= \u00CE\u00BB. However, through careful modelling and the analysis of specific features it may be possible to learn much more. If one can correctly devise a model, it may be possible to predict a given renormalization or even show experimental support for a given model via relationships between renor- malization parameters. Through MDC self-energy analysis, we have shown that the self-energy can be extracted along paths through (k, \u00CF\u0089) space in the low-coupling regime \u00E2\u0080\u0093 and that it is possible still to gain some insight at higher couplings. We have also shown that it is possible to infer the momen- tum dependence of the self-energy through comparison of renormalization properties. Methods like these, together with detailed models and powerful simulations, will hold the key to more thorough and quantitative analysis of the rich data supplied through ARPES. 40 Chapter 3 Chamber characterization A large effort was put toward chamber characterization as part of the com- missioning and optimization of the new angle-resolved photoemission spec- troscopy (ARPES) system at University of British Columbia (UBC). This chapter will describe efforts undertaken both to characterize the system in general and also specifically to characterize the data presented in Chapter 4. While the analyzer has a variety of angular modes, those used most fre- quently (by me), and the only ones presented in this thesis, are wide-angle mode (WAM) and low-angular-dispersion mode (LAD). WAM is the mode that admits the largest angular window, \u00C2\u00B113\u00E2\u0097\u00A6, and is most useful for map- ping entire Fermi surfaces (for example Fig. 4.3), while LAD (as its name suggests), with an acceptance of \u00C2\u00B17\u00E2\u0097\u00A6, is most useful for following electronic dispersions (for example Fig. 4.8). We will begin with measurements to characterize the angular resolution and its dependence on sample position in \u00C2\u00A7 3.1, using both photoemission from samples as well as an electron gun, and will cover energy resolution in \u00C2\u00A7 3.2. Finally, the \u00E2\u0080\u009Cresolution\u00E2\u0080\u009D of the final 2D electron detection system is discussed in \u00C2\u00A7 3.3. For clarity, a brief discussion of the functional forms used and a comparison between different measures of \u00E2\u0080\u009Cresolution\u00E2\u0080\u009D \u00E2\u0080\u0093 full width at half maximum (FWHM) vs. standard deviation vs. resolved width of a Fermi edge \u00E2\u0080\u0093 is presented in \u00C2\u00A7 3.4. 41 Chapter 3. Chamber characterization analyzer sample electrons slit array 32 mm 40 mm .1 mm 1 mm (not to scale) ~1.79\u00C2\u00B0 ~0.179\u00C2\u00B0 tan(\u00CE\u00B8) \u00CE\u00B8 0 0.20 0.2 (c) (b) Energy ~A ng le (a) Figure 3.1: Illustration of slit array and slit array data. (a) typical detec- tor image taken with slit array and electron gun, in WAM. (b) illustra- tion of the sample, emitted electrons, slit array, and analyzer, with sam- ple placed at the focal point of the analyzer. (c) plot of tan(\u00CE\u00B8) vs. \u00CE\u00B8 for 0 to .22689 radians, equivalent to the 13\u00E2\u0097\u00A6 maximum half-acceptance angle of the analyzer. This represents the total deviation from linearity in an- gular spacing of the electrons that pass the slit array, owing to its being straight with evenly spaced slits (rather than curved, with centre of curva- ture at the sample). The largest deviation, at the far edge of the scale, is tan(\u00CE\u00B8max) \u00E2\u0088\u0092 \u00CE\u00B8max \u00E2\u0089\u0088 .00398 \u00E2\u0088\u00BC .228\u00E2\u0097\u00A6. Assuming a point source, the angu- lar spacing of slits passing through the centre of the analyzer will be \u00E2\u0088\u00BC1.79\u00E2\u0097\u00A6 with width \u00E2\u0088\u00BC0.179\u00E2\u0097\u00A6, while at the edge of detection (13\u00E2\u0097\u00A6 from centre) spacing is \u00E2\u0088\u00BC1.71\u00E2\u0097\u00A6 with width \u00E2\u0088\u00BC0.170\u00E2\u0097\u00A6 owing to this effect. 42 3.1. Angular resolution 3.1 Angular resolution 3.1.1 Setup In order to measure the angular resolution of the analyzer in situ, a slit array can be moved in between the sample and the analyzer entrance slit to generate a data set with features evenly spaced in angle, as shown in Fig. 3.18. This can be done using an electron gun (nominal FWHM spot size is 50\u00C2\u00B5m for the SPECS EQ 22/25 used) scattering from a thin wire (thickness 95\u00C2\u00B5m) or gold target to measure the fundamental limit of angular resolution, or directly with photoemission from the sample measured during an experiment (nominal light source spot size \u00E2\u0088\u00BC500\u00C2\u00B5m). We present results from both these methods here, which yield different estimates for the angular and effective angular resolution attainable with our setup. 3.1.2 Electron gun tests In order to determine the optimal sample position within the three positional degrees of freedom of the cryostat as built, data was taken as a function of cryostat position using an electron gun (spot size 50\u00C2\u00B5m) on a thin tungsten wire (thickness 95\u00C2\u00B5m) mounted to the bottom of the cryostat. The goal was to optimize the angular resolution by moving the cryostat and thereby find the focal point of the analyzer. Unfortunately, this did not prove feasible for two reasons. Primarily, the angular resolution was not found to depend strongly or reliably on sample position. The lack of reliability was likely related to the second reason \u00E2\u0080\u0093 a 95\u00C2\u00B5m wire is difficult to reproducibly hit with a 50\u00C2\u00B5m electron beam (especially when moving the cryostat by mil- limetres), and partial illumination of the wire could also cause a change in perceived resolution. Eventually the optimal cryostat location was physi- cally measured during a chamber venting for routine maintenance, based on the nominal focal point of the analyzer. Both the wire (because of difficul- ties the wire caused inserting and removing the cryostat) and the electron 8This is the same configuration used to correct for image warping in the angular direc- tion (see \u00C2\u00A7B.1). 43 3.1. Angular resolution 0.22 0.23 0.24 0.25 0.26 0.27 548-563 400-415 218-233 Angle (deg) In te ns ity (a .u .) Rectangular Fence Fit Data R es ol ut io n FW H M (d eg ) (a) (b) -10 -5 0 5 10 0 200 400 600 800 x = -0.4 .. -2.0 mm y = -2.525 mm z = 322.5 mm x = -0.2 mm y = -2.5 .. -3.3 mm z = 322.5 mm x = -1.0 mm y = -3.0 mm z = 322.5 mm x = -0.2 .. 0.6 mm y = -2.525 mm z = 322.5 mm x = -0.2 mm y = -2.25 .. -1.5 mm z = 322.5 mm Figure 3.2: Angular resolution tests in WAM, using an electron gun. (a) example angular dispersion of the slit image, integrated from pixels 548\u00E2\u0080\u0093 563 (thick red), as well as infinite resolution rectangular fence with back- ground (thin black) and final instrumental broadened fit (thick black). (b) the energy resolution derived from such fits for varying cryostat position, integrated over three detector energy regions: pixels 200\u00E2\u0080\u0093210, typical mid- occupied state (blue); 400\u00E2\u0080\u0093410, typical Fermi edge (green); and 500\u00E2\u0080\u0093510, typically just on the unoccupied side of the Fermi energy (red). Note that the camera used in these measurements has since been upgraded from 640 to 1376 pixels in this direction. The data point corresponding to panel (a) is the filled shape in (b). 44 3.1. Angular resolution -6 -4 -2 0 2 4 6 Angle (deg) 20 40 60 80 In te ns ity (a .u .) Rectangular Fence Fit Data 0.12 0.14 0.16 0.18 0.20 R es ol ut io n FW H M (d eg ) pixels 200-210 pixels 400-410 pixels 500-510 (a) (b) x = -1 mm y = -3 .. -4 mm z = 322.5 mm x = -.5 mm y = -4 .. -2 mm z = 322.5 mm x = -1.5 mm y = -2 .. -4 mm z = 322.5 mm x = -2 mm y = -4 .. -2 mm z = 322.5 mm x = -2.5 mm y = -2 .. -4 mm z = 322.5 mm Figure 3.3: Angular resolution tests in LAD, using an electron gun. (a) example angular dispersion of the slit image, integrated from pixels 400\u00E2\u0080\u0093 410 (thick green), as well as infinite resolution rectangular fence with back- ground (thin black) and final instrumental broadened fit (thick black). (b) the energy resolution derived from such fits for varying cryostat position, integrated over three detector energy regions: pixels 200\u00E2\u0080\u0093210, typical mid- occupied state (blue); 400\u00E2\u0080\u0093410, typical Fermi edge (green); and 500\u00E2\u0080\u0093510, typically just on the unoccupied side of the Fermi energy (red). Note that the camera used in these measurements has since been upgraded from 640 to 1376 pixels in this direction. The data point corresponding to panel (a) is the filled shape in (b). 45 3.1. Angular resolution gun (in order to minimize possible sources of stray fields) have since been removed from the chamber. Despite failing to precisely determine the focal point (instead discover- ing that it was not required to be precisely at the focal point), owing to the small spot size of the electron gun this wire-test data set remains a good indicator of the fundamental angular resolution of the analyzer. In or- der to determine the resolution for each image it was first transformed into angle-space (see \u00C2\u00A7B.1) to reduce warping and to set the scale from pixels to angle9. Each image was then least-squares [53] fit to a rectangular \u00E2\u0080\u009Cfence\u00E2\u0080\u009D with polynomial background and convolved with a Gaussian resolution. The background, individual rectangle location/height, and instrumental resolu- tion were fit parameters, while the rectangular width was kept fixed at 0.18\u00E2\u0097\u00A6, which corresponds to the width expected from a point source with infinite resolution. The result of a single fit is shown in panel (a) of Fig. 3.2 for WAM and Fig. 3.3 for LAD, with the results of all fits shown in (b). These test results show that the resolution does not strongly depend on sample position relative to the focal point, although there is some vari- ation across the detector. Comparing all measurements near the typical Fermi edge location, where angular resolution is most important, gives FWHMWAM = 0.230 \u00C2\u00B1 0.005\u00E2\u0097\u00A6 and FWHMLAD = 0.141 \u00C2\u00B1 0.008\u00E2\u0097\u00A6 for the resolution and its standard deviation across the multiple measurements. If we assume that the instrumental resolution combines in quadrature with the electron gun spot size (FWHM 0.050 mm at 32 mm from the slit corresponds to 0.090\u00E2\u0097\u00A6), then we obtain FWHMWAM = 0.212\u00C2\u00B1 0.005\u00E2\u0097\u00A6, FWHMLAD = 0.109\u00C2\u00B1 0.008\u00E2\u0097\u00A6. (3.1) 46 3.1. Angular resolution Angle (deg) Angle (deg) In te ns ity (a .u .) In te ns ity (a .u .) Rectangular Fence Fit Data (a) (b) -10 -5 0 5 10 0 5 10 15 -4 -2 0 2 4 0 4 8 12 16 Figure 3.4: Angular resolution tests in WAM, on Sr2RuO4, showing the image integrated from \u00E2\u0088\u0092100 to 0 meV (thick red), as well as infinite resolu- tion rectangular fence with background (thin black) and final instrumental broadened fit (thick black). In (a) the full angular width of the detector is shown, while (b) presents a close-up of the centre region. 47 3.1. Angular resolution -6 -4 -2 0 2 4 6 Angle (deg) In te ns ity (a .u .) Rectangular Fence Fit Data 0 1 2 3 Figure 3.5: Angular resolution tests in LAD, on Sr2RuO4, showing the image integrated from \u00E2\u0088\u009235 to 0 meV (thick red), as well as infinite resolu- tion rectangular fence with background (thin black) and final instrumental broadened fit (thick black). 48 3.1. Angular resolution 3.1.3 Sample tests Although the angular resolution was not observed to strongly change as a function of position during the electron gun tests, the image warping was. This suggests that, although in principle the analyzer should map all parallel electrons to the same spot on the detector regardless of their point of origin, photon spot (\u00E2\u0088\u00BC500\u00C2\u00B5m) and sample (\u00E2\u0088\u00BC0.5 to 2 mm) size can affect the final angular resolution of ARPES measurements on a real sample. For this reason, and to ensure the accuracy of the pixel-to-angle conversion, slit array measurements are taken for every data set measured. It is useful to make an estimate of the instrumental resolution from these measurements as well. In Figs. 3.4 and 3.5 we present such tests using data sets presented in Chapter 4 for WAM and LAD, respectively. Comparing panels (a) and (b) in Fig. 3.4 for the WAM test, we find that the apparent angular resolution is not uniform across the detector, which is typical of slit images taken using a real sample in this mode. In Fig. 3.5 we find that the apparent angular resolution in LAD is much worse, despite the narrower acceptance angle, but that it is uniform across the detector. By fitting these data sets, as was done in \u00C2\u00A7 3.1.2, we find the resolutions, with uncertainly based on chi- squared (which is much less reliable than the large data sets used in the electron gun tests), to be FWHMWAM, whole detector = 0.259\u00C2\u00B1 0.005\u00E2\u0097\u00A6, FWHMWAM, centre region = 0.220\u00C2\u00B1 0.003\u00E2\u0097\u00A6, FWHMLAD = 0.850\u00C2\u00B1 0.005\u00E2\u0097\u00A6. (3.2) It is interesting that, now with a larger spot size, the apparent angular resolution in WAM has not changed much, whereas in LAD it has changed drastically. Owing to the smaller angular acceptance, which has the effect 9Moving to angle space assuming the sample is at the focal point also has the effect of eliminating a change in perceived resolution owing to expansion/contraction of the entire pattern caused by deviation from the true focal point, although it changes the perceived total angular acceptance angle. 49 3.1. Angular resolution of \u00E2\u0080\u009Czooming\u00E2\u0080\u009D in angle space on the detector, one would expect the angular resolution in LAD to be superior to that of WAM; however, when measuring photoemission on actual samples, WAM, in our chamber as built, appears to have superior angular resolution. Although this discrepancy between modes is very reproducible10, how it exists is not clear, but it extends from low- to medium- and high-angular-dispersion modes, which (although not fully characterized here) all show visually similar apparent resolutions to LAD. Indeed, given a 500\u00C2\u00B5m spot size (which is likely, as Sr2RuO4 cleaves well and often photoemits from the entire cleaved surface, making the light source the limiting factor) at 32 mm, one should even expect a .895\u00E2\u0097\u00A6 apparent broadening in the slit image, so it is not clear how WAM can do so well. 3.1.4 Conclusions As mentioned at the beginning of the last section, in principle the analyzer should map all parallel electrons to the same spot on the detector regard- less of their point of origin. This means that the apparent resolutions as measured with a large spot size using the slit array (which blocks electrons, creating features in angle that depend on the spot size) do not necessar- ily reflect the ability of the analyzer to resolve fundamental photoemitted features of the electronic state in a sample (these features are emitted in parallel, from all points on the sample surface). However, it has been ob- served (especially during the electron gun tests) that the image warping (not just position) changes when moving the sample, indicating that the analyzer does not quite map all parallel electrons to the same spot on the detector when they originate from different locations. For this reason the effective resolution will lie somewhere in between the values presented in Eqs. 3.1 and 3.2. It is recommended to always check the apparent angular resolution by analyzing a slit image for every data set. In practice, as suggested by the apparent resolution in slit-array data measured on real samples, WAM does have a consistently better angular resolution compared with LAD (demonstrated by ability to separate features 10Reproducible with photoemission from samples \u00E2\u0080\u0093 the electron gun tests have not been repeated, and cannot be repeated without re-installing an electron gun on the chamber. 50 3.2. Energy resolution which are close together in angle space). However, it is still not practical to use WAM to perform dispersion analysis, because it is not possible to set the pass-energy of the analyzer low enough to obtain sufficient energy resolution in this mode; it remains a Fermi surface mapping tool. 3.2 Energy resolution The discrimination in the energy direction for our system is outlined in Fig. 3.6. The selectable entrance slit allows for finer resolution at the cost of restricting electrons with a small slit or provides higher electron yield at the cost of reduced resolution11, while the pass energy Epass varies the overall scale of the image (energy/pixel). The energy resolution of the analyzer is therefore controlled by a combination of the pass energy and entrance slit used and nominally obeys the following relation: FWHMEnergy Resolution = Epass \u00C2\u00B7 Slit Width 300 mm , (3.3) where the (curved) slits available may be selected from 0.8, 0.5, 0.2, and 0.1 mm and the pass energies available depend on both the angular lens mode as well as the kinetic energy chosen. Along with every data set a measurement is also performed on polycrys- talline12 gold, held at sample temperature, in order to measure the energy resolution in situ. This gold sample reference is a part of the cryostat, held a known distance from the sample, allowing for quick and precise switching between sample and gold. Examining the sharpness of the Fermi edge on 11In addition to size affecting resolution, the entrance slit\u00E2\u0080\u0099s shape affects the shape of the final image. Experiments presented here were performed with a concentrically curved entrance slit, which causes electrons of the same energy to fall on concentric rings on the detector. Corrections for this effect are discussed in \u00C2\u00A7B.2. A straight slit, available in our analyzer as built, causes electrons of the same energy to fall in rings with twice the curvature. A slit with twice the curvature, not available on our analyzer as built, should cause electrons of the same energy to fall in straight lines on the detector \u00E2\u0080\u0093 but would also be more susceptible to non-uniform warping from the non-radial fields caused by the existence of the slit itself. Such warping would be difficult, if not impossible, to properly correct. 12Polycrystalline so it has no angular structure. 51 3.2. Energy resolution sample electrostatic lenses detector nose slit selectable entrance slit e- path for Ein hemisphere > Epass , Ephotoemission > Ek e- path for Ein hemisphere = Epass , Ephotoemission = Ek e- path for Ein hemisphere < Epass , Ephotoemission < Ek hemispherical analyzer Figure 3.6: Illustration of energy discrimination through the hemispherical analyzer. This view is perpendicular to that which discriminates angle, presented in Fig. 3.1. In this direction electrons first pass through a slit on the analyzer nose, which admits electrons over \u00E2\u0088\u00BC1\u00E2\u0097\u00A6 (as inferred from the apparent angular resolution taken over multiple scans at different sample angles \u00E2\u0080\u0093 for example, in Fig. 4.3). These electrons are then focused on the entrance slit to the hemispheres and retarded (or accelerated) such that electrons that entered the lenses with kinetic energy Ek will exit them and enter the hemispheres with energy Epass. Passing through the selectable entrance slit, electrons are then discriminated in energy via the radial field of the hemispheres; electrons with energy Epass will travel through the centre of the hemispheres, while those with more or less energy will drift either to the outside or inside (respectively). 52 3.2. Energy resolution -60 -40 -20 0 20 40 60 Energy (meV) 0 1 In te ns ity (a .u .) Data Gaussian resolution fit Lorentzian resolution fit Figure 3.7: Energy resolution test in LAD, on gold at 9.7 K, showing the Fermi edge integrated across the detector after image warping corrections (black) as well as fit results assuming Gaussian (red) and Lorentzian (green) resolutions. 53 3.2. Energy resolution -60 -40 -20 0 20 40 60 Energy (meV) -60 -40 -20 0 20 40 60 Energy (meV) In te ns ity (a .u .) In te ns ity (a .u .) Data Gaussian resolution fit Lorentzian resolution fit Data Gaussian resolution fit Lorentzian resolution fit 0.0 0.5 1.0 1.5 0 5 10 15 (a) (b) Figure 3.8: Energy resolution test in WAM, on gold, showing the Fermi edge integrated across the detector after image warping corrections (black) as well as fit results assuming Gaussian (red) and Lorentzian (green) resolutions. (a) shows the tests corresponding to data shown in Fig. 4.3 on gold at 8.5 K; (b) shows the tests corresponding to data shown in Fig. 4.10 on gold at 5.0 K. 54 3.2. Energy resolution the gold data thereby provides a direct measurement of the instrumental resolution directly at the Fermi energy for the exact analyzer parameters and sample position as used. The gold data taken to characterize the energy resolution for Figs. 4.8 and 4.9 is presented in Fig. 3.7. This data was taken in LAD, with slit 6 (0.5 mm, curved) and Epass = 3 eV; this corresponds to a nominal energy resolution of 5 meV, according to Eq. 3.3. At the sample temperature of 9.7 K13 the Fermi edge should have an inherent width from 90% to 10% of 3.67 meV. In order to determine the instrumental resolution, a least-squares [53] fit was performed on the data, where the model function consisted of a linear occupied state times a 9.7 K Fermi function with background, convolved with instrumental resolution of both Gaussian (red curve) and Lorentzian (green curve) form. The background, slope of the occupied states, and instrumental resolution were fit parameters, with the temperature held fixed. From the fit results it appears that the instrumental energy resolution in this mode may be better described by a Lorentzian than a Gaussian form, which is interesting, as Gaussian resolutions are usually assumed. It is not known what causes this anomaly, although it should be noted that some other resolution functions (namely square and triangular) were also attempted but did not fit the data well at all. The FWHM resolution values thus extracted were 9 meV, assuming Lorentzian, or 16 meV, assuming Gaussian. By either measure the resolution is greater than that nominally specified in the manual, by a factor of two to three. In Fig. 3.8 we present some of the gold Fermi edges used to characterize energy resolution for the WAM Fermi surfaces presented in Chapter 4. As for the LAD test the gold image was first corrected and then integrated across the detector. Panel (a) is the measurement taken along with Fig. 4.3, on gold at 8.5 K, while (b) shows that corresponding to Fig. 4.10, on gold at 5.0 K. Both images were taken using slit 6 (0.5 mm, curved) and a pass energy of Epass = 12 eV for a nominal resolution of 20 meV. In this case we find that the Gaussian provides a better-looking fit, although again neither form is perfect. For (a) resolutions found from these fits are 28 meV (15 meV); 139.7 K times the Boltzmann constant is \u00E2\u0088\u00BC0.84 meV. 55 3.3. Event counting (b) gives 24 meV (13 meV) for Gaussian (Lorentzian) forms. Other gold measurements taken in WAM give similar results for resolution values and the appearance of Gaussian vs. Lorentzian forms. From these (somewhat limited) measurements it appears that, for nomi- nal energy resolutions \u00E2\u0088\u00BC20 meV in WAM, the resolution is close to Gaussian and similar to that expected from the slit selection and analyzer pass energy. However, in LAD and with nominal energy resolutions \u00E2\u0088\u00BC5 meV, the energy resolution may not be as expected in either form or width. Regardless of specified resolution, it is good practice to verify the resolution with a gold measurement for every data set. 3.3 Event counting The final electron detection of our system consists of a double-layer multi- channel plate14 to amplify the electron signal, followed by a phosphor screen and digital camera. An electron entering the multichannel plate sets off a cascade of further electrons, which lights the phosphor screen and is ulti- mately recorded by the camera. An image highlighting the properties of single-electron events is pre- sented in Fig. 3.9 (this data was taken using gas cell photoemission by SPECS to test energy resolution during the initial analyzer commission- ing). Ideally, each electron that hits the detector would be recorded as a single event at a precise location. However, owing to variations (perhaps in how far electrons make it into the multichannel plate before triggering a cascade) the overall scale of each electron event is generally recorded with a different brightness, which can be seen in the inset of panel (a) and again in Fig. 3.10(a). Further, each event generates a dim halo with a diameter of approximately 20 pixels15. This halo can be seen in Fig. 3.9 (a) around the main spectral feature and also in (b) as the \u00E2\u0080\u009Ctails\u00E2\u0080\u009D extending on both sides of the integrated image. Although the origin of these tails was initially 14Arranged in a chevron shape, such that electrons cannot pass all the way through without hitting the side of a channel. 15Diameter \u00E2\u0088\u00BC20 pixels on the old 640\u00C3\u0097 480 camera, which has since been upgraded. 56 3.3. Event counting } 0 100 200 300 400 500 600 0 50 100 150 200 250 300 In te ns ity (a .u .) Pixels (a) (b) In te ns ity (a .u .) Figure 3.9: Gas cell photoemission image demonstrating the form of electron events. (a) entire image, with colour scale stretched such that the \u00E2\u0080\u009Ctail\u00E2\u0080\u009D from electron events, when integrated over many counts, is visible as a faint halo around the main spectral feature. Numerous single-electron events are visible on the detector screen, with three showing different intensity levels highlighted in the inset. (b) integration across the detector, with an orange brace marking the \u00E2\u0080\u009Ctails\u00E2\u0080\u009D around the main spectral feature. 57 3.3. Event counting (a) (b) Figure 3.10: A zoom-in on a group of events before (a) and after (b) the application of a single-event filter. 58 3.3. Event counting unclear, they have remained with similar size on both cameras used on the system and additionally could be observed to occur around the large number of events generated by a flaw that later developed on the multichannel plate (thought to be a speck of dust). For these reasons the tails must be due to either the multichannel plate or the phosphor screen. Regardless of their origin, the presence of these \u00E2\u0080\u009Ctails\u00E2\u0080\u009D can cause spurious data, especially when trying to characterize a Fermi edge or spectral feature shape. In order to eliminate this problem, a single event filter was developed and applied to single images taken with fast acquisition times, such that few events were recorded to overlap. This routine would transform the image into islands of equal value for everything above a given threshold, and then further reduce it to single values of 1 at the centre of mass of each island. The results of this filter are illustrated in Fig. 3.10. Although occasional events were missed, either because they were too faint or because two events touched, this filter effectively reduced the camera image to a map of where each electron fell. Unfortunately, although the filter itself was reasonably fast, getting images from the camera itself proved to be a major bottleneck: with reasonable count rates the camera could not provide images fast enough, and camera dead-time accounted for a reasonable fraction of total experiment time. Given that many ARPES experiments are limited by sample life, this technique was not an option. Instead the camera was able to threshold multiple exposures and sum them at the hardware level, before sending a final image representing these multiple exposures. Given the correct choice of dwell time per exposure (such that not too many tail features overlapped) and threshold (such that only the tails were removed) the final image sent by the camera would be free of tails, and, with high enough statistics, the differing relative brightness of each electron event would average out. All data presented in this thesis (with the exception of that to illus- trate the tails) was taken such that the problem of tails was eliminated through camera thresholding. It is important, especially if detailed infor- mation about spectral shape is desired, to verify on a sample image that each exposure taken will be in the non-overlap regime and have the proper 59 3.4. Functional forms and a comparison of different measures threshold applied. 3.4 Functional forms and a comparison of different measures Sometimes, for clarity, it is best to simply set out the functional forms used. The Gaussian form used is as follows: G(x) = A \u00CF\u0083 \u00E2\u0088\u009A 2pi e\u00E2\u0088\u0092 (x\u00E2\u0088\u0092x0)2 2\u00CF\u00832 , (3.4) which is centred at x0, has area A (set to unity, based on step size, when doing a numeric convolution), and has standard deviation \u00CF\u0083. For a Gaus- sian, \u00CF\u0083 is related to the FWHM and half width at half maximum (HWHM) according to FWHM = 2 \u00E2\u0088\u009A 2 ln 2\u00CF\u0083 \u00E2\u0089\u0088 2.35\u00CF\u0083 HWHM \u00E2\u0089\u0088 1.18\u00CF\u0083 \u00CF\u0083 \u00E2\u0089\u0088 0.426 FWHM \u00CF\u0083 \u00E2\u0089\u0088 0.847 HWHM. (3.5) The Lorentzian form used is as follows: L(x) = A pi \u00CE\u0093/2 (x\u00E2\u0088\u0092 x0)2 + (\u00CE\u0093/2)2 , (3.6) which is centred at x0, has area A, and has FWHM \u00CE\u0093 (the standard deviation of a Lorentzian is undefined). Although a true convolution was used in this chapter, an energy reso- lution rule of thumb exists: often, to get a rough estimate of instrumental energy resolution, the width of a (low-temperature) Fermi function from 10% to 90% is taken to be the FWHM resolution. Convolving a Gaussian with a step edge shows that, for a Gaussian, this is not a bad approximation. For a Gaussian with FWHM= 1, 90%/10% happens at \u00C2\u00B10.544, for a difference of \u00E2\u0088\u00BC1.09 (an overestimation of \u00E2\u0088\u00BC9%). However, the same treatment with a 60 3.4. Functional forms and a comparison of different measures -4 -2 0 2 4 -4 -2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (a) (b) Gaussian Lorentzian Gaussian Step function convolved with: Lorentzian Figure 3.11: A comparison of Gaussian (red) and Lorentzian (green) func- tions. (a) Gaussian and Lorentzian functions with unit area and FWHM, with the FWHM outlined in black. (b) the results of convolving these func- tions with a unit step function, with lines marking 10% and 90% height. 61 3.4. Functional forms and a comparison of different measures Lorentzian (which has much larger tails) results in \u00C2\u00B11.54, for a difference of \u00E2\u0088\u00BC3.08 and an overestimation by a factor of 3. This is illustrated in Fig. 3.11. 62 Chapter 4 ARPES on Sr2RuO4 Since its discovery [13], superconductivity in Sr2RuO4 has received enormous attention in the attempt to understand not only the superconducting but also the normal state from which superconductivity arises. In the study of the normal state a unified picture of the Fermi surface (FS) was obtained from bulk-sensitive de Haas\u00E2\u0080\u0093van Alphen (dHvA) measurements [14, 54, 55], surface-sensitive angle-resolved photoemission spectroscopy (ARPES) [56], and local-density approximation (LDA) band structure calculations [17, 57]. Agreement with the latter could only be obtained with the inclusion of spin\u00E2\u0080\u0093 orbit (SO) coupling, establishing its importance in the description of the low-energy electronic band structure16. This agreement between bulk and surface probes was not easily reached \u00E2\u0080\u0093 ARPES initially revealed a different electronic structure than the bulk [56, 58], a result explained by a surface reconstruction also observed via low-energy electron diffraction (LEED) [59\u00E2\u0080\u0093 61]. However, in our own ARPES measurements here at the University of British Columbia (UBC) with improved angular resolution and vacuum conditions, we began to regularly detect additional dispersive features not explained by this model \u00E2\u0080\u0093 a result also found in another recent ARPES study that presented no definitive explanation but suspected a surface state driven by electronic interactions [16]. As in the case of topological insulators, the presence of SO coupling raises the possibility that these features may be due to novel topological states, Dirac or Rashba-type, that might exist on the (001) cleaving surface of Sr2RuO4 in the normal state (in addition to the chiral superconductivity topological edge-states detected below Tc on the side surfaces [62]). These new revelations and possibilities demanded that 16SO coupling may also lie at the root of the larger question regarding the nature of the pairing in Sr2RuO4, which is explored in more detail in Chapter 5. 63 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands this simple model of surface reconstruction be revisited. In this chapter I will first discuss these additional features in contrast to those attributed to the previously accepted electronic structure from a bulk and reconstructed surface in \u00C2\u00A7 4.1, before detailing the efforts made by our group to unravel their nature. These efforts included high-resolution mapping to determine the extent of previously unnoticed features and their characteristics, discussed in \u00C2\u00A7 4.1.1; ruling out the possibility of a relation- ship to ruthenium metal inclusions17 discussed in \u00C2\u00A74.1.2; time resolved sur- face degradation discussed in \u00C2\u00A74.1.3; the use of polarized light to determine the symmetry of the underlying wave-functions discussed in \u00C2\u00A74.1.4; and fi- nally a comparison to related folding in the Bi-cuprates in discussed \u00C2\u00A74.1.5. These efforts were combined with the examination of a variety of different possible structural distortions through LDA bulk and slab band structure calculations discussed in \u00C2\u00A74.2. The culmination of these efforts allowed us to rule out novel phases driven by electronic interactions such as Dirac and Rashba-type states or surface magnetism and instead conclude that a sur- face reconstruction was still responsible for the additional features but that this reconstruction was not limited to the surface layer alone. We found that there exists a progressive reconstruction whereby both the surface and (at a minimum) the sub-surface layer both undergo similar reconstructions, differing in magnitude as the crystal structure relaxes progressively toward that of the bulk. This work resulted in a publication [6] that drew on many aspects of the work presented here but left out much detail in the avenues explored for brevity\u00E2\u0080\u0099s sake. 4.1 Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands The bulk crystal structure of Sr2RuO4 is precisely known from x-ray [63], as well as both single-crystal [64] and powdered [65\u00E2\u0080\u009367] neutron diffraction. Sr2RuO4 is the single-layer member of its perovskite family (SrRuO3 being 17The so-called 3K phase \u00E2\u0080\u0093 a common growth defect. 64 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands \u00CE\u0093 ZM X \u00CE\u00B1 \u00CE\u00B2 \u00CE\u00B3 (a) Z \u00CE\u0093 M X \u00CE\u0093 0 1 -1 (c) \u00CE\u00B1\u00CE\u00B2 \u00CE\u00B3 En er gy (e V ) \u00CE\u00B4 (b) dxz/dyz dxy \u00CE\u0093 ZM X Figure 4.1: The near-FS bulk band structure of Sr2RuO4 from LDA+SO, with \u00CE\u00B1, \u00CE\u00B2, \u00CE\u00B3, and \u00CE\u00B4 bands as well as \u00CE\u0093, M, X and Z BZ positions labelled. (a) the kz = 0 FS across multiple BZs in black, with the primitive and conventional BZ boundaries shown in blue and red, respectively; the inset shows the shapes of the primitive BCT and conventional unit cell in 3D momentum space. (b) orbital character of the \u00CE\u00B1, \u00CE\u00B2, and \u00CE\u00B3 bands at the kz = 0 crossing. (c) band dispersions along the high-symmetry cuts through momentum space. 65 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands (a) (b) \u00CE\u00B81 \u00CE\u00B82 r1 r2 Figure 4.2: Laue diffraction pattern for sample alignment on Sr2RuO4. Panel (a) shows a single crystal aligned with the a and b axis horizontal and vertical. This orientation will produce ARPES oriented similar to Fig. 4.3 (for example). To achieve this orientation (as opposed to 45\u00E2\u0097\u00A6 rotated) note that r1 < r2 and \u00CE\u00B81 > \u00CE\u00B82. Panel (b) shows a multi-faceted crystal, with ad- ditional diffraction patterns visible, as indicated with arrows. Such a sample should not be used. 66 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands the infinite-layer and Sr3Ru2O7 the bi-layer compound). It has a single layer of (vertically) distorted oxygen octahedra centred on ruthenium atoms, with strontium in the gaps formed by their apical oxygen. These layers are offset- stacked, forming a natural (and charge-neutral) cleavage plane in between the Sr\u00E2\u0080\u0093O layers, which is sketched in Fig. 4.15. The near-FS bulk electronic structure derives from the Ru t2g orbitals, and is occupied by four electrons per unit cell. The quasi-1D dxz and dyz bands hybridize at their crossings to create the electron-like \u00CE\u00B2 and hole-like \u00CE\u00B1 pockets, while the roughly circular \u00CE\u00B3 derives from the quasi-2D dxy band. Just above the Fermi energy at the \u00CE\u0093 point there lies the \u00CE\u00B4 pocket of dx2\u00E2\u0088\u0092y2 character, as illustrated in Fig. 4.1. Because of the body-centred tetragonal (BCT) crystal structure, the primitive Brillouin zone (BZ) unit cell has an offset stacking as illustrated in Fig. 4.1 (blue); however, for simplicity a conventional cubic BZ (red) is generally used, which results in neighbouring cells having different kz values in the same kx \u00E2\u0088\u0092 ky plane. [For example, at kz = 0 we find the \u00CE\u0093 point at (0, 0) and the Z point at (2pi, 0).] In this thesis I will use the conventional unit cell, as is done in most published works on Sr2RuO4. Although, in the bulk, there is no tilting or rotation of the oxygen octa- hedra (which would be seen by diffraction experiments) those in the surface layer are known to rotate about the c-axis, which causes a ( \u00E2\u0088\u009A 2\u00C3\u0097\u00E2\u0088\u009A2)R45\u00E2\u0097\u00A6 reconstruction of the primitive cell, as seen by surface-sensitive experiments such as ARPES [6, 56, 58, 68, 69] and LEED [59\u00E2\u0080\u009361, 70, 71].Additionally, it is likely that there are dimensional distortions along the c-axis in (at least) the topmost layer. As always, estimates of the true value for all these dimensions vary. ARPES experiments were performed at the University of British Columbia using our SPECS Phoibos 150 analyzer and the 21.22eV He I spectral line from a monochromatized UVS300 discharge lamp, over a variety of polar- izations. For dispersion mappings typical energy resolution is measured as 9 meV and angular resolution as 0.4\u00E2\u0097\u00A6 over a window of \u00E2\u0088\u00BC 14.5\u00E2\u0097\u00A6 (equivalent to \u00E2\u0088\u00BC 0.04 pi\u00E2\u0088\u009A 2a ). For Fermi surface mapping typical energy resolution was measured to be 17meV and angular resolution 0.1\u00E2\u0097\u00A6 over a window of \u00E2\u0088\u00BC29\u00E2\u0097\u00A6 67 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands (equivalent to \u00E2\u0088\u00BC 0.01 pia ). Samples were examined for defects and oriented using Laue diffraction (see Fig. 4.2) and cleaved in situ. During measure- ment orientation was controlled by our two-rotation-axis cryogenic manip- ulator (rotation available parallel and perpendicular to analyzer entrance slit), with temperatures held between 5 and 10 K and pressures varied in the 5 \u00E2\u0088\u0092 9 \u00C3\u0097 10\u00E2\u0088\u009211mbar range. The high-quality single crystals used in this study were grown by the floating zone method with Ru self-flux [72, 73] (see also \u00C2\u00A7 4.1.2). 4.1.1 Additional features found We begin our description of the new features with a brief review of the FS one would expect to measure based on the previously accepted electronic structure from a bulk and reconstructed surface. In Fig. 4.3(a) these bands are shown as calculated by LDA+SO; these predictions has been verified by both ARPES [56] and dHvA [14, 54, 55, 74, 75]. This is a portion of the same FS, as shown earlier in Fig. 4.1. If the oxygen octahedra are rotated, as has been observed for the surface layer of Sr2RuO4 by LEED [59, 60], this causes a ( \u00E2\u0088\u009A 2\u00C3\u0097\u00E2\u0088\u009A2)R45\u00E2\u0097\u00A6 reconstruction of the BZ [shown in Fig. 4.3(b) for a 5.5\u00E2\u0097\u00A6 rotation, which demonstrates the observed surface topology using bulk LDA calculations]. This reconstruction causes folded replicas (dotted line) of each original unfolded band (solid line) to appear mirrored across the new BZ boundary (dashed line). Because of the shallow dispersion at (pi, 0) (and points related by symmetry), this modulation also changes the topology of the \u00CE\u00B3 sheet from circular electron-like to lobed hole-like. In these panels we also begin using the subscript s to refer to surface-like states, b to refer to bulk-like states, and primes to indicate folded bands. Measuring with ARPES, one might expect a superposition of panels (a) and (b), as shown in panel (c); with sufficient resolution, bands from both surface and bulk would be visible as an apparent doubling for the case of unfolded bands (and unfolded bands only, as there should exist no folded bulk bands). In panel (d) we present a FS map over the entire upper right quadrant of the BZ from a pristine cleave of Sr2RuO4, obtained by integrating ARPES 68 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands (d) \u00E2\u0088\u009A2 BZ (a) (e) (b)\u00CE\u00B1b \u00CE\u00B2b \u00CE\u00B3b \u00CE\u00B1s \u00CE\u00B3s \u00CE\u00B2s \u00CE\u00B1's \u00CE\u00B2's \u00CE\u00B3's (f) (c) expected observed Figure 4.3: Expected vs. measured Fermi surface of Sr2RuO4. (a\u00E2\u0080\u0093c) the previously expected FS of Sr2RuO4 as would be measured by ARPES based on LDA+SO at kz=0. The FS of the (a) undistorted bulk and (b) bulk with RuO6 octahedra rotated 5.5 \u00E2\u0097\u00A6 causing a ( \u00E2\u0088\u009A 2\u00C3\u0097\u00E2\u0088\u009A2)R45\u00E2\u0097\u00A6 reconstruction (new BZ \u00E2\u0080\u0093 dashed; unfolded FS \u00E2\u0080\u0093 solid; folded replica FS \u00E2\u0080\u0093 dotted) should combine to give (c). (d) shows the upper right quadrant of the FS as measured by ARPES, in p-polarization (parallel to incidence/emission plane), on a fresh low-temperature cleave. (e) reconstructs the entire FS by folding this image, while (f) shows a phenomenological FS based on (e), Figs. 4.4, 4.8, and 4.9. The most obvious outstanding unexplained feature, the double-folded \u00CE\u00B2 band, is highlighted with a rectangle in (d). 69 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands In te ns ity (a .u .) 0 kx (\u00CF\u0080/a)kx (\u00CF\u0080/a) kx (\u00CF\u0080/a) kx (\u00CF\u0080/a) \u00CE\u00B2's\u00CE\u00B2'b\u00CE\u00B2s \u00CE\u00B2b \u00CE\u00B1s \u00CE\u00B1b \u00CE\u00B1's \u00CE\u00B1'b(c)(b) (d) (e) (a) 0.6 0.70.3 0.40.6 0.7 1.6 1.7 (0,0) (2\u00CF\u0080,0) (\u00CF\u0080,\u00CF\u0080) eb c d Figure 4.4: MDC cuts on the Fermi surface of Sr2RuO4. (a) Sr2RuO4 FS maps (greyscale) marking the location of MDC cuts at the Fermi energy, showing clear doubling in the (b) \u00CE\u00B2 sheet, (c) its folded replica \u00CE\u00B2\u00E2\u0080\u00B2, (d) \u00CE\u00B1, and (e) folded \u00CE\u00B1\u00E2\u0080\u00B2. (b\u00E2\u0080\u0093e) Data is shown in colour, with red and blue referring to surface and bulk-like features, respectively, while fit results (see text) are shown in black (dashed lines for folded bands). 70 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands intensity over a 7meV region. This data set was taken in less than 3 hours after the cleave, during which time the sample was maintained at low tem- perature in ultra-high vacuum (6\u00C3\u009710\u00E2\u0088\u009211mbar) and no significant aging was observed. The data was integrated over a 7 meV region around the Fermi en- ergy and transformed into k-space without symmetrization. The size of the BZ was then determined from a wider-angle-scan which included data across more than one complete BZ at lower resolution and statistics, allowing the high-resolution image to be cropped and reflected about (\u00E2\u0088\u0092pi, 0)\u00E2\u0080\u0093(pi, 0) and (0,\u00E2\u0088\u0092pi)\u00E2\u0080\u0093(0, pi) to reproduce the entire BZ as shown in panel (e). The ana- lyzer entrance slit was oriented horizontally with respect to the final image, giving greater angular resolution in that direction. One can immediately see an apparent doubling of many features, as well as \u00CE\u00B3 bands with similar topology to both \u00CE\u00B3b and \u00CE\u00B3s [with \u00CE\u00B3s showing clearly near (0, pi) and \u00CE\u00B3b near (pi, 0) owing to varying matrix element dependence]. Much of this FS could be understood using the existing model of photoemission from both bulk and a reconstructed surface layer. However, contrary to this explanation, the folded \u00CE\u00B2\u00E2\u0080\u00B2 is clearly doubled as well, with both a \u00CE\u00B2\u00E2\u0080\u00B2b and \u00CE\u00B2 \u00E2\u0080\u00B2 s visible, as highlighted with a rectangle in panel (d). We will show in Fig. 4.4 (and additionally in Fig. 4.8) that in fact all bands are doubled \u00E2\u0080\u0093 both unfolded and folded \u00E2\u0080\u0093 which cannot be explained by the simple model of a recon- structed surface. These observed doublings are indicated in a phenomeno- logical FS plot in panel (f), where we mark the position of all features visible to ARPES, based on all data collected. In Fig. 4.4 we confirm the doubling of features with momentum distri- bution curves (MDCs) obtained by integrating a 10 meV region just above EF : panels (b\u00E2\u0080\u0093e) show the doubling of \u00CE\u00B2, \u00CE\u00B2 \u00E2\u0080\u00B2, \u00CE\u00B1, and \u00CE\u00B1\u00E2\u0080\u00B2, respectively, ob- served at the k-locations indicated in the alignment FS of (a). Note that the splitting between \u00CE\u00B1\u00E2\u0080\u00B2s and \u00CE\u00B1\u00E2\u0080\u00B2b is more clearly resolved in the second BZ [Fig. 4.4(e)], perhaps because of matrix element effects and/or the higher ef- fective angular resolution in the second BZ. By performing a least-squares fit to two Lorentzian peaks with a second-order polynomial background (black curves), we can estimate the observed splittings as 0.057, 0.065, 0.027, and 0.029\u00C2\u00B10.003 pia for \u00CE\u00B2, \u00CE\u00B2\u00E2\u0080\u00B2, \u00CE\u00B1, and \u00CE\u00B1\u00E2\u0080\u00B2, respectively. Although it is difficult to 71 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands compare these values directly, because of the slight difference in angles and k-space locations between them, we note that the splittings between \u00CE\u00B2s/\u00CE\u00B2b and \u00CE\u00B2\u00E2\u0080\u00B2s/\u00CE\u00B2\u00E2\u0080\u00B2b are very close and approximately double those between \u00CE\u00B1s/\u00CE\u00B1b and \u00CE\u00B1\u00E2\u0080\u00B2s/\u00CE\u00B1\u00E2\u0080\u00B2b, which are almost identical. These doublings \u00E2\u0080\u0093 particularly the existence of \u00CE\u00B1\u00E2\u0080\u00B2b, \u00CE\u00B2 \u00E2\u0080\u00B2 b, and \u00CE\u00B3 \u00E2\u0080\u00B2 b (demon- strated later) \u00E2\u0080\u0093 do not fit into the previous model of a bulk and reconstructed surface. In the rest of this section we will explore the dependence of these features on the absence/presence of the 3K phase in \u00C2\u00A74.1.2, their degrada- tion over time in \u00C2\u00A74.1.3, their polarization dependence in \u00C2\u00A74.1.4, and we will compare them to similar features in the Bi-cuprates in \u00C2\u00A74.1.5. 4.1.2 Exploring the 3K phase The high-quality Sr2RuO4 crystals used in this work were grown using the floating zone method [73]. This method involves melting one end of a feed rod manufactured from RuO2 and SrCO3 powders in an image furnace, seeding this molten zone with an existing Sr2RuO4 single crystal, and then passing it slowly through the furnace while rotating the seed crystal and feed rods in opposite directions. Thus, a narrow molten zone travels along the feed rod, where it solidifies as a single crystal with the same orientation as the seed. The quality of the resulting crystal relies heavily on a nearly endless variety of growth parameters, including stoichiometry of the feed rod; growth atmosphere temperature, pressure, and composition; rotation and draw rates of both seed and feed rods; as well as melt temperature and size \u00E2\u0080\u0093 in addition to endless possible post-processing operations. Crystal growth is its own field of research. Of the observed defects that may be present in Sr2RuO4 crystals, the one that has arguably the greatest effect on its electronic structure is the so-called 3K phase, which is caused by small (on the order of microns) Ru metal inclusions embedded in the final crystal. In order to compensate for the relative volatility of Ru, higher-than-stoichiometric quantities are used in the growth rod. This can result in a gradient of Ru concentration in the melt, which produces pure Sr2RuO4 near the outer edges of the final rod 72 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands Cleaving plane Surface polished Gr ow th dir ec tio n c-axis Figure 4.5: Orientation of the Sr2RuO4 single crystal that was polished to produces pieces with and without the presence of the 3K phase. The original growth rod was cut with a saw perpendicular to the growth direction and cleaved along the c-axis to mark that direction before polishing. 73 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands and a eutectic mix of Sr2RuO4 and Ru metal lamellae in the centre. These lamellae are approximately 1\u00C2\u00B5m thick and on the order of 10\u00C2\u00B5m in plane. The lamellae also tend to order into stripes, which can have long-range order over distances of millimetres, although the orientation with respect to the crystal axis of the underlying Sr2RuO4 is not unique and is known to vary within single-crystal samples [76]. In Sr2RuO4 samples with a high concentration of Ru metal18 lamellae, the superconducting temperature is enhanced from a sharp transition at 1.5 K to a broad transition up to 3.5 K, although the method of this enhancement remains unresolved [72, 76\u00E2\u0080\u009378]. In order to ensure that the features observed in \u00C2\u00A74.1.1 were not somehow reliant on the presence or absence of the 3K phase, it was necessary to measure samples where the presence and quantity of 3K phase was known. It is possible to directly observe the 3K phase on polished samples using polarized light microscopy. In order to facilitate performing measurements on samples that would be identical other than the inclusion of 3K phase, we polished a cleaved growth rod end-on (on a face perpendicular to the growth direction, as shown in Fig. 4.5). The sample was glued to a polishing jig with crystal bond, which allowed the pressure used to be fine-tuned while holding the sample precisely perpendicular to the lapping surface. After polishing with progressively finer alumina grit suspended in water (down to 0.3\u00C2\u00B5m) the 3K phase became visible, as shown in Fig. 4.6. As expected, the Ru metal inclusions were concentrated toward the centre of the growth rod. This allowed the sample to be split into pieces and measured both with and without the 3K phase, as marked in Fig. 4.6. In Fig. 4.7 we present ARPES FS data taken on these pieces. These data sets were taken in under 4.5 hours from the cleave in vacuum of 6 \u00C3\u0097 10\u00E2\u0088\u009211mbar, during which time no significant aging was observed. These FS images were generated by integrating a 4 meV region near the Fermi energy and transforming this data into k-space. In these images we can clearly see the presence of the mysterious doubling in both the pure and 3K phase samples, ruling out the possibility that the 3K phase is responsible. In fact, it is difficult (or perhaps impossible, although this has not been fully 18Tc Ru = 0.49 K. 74 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands 100\u00CE\u00BCm 10\u00CE\u00BCm (b) 1mm 100\u00CE\u00BCm (a) Figure 4.6: Polarized light microscopy image of the 3K phase (bright spots) in Sr2RuO4, showing (a) the entire polished face and (b) a zoom-in on the region marked by a pink circle in (a). The locations of samples cleaved for measurements with and without 3K phase are marked in white on panel (a). Colour in these images is due to the use of a birefringent wave plate to increase contrast; under normal light the crystal is black. 75 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands (a) (b) (0,0) (0,0) (\u00CF\u0080,\u00CF\u0080) (\u00CF\u0080,\u00CF\u0080) kx ky kx ky 0 Intensity max Figure 4.7: FS map of Sr2RuO4 as measured by ARPES on fresh samples both (a) without and (b) with the presence of the 3K phase. The discon- tinuity in (b) was caused by a discontinuity in the \u00CF\u0086 (\u00E2\u0088\u00BC ky) motion of the manipulator, not the sample; it was not reproduced in a smaller subsequent sweep through that region alone. 76 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands explored) to tell samples with or without the 3K phase apart using ARPES (at least at the quantity of ruthenium metal inclusions tested here). Despite the observed null-effect of the 3K phase on the existence of these features, all ARPES data presented in this thesis (aside from Fig. 4.7) was taken on samples inspected and known not to contain any 3K phase, lest it have some other unnoticed effect. 4.1.3 Time-dependent surface degradation and dispersion analysis Thus far, with a doubling observed for all \u00CE\u00B1 and \u00CE\u00B2 bands, given no de- pendence on the 3K phase, and in light of the bulk magnetism of many Ru-oxides, one might be drawn to the possibility of Rashba-type effects [16] or even surface ferromagnetism [59]. If any of these were the case, one would expect that \u00E2\u0080\u0093 under surface degradation \u00E2\u0080\u0093 the surface states\u00E2\u0080\u0099 splitting would shrink and/or the intensity would decay uniformly over time for all of them, eventually revealing the underlying bulk electronic structure. In Figs. 4.8 and 4.9 we show the results of such a time-dependent experi- ment performed over 16 hours on the band dispersion along the (0, 0)\u00E2\u0088\u0092(pi, pi) direction, where all features and their folded replicas are visible simultane- ously. This sample was maintained at 9.5 K while raising the base pressure to about 8.5\u00C3\u009710\u00E2\u0088\u009211 mbar. The time evolution of MDCs over the first 14 hours, integrated over 4 meV just above EF in 100-minute intervals, is presented in Fig. 4.8(a). Panel (b) shows the location of the cut on the alignment FS, while (c,e,g) and (d,f,g) present the MDC, band map, and band map zoomed to the near-EF region for the first and last 100-minute intervals. Fig. 4.9 shows the entire band map for each of these intervals, as well as an additional one showing the near-saturation of further degradation. While some features remain at a similar level of intensity during the degradation, others age and have their intensity suppressed, without any overall change in slope or shape. Particularly, comparing the panels representing the first and last 100-minute intervals in both Fig. 4.8 and Fig. 4.9, we can infer that (i) all bands and their folded replicas are doubled for a total of 12 bands 77 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands (as labelled), including the \u00CE\u00B3 bands not previously demonstrated in Fig. 4.4 and (ii) based on the observed difference in degradation rate, the electronic states do not all originate from the very same surface layer and can instead be classified as surface- (fast degradation) or bulk-like (slow degradation). 78 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands (0,0) (\u00CF\u0080,\u00CF\u0080)(a) Int en sit y ( a.u .) En erg y ( me V) 0 50 100 150 200 En erg y ( me V) 0.3 0.4 0.5 0.6 0.7 Int en sit y ( a.u .) kx = ky (\u00CF\u0080/a)~ 0.3 0.4 0.5 0.6 0.7 kx = ky (\u00CF\u0080/a)~ \u00CE\u00B1' s + \u00CE\u00B1' b \u00CE\u00B3' s + \u00CE\u00B3' b \u00CE\u00B2' b \u00CE\u00B2' s \u00CE\u00B2 s \u00CE\u00B2 b \u00CE\u00B3 s + \u00CE\u00B3 b \u00CE\u00B1 s + \u00CE\u00B1 b \u00CE\u00B1's \u00CE\u00B3's \u00CE\u00B2'b \u00CE\u00B2's \u00CE\u00B2s \u00CE\u00B2b \u00CE\u00B3s \u00CE\u00B1s (e) (c) (g) (f) (d) 0 10 20 30 (h) \u00CE\u00B1' b \u00CE\u00B3' b \u00CE\u00B2 ' b \u00CE\u00B2 b \u00CE\u00B3 b \u00CE\u00B1 b \u00CE\u00B1'b \u00CE\u00B3'b \u00CE\u00B2'b \u00CE\u00B2b \u00CE\u00B3b\u00CE\u00B1b (b) 0 1 2 0 1 2 3 4 5 0 1 2 3 4 5 40-140 min 140-240 min 340-440 min 440-540 min 640-740 min 740-840 min mi n ma x Figure 4.8: (a) time evolution of a 45\u00E2\u0097\u00A6 MDC cut at EF ; its location is marked on the alignment FS in (b). (c,e,g) plot of MDC at EF , band map of features on a large energy range, and zoom-in on the Fermi crossings, respectively, for the first 100 minutes after cleave; (d,f,h) the same for the final 100 minutes of acquisition. In (g,h), MDC maxima (black circles) and MDC-dispersion fits (blue \u00E2\u0080\u0093 bulk; red \u00E2\u0080\u0093 surface) are also shown. 79 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands En er gy (m eV ) 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 0.3 0.4 0.5 0.6 0.7 kx = ky (\u00CF\u0080/a)~ (a) m in m ax (b) (c) (d) (e) (f) (g) 0.3 0.4 0.5 0.6 0.7 kx = ky (\u00CF\u0080/a)~ (h) (i) (j) (k) (l) (m) (n) Time since cleave 40-140 min 140-240 min 340-440 min 440-540 min 640-740 min 740-840 min 840-940 min \u00CE\u00B1'b \u00CE\u00B3'b \u00CE\u00B2b \u00CE\u00B3b \u00CE\u00B1b\u00CE\u00B2'b\u00CE\u00B1's \u00CE\u00B3's \u00CE\u00B2's \u00CE\u00B2s \u00CE\u00B3s \u00CE\u00B1s Figure 4.9: Band maps for all data in Fig. 4.8 both with guides to the eye (a\u00E2\u0080\u0093g) and without (h\u00E2\u0080\u0093n). Shown in (g,n) is an additional band map, which was omitted from Fig. 4.8 for clarity, as it is almost identical to (f,m). 80 4.1 . B u lk b an d stru ctu re, F erm i su rface, an d \u00E2\u0080\u009Cex tra\u00E2\u0080\u009D b an d s Band nLDA+SOe v LDA+SO F (eV A\u00CC\u008A) n ARPES e v ARPES F (eV A\u00CC\u008A) v LDA+SO F /v ARPES F n dHvA e \u00CE\u00B1s, \u00CE\u00B1 \u00E2\u0080\u00B2 s 1.668 \u00C2\u00B1 0.007 2.30 \u00C2\u00B1 0.01 1.721 \u00C2\u00B1 0.040 0.56 \u00C2\u00B1 0.05 4.1 \u00C2\u00B1 0.4 \u00E2\u0080\u0093 \u00CE\u00B2s, \u00CE\u00B2 \u00E2\u0080\u00B2 s 0.864 \u00C2\u00B1 0.006 2.14 \u00C2\u00B1 0.01 0.757 \u00C2\u00B1 0.020 0.59 \u00C2\u00B1 0.05 3.6 \u00C2\u00B1 0.3 \u00E2\u0080\u0093 \u00CE\u00B3s, \u00CE\u00B3 \u00E2\u0080\u00B2 s 1.340 \u00C2\u00B1 0.011 2.18 \u00C2\u00B1 0.02 1.396 \u00C2\u00B1 0.060 0.42 \u00C2\u00B1 0.05 5.2 \u00C2\u00B1 0.6 \u00E2\u0080\u0093 Total 3.872 \u00C2\u00B1 0.014 \u00E2\u0080\u0093 3.874 \u00C2\u00B1 0.120 \u00E2\u0080\u0093 \u00E2\u0080\u0093 \u00E2\u0080\u0093 \u00CE\u00B1b, \u00CE\u00B1 \u00E2\u0080\u00B2 b 1.740 \u00C2\u00B1 0.003 2.14 \u00C2\u00B1 0.02 1.760 \u00C2\u00B1 0.040 1.30 \u00C2\u00B1 0.60 1.6 \u00C2\u00B1 0.8 1.781 \u00CE\u00B2b, \u00CE\u00B2 \u00E2\u0080\u00B2 b 0.967 \u00C2\u00B1 0.022 2.60 \u00C2\u00B1 0.25 0.903 \u00C2\u00B1 0.020 0.80 \u00C2\u00B1 0.10 3.3 \u00C2\u00B1 0.5 0.921 \u00CE\u00B3b, \u00CE\u00B3 \u00E2\u0080\u00B2 b 1.252 \u00C2\u00B1 0.014 2.21 \u00C2\u00B1 0.07 1.280 \u00C2\u00B1 0.060 0.76 \u00C2\u00B1 0.10 2.9 \u00C2\u00B1 0.4 1.346 Total 3.959 \u00C2\u00B1 0.026 \u00E2\u0080\u0093 3.943 \u00C2\u00B1 0.120 \u00E2\u0080\u0093 \u00E2\u0080\u0093 4.048 Table 4.1: A comparison between Fermi surface volumes, velocities, and their renormalizations as measured by ARPES, LDA+SO, and dHvA for Sr2RuO4. Carrier concentrations counting electrons, ne=2\u00C3\u0097AFS/ABZ (with 2 accounting for the spin degeneracy), and Fermi velocities along the (0, 0) \u00E2\u0086\u0092 (pi, pi) direction, vF , as determined by our ARPES and LDA+SO slab calculations for surface- and bulk-like electronic structure. ARPES-FS volume estimates are from the phenomenological FS in Fig. 4.3; LDA+SO results were obtained for the 24 meV shifted chemical potential to match the average electron counting of surface and sub-surface FSs as determined by ARPES. The dHvA results from Ref. 55, representative of the bulk Luttinger\u00E2\u0080\u0099s counting, are also shown. 81 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands These observations can be made more quantitative by doing an MDC- dispersion fit analysis, which allows us to distinguish between surface and bulk-like bands based on their remarkably different Fermi velocity vF (see Tab. 4.1). While the fresh data in Fig. 4.8(g) are dominated by the surface \u00CE\u00B1s/\u00CE\u00B1 \u00E2\u0080\u00B2 s, \u00CE\u00B2s/\u00CE\u00B2 \u00E2\u0080\u00B2 s, and \u00CE\u00B3s/\u00CE\u00B3 \u00E2\u0080\u00B2 s, with the addition of the well-separated \u00CE\u00B2b/\u00CE\u00B2 \u00E2\u0080\u00B2 b pair, the aged data of Fig. 4.8(h) show only the much steeper bulk-like \u00CE\u00B1b/\u00CE\u00B1 \u00E2\u0080\u00B2 b, \u00CE\u00B2b/\u00CE\u00B2 \u00E2\u0080\u00B2 b, and \u00CE\u00B3b/\u00CE\u00B3 \u00E2\u0080\u00B2 b (\u00CE\u00B1 \u00E2\u0080\u00B2 b, although visible, could not be fit and is shown with a dashed line as a guide to the eye). Since all s states are suppressed evenly and independently from the b states, we must conclude that no electronic interaction \u00E2\u0080\u0093 ferromagnetism or Rashba coupling \u00E2\u0080\u0093 is responsible for the apparent band doubling. Similarly, we must also rule out possible surface patches with different electronic struc- ture or effective doping. The most natural explanation would be that the top surface layer is heavily modulated by the rotation of the RuO6 octahedra and is responsible for all the s states, while the sub-surface layer is also mod- ulated in a similar yet weaker fashion and is responsible for the b states; this lesser modulation would be enough to break the symmetry of the BZ and cause a visible folding but not enough to strongly modify FS topologies and volumes. Measured FS volumes [Fig. 4.3(c,d)] support this model: as shown in Tab. 4.1, while the s FS electron counting is somewhat reduced (nse'3.87), the b FS electron counting is much closer to the dHvA results, adding up to nbe'3.94, i.e., almost four electrons, as expected in the Ru t2g orbitals (this still seems to suggest the possibility that these near-surface layers are slightly hole-doped, possibly owing to the cleaving-induced Sr-vacancies revealed by scanning tunnelling microscopy [79]). Finally photoemission from a second layer, also folded but otherwise similar to the bulk, could even explain the so-called shadow band that remained after high-temperature cleaving had suppressed the surface states, in the original work uniting the dHvA and ARPES pictures of the FS of Sr2RuO4 [56]. 82 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands Horizontal (p-polarization) 45\u00C2\u00B0 Vertical (s-polarization) Polarization (a) (b) (c) Approximate hours since cleave 1.5 through 4.5 5 through 8 8.5 through 11.5 (0,0) (\u00CF\u0080,\u00CF\u0080) (0,0) (\u00CF\u0080,\u00CF\u0080) (0,0) (\u00CF\u0080,\u00CF\u0080) 0 10 In te ns ity (a .u .) kx (\u00CF\u0080/a) kx (\u00CF\u0080/a) 0 1 (d) (e) ky = \u00CF\u0080/a ky = 0~ ~ \u00CE\u00B2's\u00CE\u00B2'b \u00CE\u00B1s \u00CE\u00B3s \u00CE\u00B3s\u00CE\u00B3b \u00CE\u00B3b \u00CE\u00B1b \u00CE\u00B1's\u00CE\u00B1'b \u00CE\u00B2s\u00CE\u00B2b p 45\u00C2\u00B0 s p 45\u00C2\u00B0 s Figure 4.10: Polarization dependence of the ARPES FS of Sr2RuO4 us- ing light in (a) p-polarization, (b) intermediate 45\u00E2\u0097\u00A6 polarization, and (c) s- polarization. Panel (d) shows the high-symmetry cut from these FSs along ky = pi/a, while panel (e) shows the high-symmetry cut along ky = 0, with locations where features cross these cuts marked. Acquisition times after the cleave are noted, with the sample held between 5 and 6 K in 6\u00C3\u0097 10\u00E2\u0088\u009211 mbar vacuum. 83 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands kx (\u00CF\u0080/a) In te ns ity (a .u .) In te ns ity (a .u .) (b)(a) (2\u00CF\u0080,0)(\u00CF\u0080,0) \u00CE\u00B1's \u00CE\u00B1'b 1.55 1.60 1.65 1.700 \u00CE\u00B1' s + \u00CE\u00B1' b \u00CE\u00B3' s + \u00CE\u00B3' b \u00CE\u00B2' b \u00CE\u00B2 ' s \u00CE\u00B2 s \u00CE\u00B2 b \u00CE\u00B3 s + \u00CE\u00B3 b \u00CE\u00B1 s + \u00CE\u00B1 b 0.3 0.4 0.5 0.6 0.7 0 kx = ky (\u00CF\u0080/a)~ p/0.5h 45\u00C2\u00B0/1.0h s/1.5h (d)(c) (0,0) (\u00CF\u0080,\u00CF\u0080) p/1.5 - 2h 45\u00C2\u00B0/4 - 4.5h s/5.5 - 6h Figure 4.11: Polarization dependence of selected FS features of Sr2RuO4; (a) the folded \u00CE\u00B1\u00E2\u0080\u00B2 pocket in the second BZ, as marked in the alignment FS (b), as well as (c) a cut slightly offset from the (0, 0) \u00E2\u0086\u0092 (pi, pi) high- symmetry direction, which crosses all bands, as marked in the alignment FS (d). Acquisition time for each cut in (a) was \u00E2\u0088\u00BC30 minutes, with the time since the cleave noted; the sample was held between 9 \u00E2\u0080\u0093 18 K in 8 \u00C3\u0097 10\u00E2\u0088\u009211 mbar vacuum . For (c) acquisition time was on the order of minutes owing to higher intensity, and the sample was held at 5.6 K in 4\u00C3\u009710\u00E2\u0088\u009211 mbar vacuum. 84 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands 4.1.4 Polarization dependence Given the persistence of the b states compared with the s states in the time- dependent surface degradation measurements, it is certain that the b states originate from below the surface. However, it is difficult to definitively de- termine the source of the folding in the b states. There are two possible scenarios (given that the bulk is known not to reconstruct): either a recon- struction exists in the sub-surface layer, or bulk photoemission is scattering off a reconstructed surface. In the latter case the surface must retain enough of its structural modulation to continue to perform this scattering function, despite its being degraded enough to completely suppress photoemission. In the next two sections, first here through polarization dependence and then through a comparison with the Bi-cuprates, we explore the feasibility of the bulk-scattering scenario. It is possible to measure the symmetry of the underlying wave-functions for various features by examining their ARPES intensity as a function of light polarization. Because of the monochromator between the He lamp and chamber, all light used in our lab at UBC is linearly polarized. This po- larization can be changed by rotating the entire lamp and monochromator. Any detected electrons cannot have an odd final state with respect to the plane defined by the analyzer slit and the incoming photon. For this reason, using photons with p-polarization (electric field parallel, and hence even, to incidence/emission plane) will allow the detection of even (with respect to the incidence/emission plane) initial states; similarly, using s-polarization (from senkrecht, German for perpendicular) will allow the detection of odd initial states. The effects are the most drastic along high-symmetry direc- tions, where the initial state is likely to be of pure even or odd character. In principle, from the LDA calculations, it is possible to determine not only the band structure but also the orbital character and hence polarization dependence that should be measured by ARPES in any geometry. Unfor- tunately, for this work on Sr2RuO4, because of orbital character mixing between bands as well as uncertainty in the kz position of the measure- ment, I was unable to make such rigorous advances on the nature of these 85 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands additional bands using their polarization dependence. Additionally, the pro- cess of changing polarization is time consuming because the rotation must be slow in order to minimize the gyroscopic forces on the bearings in the turbo pumps. This time delay means that polarization-dependent measure- ments are often, effectively, also time-dependent measurements. Depending on sample temperature and chamber pressure, this may also have a large effect, owing to sample degradation. Nevertheless, from the polarization data presented in this section it becomes clear that the polarization depen- dence of both halves of many splittings are independent from each other. Likewise, the polarization dependence of the folded (primed) bands is of- ten different from that of the primary (unprimed) bands. Their underlying wave-functions must therefore be of different character, either because they originate from either the surface or sub-surface (which may have different kz locations) or because they are originating from k points on opposite sides of a mirror-plane (and are thus an initial state effect, rather than scattering through a reconstructed surface layer). In Fig. 4.10 we present high-statistic FS measurements, integrated in a 7 meV region near the Fermi energy, for both p- and s-polarization as well as an intermediate polarization at 45\u00E2\u0097\u00A6. Additionally, we highlight momentum distribution curve (MDC) cuts along the two high-symmetry locations ac- cessed during this FS scan. Unfortunately, despite good vacuum and low temperature, some slight sample degradation during this time is visible as a general blurring of features in the s-polarization data. However, some information on the polarization dependence is still clear. In panel (d) we see that there is not strong polarization dependence in the (0, pi) \u00E2\u0086\u0092 (pi, pi) cut, although it appears that the b half of the splitting may be stronger in the \u00CE\u00B1 band at p-polarization, with both equally strong in s-polarization. In panel (e) we note strong polarization dependence, including the following characteristics: overall enhancement of both halves of the folded \u00CE\u00B1\u00E2\u0080\u00B2 band in s-polarization, while in p-polarization \u00CE\u00B1\u00E2\u0080\u00B2b appears stronger than \u00CE\u00B1 \u00E2\u0080\u00B2 s (which may not appear at all in this polarization); overall enhancement of both halves of the \u00CE\u00B2 band in p-polarization, while in s-polarization only \u00CE\u00B2s ap- pears strongly; an increase in \u00CE\u00B3s while \u00CE\u00B3b appears to maintain equal inten- 86 4.1. Bulk band structure, Fermi surface, and \u00E2\u0080\u009Cextra\u00E2\u0080\u009D bands sity. In both these sets we see different polarization dependence for different halves of each splitting. In Fig. 4.11 we present high-statistic MDC cuts integrated 6 meV near the Fermi energy, on which only a fast FS was taken for alignment pur- poses. This method allowed for quicker measurements with less opportunity for sample degradation. Unfortunately, for panel (a), helium flow problems resulted in warmer sample temperatures and higher pressures (see caption), and longer acquisition times were required owing to the relatively low in- tensity of the bands; therefore some sample degradation cannot be ruled out. Nevertheless in Fig. 4.11 panel (a) we see [similar to panel (e) from Fig. 4.10] that \u00CE\u00B1\u00E2\u0080\u00B2b appears stronger than \u00CE\u00B1 \u00E2\u0080\u00B2 s under p-polarization, but unlike in Fig. 4.10 we do not note an overall enhancement of both halves of the splitting in s-polarization. This could be due to either sample degradation or the differing location in k-space. In Fig. 4.11 panel (c) we cut across all bands and see a large polarization dependence. Most notable is that in s- polarization there is symmetry between the primary bands (unprimed) and their folded replicas (primed) and that, overall, the intensity of the folded replicas are strongly reduced under p-polarization, while the primary (un- primed) bands are mostly enhanced. However, there are exceptions to this trend \u00E2\u0080\u0093 notably \u00CE\u00B2b and \u00CE\u00B2 \u00E2\u0080\u00B2 b, which, while they still show a great difference between s- and p-polarization, show the opposite enhancement/reduction between folded and primary. Also, while an overall effect is clear, the over- lap of b and s in the \u00CE\u00B1 and \u00CE\u00B3 bands makes it difficult to discern which components of these peaks might be enhanced or reduced. As we conclude the examination of the polarization data, two things are evident. First, the polarization dependence of each half of the splittings (b vs. s) can be different for some bands. This supports the hypothesis that they may originate from the surface and sub-surface layer, as the underlying wave-functions in this case would have slightly different structure, especially with respect to kz (which is poorly defined for a single layer). Additionally the polarization dependence of both halves of each folding (primed vs. un- primed) also shows different polarization dependence in many places. This suggests the bands must be of differing origin \u00E2\u0080\u0093 i.e., not simply replicas 87 4.2. LDA calculations and structural distortions scattered through a reconstructed surface layer. 4.1.5 A comparison with folded features in the cuprates While the results of surface degradation and polarization dependence are strong evidence against the diffraction of bulk photoelectrons by the sur- face layer, it is nevertheless important to explore the parallel between the additional features observed in Sr2RuO4 and the diffraction proposed at one time to occur in the Bi-cuprates. There, two sets of unexpected fea- tures are detected in addition to the single band of the ideal CuO2 square plane [80]: these have been referred to in the literature as \u00E2\u0080\u009Cshadow bands\u00E2\u0080\u009D and \u00E2\u0080\u009Cumklapp bands\u00E2\u0080\u009D [33]. It is the umklapp bands that were initially suspected to be due to the diffraction of the CuO2-plane photoelectrons travelling through the modulated BiO surface layer, since they appeared to exhibit multiple positive and negative high-order replicas, consistent with a diffraction scenario. However, this was later shown not to be the case: all crystal planes present the same incommensurate modulations as the BiO planes, and the umklapp bands therefore stem from initial states intrinsic to the modulated CuO2 plane and not from extrinsic photoelectron diffrac- tion [81\u00E2\u0080\u009384]. Additionally, it is not with the umklapp but with the shadow bands that the sub-surface folded bands in Sr2RuO4 share their form of re- construction. Importantly, contrary to the umklapp bands, these shadow bands in Bi-cuprates were deemed inconsistent with a photoelectron diffrac- tion scenario; by analogy, the same would hold for Sr2RuO4. While initially attributed to antiferromagnetic fluctuations giving rise to a ( \u00E2\u0088\u009A 2\u00C3\u0097\u00E2\u0088\u009A2)R45\u00E2\u0097\u00A6 larger unit cell and a correspondingly folded Brillouin zone [80], the shadow bands in Bi-cuprates were shown to arise from the presence of two struc- turally inequivalent Cu atoms per orthorhombic unit cell [81\u00E2\u0080\u009384]. 4.2 LDA calculations and structural distortions In order to explore the possibility of the additional features being due to a structural distortion in both the surface and sub-surface layers, a variety 88 4.2. LDA calculations and structural distortions of LDA calculations were performed both with and without SO coupling using the linearized augmented-plane-wave method in the WIEN2k pack- age [85]. As discussed earlier, it is known that oxygen octahedra in the surface layer rotate about the c-axis, which causes a ( \u00E2\u0088\u009A 2\u00C3\u0097\u00E2\u0088\u009A2)R45\u00E2\u0097\u00A6 recon- struction of the primitive cell as seen by surface-sensitive experiments such as ARPES [6, 56, 58, 68, 69] and LEED [59\u00E2\u0080\u009361, 70, 71]. Additionally, it is likely that there may be dimensional distortions along the c-axis in (at least) the topmost layer. It is also known that the bulk crystal does not undergo such a reconstruction, as this is not detected in any x-ray [63] or neutron diffraction experiments (on either single-crystals [64] or powders [65\u00E2\u0080\u009367]), nor is the resultant reconstruction of the BZ seen in dHvA measurements [14, 54, 55]. However, the precise nature of the sub-surface layers is not known, and is difficult to probe directly, although ARPES may come close in that it can be sensitive to a superposition of the electronic structure of surface, sub-surface, and bulk-like layers. If the sub-surface layer were to undergo a similar rotation to that of the surface, one would expect that photoemission from that layer would display folded replicas in the same manner \u00E2\u0080\u0093 thus explaining all visible features. However, as shown by bulk calculations, simply modulating the same struc- tural distortion cannot entirely explain the features as observed. Namely, the change in volume of the \u00CE\u00B2 splitting (distance between \u00CE\u00B2s and \u00CE\u00B2b seen, for example, in Fig. 4.3 or 4.4) is not reproduced. It is possible that the surface itself, owing to the disruption of the periodic potential seen by electrons, either alone or in combination with a structural distortion, may cause these discrepancies. For this reason slab calculations are necessary in order to model the existence of the vacuum, surface, and sub-surface layers and any possible interplay between them. In such calculations the unit cell is ex- panded along the c-axis to include multiple, possibly different, atomic layers and a layer of vacuum is also included. In this manner one can simulate both surfaces with a bulk-like layer, at the cost of finite size effects (for the bulk-like layers) and computational time because of to the increased number of atoms (which scales non-linearly). In order to decide on the parameters to be used for the computationally expensive slab calculations, in particu- 89 4.2. LDA calculations and structural distortions lar the unknown sub-surface layer, a series of comparatively fast bulk LDA calculations were performed, both with and without SO coupling. I will first discuss the results of these preliminary calculations in \u00C2\u00A74.2.1, before describing the slab choice (\u00C2\u00A74.2.2) and finally the results (\u00C2\u00A74.2.3) of those calculations. 4.2.1 Preliminary bulk calculations In order to determine the structure to use for slab calculations, in particular the sub-surface layer, LDA bulk band structure calculations were performed for varying rotations of the oxygen octahedra as well as a possible buckling mode. Additionally, the effects of adding Coulomb repulsion U were exam- ined. This allowed the exploration of a wide parameter space without the computational cost of many slab-based calculations. As Ru oxygen octahedra rotation is known to occur in the surface layer, a range of crystal structures with different rotations (unit cell dimensions kept constant) were studied with bulk LDA+SO calculations. The structure can be visualized in Fig. 4.15, keeping in mind that it is a bulk calculation with all layers rotated. In Fig. 4.12(a) we see that, for rotations under \u00E2\u0088\u00BC3\u00E2\u0097\u00A6, no significant change in Fermi surface volume (average 3%) is observed, with, in particular, no change in the topology of the \u00CE\u00B3 Fermi surface. [Correspond- ingly the \u00CE\u00B3 band Van Hove singularity at the M point remains above the Fermi level in Fig. 4.12(b).] For rotations above \u00E2\u0088\u00BC5.5\u00E2\u0097\u00A6 the topology of the \u00CE\u00B3 band changes to match that of the surface, and \u00E2\u0088\u00BC7.5\u00E2\u0097\u00A6 introduces also the \u00CE\u00B4 band at \u00CE\u0093/X. Comparing these calculations to experiments [for example, Fig. 4.3(f), and values in Tab. 4.1] then suggests that any sub-surface rota- tion should be .3\u00E2\u0097\u00A6, while the surface rotation should be &5\u00E2\u0097\u00A6. (As discussed below, the \u00CE\u00B4 crossing moves above the Fermi level with the inclusion of U , so \u00E2\u0088\u00BC7.5\u00E2\u0097\u00A6 is not a hard upper limit.) While tilting of the oxygen octahedra occurs in the related family of compounds with Ca substitution, Ca2\u00E2\u0088\u0092xSrxRuO4, it is known from the sym- metry of the Sr2RuO4 surface that no tilting of the octahedra occurs [60, 71], as moving the in-plane (with Ru) O atoms out of plane would change the 90 4.2. LDA calculations and structural distortions \u00CE\u0093 ZM M X \u00E2\u0080\u009CX1\u00E2\u0080\u009D En er gy (e V ) EF 1 2 -1 -2 \u00CE\u0093 \u00E2\u0080\u009CX1\u00E2\u0080\u009D \u00CE\u0093 (b) (a) 0.0\u00C2\u00B0 3.0\u00C2\u00B0 5.5\u00C2\u00B0 7.5\u00C2\u00B0 Figure 4.12: LDA+SO bulk band structure of Sr2RuO4, with incremental octahedra rotations. (a) Fermi surface and (b) high-symmetry cut. Primary bands are shown solid, with their folded replicas shown dotted in (a). 91 4.2. LDA calculations and structural distortions (d) (b) LDA wo SO, .005\u00C2\u00B0 rotation LDA wo SO, .005\u00C2\u00B0 rotation + .126 \u00C3\u0085 buckling LDA wo SO, 6.5\u00C2\u00B0 rotation LDA wo SO, 6.5\u00C2\u00B0 rotation + .126 \u00C3\u0085 buckling En er gy (e V ) -2 -1 EF 1 2 En er gy (e V ) -2 -1 EF 1 2 \u00CE\u0093 X1 M \u00CE\u0093 \u00CE\u0093 X1 M \u00CE\u0093 \u00CE\u0093 ZM X X1 (a) (c) Figure 4.13: Bulk band structure of Sr2RuO4, with buckling in oxygen octa- hedra. (a) high-symmetry cut and (b) Fermi surface from bulk LDA (with- out SO) band structure calculations, both with (red) and without (black) elongation of alternate oxygen octahedra. (c,d) show the same but with ad- ditional rotation of the oxygen octahedra. Primary bands are shown solid, with their folded replicas shown dotted in (b,d). 92 4.2. LDA calculations and structural distortions 6.5\u00C2\u00B0 rotation, LSDA 6.5\u00C2\u00B0 rotation, LSDA + U 6.5\u00C2\u00B0 rotation, LSDA + SO 6.5\u00C2\u00B0 rotation, LSDA + SO + U En er gy (e V ) -2 -1 EF 1 2 \u00CE\u0093 X1 M Z\u00CE\u0093 En er gy (e V ) -2 -1 EF 1 2 \u00CE\u0093 X1 MZ \u00CE\u0093 (a) (b) Figure 4.14: Bulk band structure of Sr2RuO4, with electronic interaction U. High-symmetry cut from bulk LSDA band structure calculations showing the effect of adding an effective electronic interaction of 1 eV (red), compared to 0 eV (black) both without (a) and with (b) SO. symmetry group. However, a c-axis distortion of the apical O, either uni- formly or as a buckling mode, is not similarly restricted. To explore this possibility, calculations were performed where the Ru oxygen bond lengths were distorted along the c-axis by \u00C2\u00B10.063 A\u00CC\u008A. This is a rather large num- ber, as the uniform (not as a buckling mode) c-axis elongation of this bond length observed to occur at the surface by LEED is \u00E2\u0088\u00BC0.0141 A\u00CC\u008A. Despite the introduction of this rather large buckling, only small effects were observed on the LDA band structure the near the Fermi energy, as shown in Fig. 4.13 (a,b) for both the case of non-rotated19 octahedra and octahedra with a 6.5\u00E2\u0097\u00A6 rotation. The only observable effect was the lifting of certain degeneracies \u00E2\u0080\u0093 for example, the \u00CE\u00B3\u00E2\u0080\u0093\u00CE\u00B3\u00E2\u0080\u00B2 intersection along X1\u00E2\u0086\u0092M. For this reason further calculations exploring the effects of a buckling mode were not performed, and it was not considered in slab calculations. The final bulk calculations discussed here involve the inclusion of elec- tronic interactions via an on-site Coulomb interaction U and exchange in- teraction J , for an effective interaction U \u00E2\u0088\u0092 J [86\u00E2\u0080\u009388] using the local-spin- 19Nominally undistorted; however, the symmetry was pre-broken by introducing a .005\u00E2\u0097\u00A6 rotation, so the symmetry remained the same with the introduction of buckling. 93 4.2. LDA calculations and structural distortions density approximation (LSDA). Although ab-initio methods for calculating these interactions have been proposed [89], often the more empirical method of varying the effective interaction until good agreement with experiment is found has been used (for example, in related Sr2RhO4 [57]). Here we sought merely to explore the effect of adding such an effective interaction, before investing computation resources with more costly LSDA calculations and determining the best value to use for the interaction. Accordingly we chose the rather large effective coupling of 1 eV (compared with, for example, re- lated Sr2RhO4, where an effective coupling of 0.3 eV \u00E2\u0080\u0093 0.5 eV was used [57]). The results of these LSDA calculations for a bulk crystal structure with oxy- gen octahedra rotated 6.5\u00E2\u0097\u00A6, both with and without the inclusion of SO, for effective coupling 1 eV and 0 eV are shown in Fig. 4.14. Here it can be seen that the effect is on overall bandwidth, which, near the Fermi energy, causes little change other than eliminating the \u00CE\u00B4 crossing at the \u00CE\u0093 point by moving it above the Fermi level. Because adding U had little overall effect slab calculations were not performed with the inclusion of electronic interactions under this scheme. 4.2.2 Slab choice The exact slab structure chosen for the LDA+SO slab calculations is pre- sented in detail in Fig. 4.15. The surface layer was chosen to exactly match that measured by LEED, which includes a slight elongation as well as a 7.46\u00E2\u0097\u00A6 rotation of the oxygen octahedra, and, based on the preceding bulk calcula- tions, should be expected to match the overall band structure of the surface well. However, as discussed earlier, the exact nature of the sub-surface layer is unknown and cannot be probed accurately. From the preceding discussion it appears that any sub-surface rotation should be .3\u00E2\u0097\u00A6. Since the choice of any finite small rotation in the sub-surface layer would thus be largely ar- bitrary, and unnecessary in order to simply introduce a folding because of the nature of the calculations, we chose to use a pure bulk-like structure for the inner three layers as the simplest possible model. This also allowed the realization \u00E2\u0080\u0093 with only five layers \u00E2\u0080\u0093 of a bulk-like region with large size 94 4.2. LDA calculations and structural distortions 2a 7.46\u00C2\u00B0 7.46\u00C2\u00B0 0\u00C2\u00B0 0\u00C2\u00B0 a 0.2083 0.1571 2.0739 2.0739 2.2520 2.0598 2.0598 2.2501 ... 0.1892 0.1892 ... bu lk - lik e bu lk - lik e bu lk - lik e ro ta te d su rf ac e oc ta he dr a ro ta te d su rf ac e oc ta he dr a bulk unit cell 12.7394 3.8710 5.4 74 4 (a) (b) a b c a c b vacuum vacuum Sr Ru (in octahedra) O Figure 4.15: Atomic positions in the five-layer slab used for LDA+SO calcu- lations, with rotated octahedra and modified c-axis dimensions for surface (red) and bulk-like layers (blue). All dimensions in A\u00CC\u008A. Panel (a) shows the surface viewed from the vacuum, with bulk primitive cell shown with solid lines and ( \u00E2\u0088\u009A 2\u00C3\u0097\u00E2\u0088\u009A2)R45\u00E2\u0097\u00A6 reconstructed shown with dashed lines. Panel (b) illustrates c-axis dimensions for both bulk and surface layers, viewed from the side. 95 4.2. LDA calculations and structural distortions compared with the surface-like region. To include a sub-surface layer with a different (and arbitrary) rotation would necessitate 10 layers to have a similar ratio of surface-like to bulk-like layers. 4.2.3 Slab results The results of the slab calculations are presented in Fig. 4.16. In addition to allowing us to study both the structural reconstruction as well as effects due to broken symmetry in proximity to the surface in a self-consistent way we are able us to estimate the surface/bulk character of the resulting band structure by projecting the corresponding wave-functions onto the Ru or- bitals for each layer. This method was used to colour the band structure red and blue for surface and bulk-like layers, respectively. As the ARPES measured FS shows a slight hole doping (see Tab. 4.1) a variety of chemical potential shifts, panels (d\u00E2\u0080\u0093g), were considered with the constant energy con- tour at 24 meV binding energy chosen for a final comparison, panel (a). This corresponds to a FS with an averaged electron counting ne'3.91 to match the s\u00E2\u0080\u0093b averaged value experimentally observed by ARPES and, as shown in Tab. 4.1, also closely matches the relative doping measured between layers (with s containing slightly fewer electrons). From the colourized band struc- ture hybridization between layers does occur at some k-points, where sheets can be seen to change character, and although somewhat complicated by hybridization gaps appearing at all crossings, these calculations show good overall agreement with the phenomenological FS concerning the splitting, topology, and volume of the various bands (Tab. 4.1), as well as \u00E2\u0080\u0093 and most importantly \u00E2\u0080\u0093 the surface vs. bulk-like assignment of all detected features20. These calculations validate the explanation of a progression of structural dis- tortions from surface to bulk, with a corresponding progression of electronic states. 20The appearance of the \u00CE\u00B4s pocket of dx2\u00E2\u0088\u0092y2 character at (0, 0) and (pi, pi), which ex- perimentally is found just above EF [58] and not observed to cross, would lift above the FS with the addition of an electronic interaction term U , as seen in \u00C2\u00A74.2.1, but this would require more computationally expensive LSDA. 96 4.2. LDA calculations and structural distortions (a) (b) vacuum O Sr Ru (in octahedra) rotated surface broken symmetry bulk-like bulk a c b (c) (d) (e) (f) (g) ARPES ne = 3.91 0 meV ne = 4 20 meV ne = 3.92 40 meV ne = 3.84 60 meV ne = 3.76 24 meV ne = 3.91 Figure 4.16: Results of LDA+SO slab calculations. (a) FS obtained from the LDA+SO slab calculations by shifting the chemical potential 24 meV down in energy to match the slight average hole doping observed by ARPES. Surface and bulk character are shown in red and blue, respectively. (b) par- tial structure of Sr2RuO4 slab with FS sketches showing the corresponding surface and bulk-like FS derived from (a). (c) FS as measured by ARPES, for comparison. (d\u00E2\u0080\u0093g) FS from slab calculations for shifts of 0\u00E2\u0086\u009260 meV. 97 4.3. Conclusions 4.3 Conclusions To conclude, by means of ARPES and LDA+SO slab calculations we have been able to unravel a surface-to-bulk progression of the electronic structure in Sr2RuO4. We find that a ( \u00E2\u0088\u009A 2 \u00C3\u0097 \u00E2\u0088\u009A2)R45\u00E2\u0097\u00A6 reconstruction of surface and sub-surface layers provides a consistent explanation of all detected dispersive features in terms of a progression of electronic states induced by structural instabilities, with no evidence for novel phases driven by topological bulk properties, ferromagnetic ordering, or the interplay between SO coupling and the broken symmetry of the surface. This layer-by-layer approach pro- vides the most detailed information on FS volumes, Fermi velocities, as well as many-body renormalizations vARPESF /v LDA+SO F , for both surface and bulk-like bands (Tab. 4.1). In analogy with the recent work on underdoped cuprates [84, 90], which found evidence for the enhancement of electronic correlations and ordering tendencies in the surface and sub-surface region, this study of Sr2RuO4 also highlights the significantly more correlated char- acter of the top surface layer bands, which might thus more strongly benefit from an LDA+SO+U description [57]. It is likely that similar phenomena may occur, yet unnoticed, in other systems. This possibility should be considered whenever analyzing data from a surface-sensitive technique like ARPES, especially as resolution and the ability to resolve features advance. 98 Chapter 5 CPS-ARPES on Sr2RuO4 Sr2RuO4 is a well-known candidate for spin-triplet superconductivity [91]. This came after a flurry of experimental activity observed, in rapid succes- sion, spontaneous internal magnetic fields in the superconducting state, by muon spin-relaxation experiments, indicating a p-wave state [92]; a square flux-line lattice, by neutron scattering experiments, indicating that pairing occurs in the 2D \u00CE\u00B3 band [93] (although proposals exist whereby superconduc- tivity, driven by the interplay between spin and charge fluctuations, emerges from the 1D \u00CE\u00B1 and \u00CE\u00B2 bands [94]); and no change in spin susceptibility passing through the superconducting transition temperature, through 17O Knight shift measurements, indicating spin-triplet pairing [95]. Following this the description of superconductivity in Sr2RuO4 was thought to be complete \u00E2\u0080\u0093 it became a hallmark spin-triplet chiral p-wave superconductor, the 2D analogue of superfluid 3He [96]. However, despite the apparent existence of such a pairing, some later experiments have failed to support this con- clusion, suggesting that this description alone may not be sufficient. While the additional superconducting transition in magnetic fields (expected as a result of the multi-component order parameter in spin-triplet superconduc- tivity) has been measured in specific heat [97], DC magnetization [98], and AC susceptibility [99], there is an extreme, and puzzling, sensitivity to the angle of the field with respect to the ab-plane \u00E2\u0080\u0093 it is observed only for angles . 1\u00E2\u0097\u00A6 [100]. Additionally, under similar conditions, the upper critical field is found to be strongly suppressed [101]. This strange behaviour cannot be explained within a theoretical model using spin-triplet superconductivity alone [91, 102]. A solution to this conflict is offered by relativity and the spin\u00E2\u0080\u0093orbit (SO) interaction, which blurs the distinction between singlet and triplet and could 99 5.1. Single-particle predictions \u00E2\u0080\u0093 entanglement at the Fermi energy facilitate mixed-parity pairing [17, 103, 104]. However, thus far, experimen- tal confirmation has been largely circumstantial, limited to measurements of the band structure itself [17]. In this work we aim to directly probe the exis- tence of mixing between spin and orbital quantum numbers. We do this by performing spin- and angle-resolved photoemission spectroscopy (SARPES) and using the angular momentum inherent in circularly polarized light along with final state selection rules to generate spin-polarized photoemission from the mixed states (CPS-ARPES). In this chapter we will first review and expand on the theoretical pre- diction of SO-entangled bands in Sr2RuO4 by Haverkort et al. [17] and the implication for the single-particle states, in \u00C2\u00A7 5.1. In \u00C2\u00A7 5.2 we will then confirm these predictions using CPS-ARPES performed at the Swiss Light Source Complete Photoemission Experiment (COPHEE) endstation. Fol- lowing this we briefly discuss the ramifications this entanglement must have on the superconducting state and its description in this material in \u00C2\u00A7 5.3. We find that the presence of SO coupling must drive the superconducting state in Sr2RuO4 to be even more unconventional than generally assumed, with strong momentum-dependent entanglement between the spin and orbital sectors; this leaves a classification in terms of spin eigenstates meaningless. 5.1 Single-particle predictions \u00E2\u0080\u0093 entanglement at the Fermi energy The bulk band structure of Sr2RuO4 was calculated using the tight-binding (TB) linear muffin-tin orbital (LMTO) code [105], which was tested against that calculated with full-potential linearized augmented-plane-wave method using WIEN2k [85]. The TB parameters were extracted from the TB-LMTO results using the order-N muffin-tin orbital (NMTO) method [106] on the minimal basis of oxygen p and ruthenium d orbitals. Atomic spin\u00E2\u0080\u0093orbit cou- pling was then added as a local term in the tight-binding Hamiltonian, the results of which show excellent agreement with calculations performed using relativistic local-density approximation (LDA)+SO calculations in WIEN2k. 100 5.1. Single-particle predictions \u00E2\u0080\u0093 entanglement at the Fermi energy Z \u00CE\u0093 M X \u00CE\u0093 \u00CE\u0093 M X \u00CE\u0093 En er gy (e V ) 0 1 -1 0 1 -.5 .5 \u00CE\u0093 MX (a) (b) (c) (d) \u00CE\u00B1 \u00CE\u00B2 \u00CE\u00B3 \u00CE\u00B1\u00CE\u00B2 \u00CE\u00B3 \u00CE\u00B2 \u00CE\u00B3 \u00CE\u00B3 \u00CE\u00B1 -.5 .50 1 \u00CE\u00980 \u00CF\u0080 \u00CE\u00B1 \u00CE\u00B2 \u00CE\u00B3 \u00CE\u0093 -X \u00CE\u0093 -X \u00CE\u0093 -M \u00CE\u0093 -M \u00CE\u0093 -M \u00CE\u0098\u00CE\u00B2,\u00CE\u00B3 \u00CE\u0098\u00CE\u00B1 Figure 5.1: Band structure of Sr2RuO4 and expectation values of ~l \u00C2\u00B7 ~s; (a) along high-symmetry directions and (b) at the Fermi energy for kz = 0. \u00E3\u0080\u0088~l \u00C2\u00B7 ~s \u00E3\u0080\u0089 for LMTO/NMTO+SO is shown via the colour scale at top using thick lines, while that without SO is shown underneath in fine black. The value of \u00E3\u0080\u0088~l\u00C2\u00B7~s \u00E3\u0080\u0089 is shown in panels (c) and (d), and the location of CPS-ARPES measurements performed is highlighted yellow in (a) and (b). 101 5.1. Single-particle predictions \u00E2\u0080\u0093 entanglement at the Fermi energy 0 \u00CF\u0080 \u00CF\u0080 0 -\u00CF\u0080 0 \u00CF\u0080 2\u00CF\u0080 dxz/dyz dxy 0 \u00CF\u0080 \u00CF\u0080 0 -\u00CF\u0080 0 \u00CF\u0080 2\u00CF\u0080 0 \u00CF\u0080 \u00CF\u0080 0 -\u00CF\u0080 0 \u00CF\u0080 2\u00CF\u0080 (a) (b) \u00CE\u00B2 \u00CE\u00B3 \u00CE\u00B1 x x y y z z (c) (d) (e) -.5 .50 1 Figure 5.2: Orbital character (a) as well as expectation value of ~l \u00C2\u00B7 ~s (b), and its components (c\u00E2\u0080\u0093e) around the Fermi surface sheets of Sr2RuO4, as calculated by LMTO/NMTO+SO. The extended zone scheme is used to il- lustrate the conventional BZ inside the fundamentally body-centred tetrag- onal (BCT) unit cell. 102 5.1. Single-particle predictions \u00E2\u0080\u0093 entanglement at the Fermi energy In this manner the effects of the SO interaction can be easily studied on their own. In Sr2RuO4 the relativistic SO coupling is relatively small (\u00CE\u00B6 \u00E2\u0089\u0088 93 meV) with respect to the bandwidth (\u00E2\u0088\u00BC 1.5 eV). Nevertheless its influence always becomes important whenever two non-relativistic bands would be degener- ate, either by symmetry or accidentally. In Sr2RuO4 this happens at several places in the Brillouin zone (BZ), as demonstrated in Fig. 5.1(a,b) where we show a comparison of the band structure and Fermi surface calculated both with and without SO coupling included. For example, by symmetry the dxz and dyz bands would be degenerate along the entire momentum path from \u00CE\u0093 to Z. Additionally there are accidental degeneracies along the path \u00CE\u0093 to X where the bands of dxz, dyz, and dxy all cross \u00E2\u0080\u0093 the location of which varies with kz but often occurs at the Fermi level, which can be seen in Fig. 5.2 panel (a), where the Fermi sheets change orbital character. At all these locations SO coupling leads to a splitting of the bands and a non- trivial entanglement between the spin and orbital character, as shown via the expectation value of ~l\u00C2\u00B7~s (where ~l is the orbital and ~s the spin angular mo- mentum operator, and which is zero when spin and orbit are independent) near these crossings in Fig. 5.1 panels (c) and (d). If this mixing existed at a few isolated k-points on the Fermi surface, one could imagine ignor- ing SO coupling, but the importance of SO coupling is not limited to these would-be degeneracies. The SO interaction can be enhanced by correlations that either reduce the bandwidth or, as suggested in Ref. 57, increase the SO splitting in a mean-field approach if they span the Fermi level. As seen in Fig. 5.1(a), despite the large bandwidth, the bands are often separated by energies small compared with the SO interaction of \u00E2\u0089\u0088 93 meV, which indi- cates that SO coupling may be important everywhere. In Fig. 5.2 we see that this is, in fact, the case \u00E2\u0080\u0093 in panel (b) we see that \u00E3\u0080\u0088~l \u00C2\u00B7 ~s \u00E3\u0080\u0089 is non-zero around almost the entire Fermi surface sheet. Although there is a small region near the corners of the \u00CE\u00B3 band where \u00E3\u0080\u0088~l \u00C2\u00B7 ~s \u00E3\u0080\u0089 = 0, we see in panels (c\u00E2\u0080\u0093e) that it is not because ~l and ~s are independent from each other at that location but rather because the individual non-zero components sum to zero. Overall we find that LMTO/NMTO+SO predicts a strong entanglement between the 103 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement spin and orbital quantum numbers on the entire Fermi surface, for all three bands, everywhere in k-space. 5.2 Measuring spin\u00E2\u0080\u0093orbital entanglement SARPES measures the spin polarization of photoemitted electrons, while the SO-induced splitting in Sr2RuO4 is inherently not spin polarized, owing to an occupation of two degenerate states with opposite (pseudo) spin21. For this reason the angular momentum of circularly polarized light, carefully selected initial states, and final state selection rules were used together in order to transform SO splitting into spin-polarized photoemission, in a similar manner to that done on GaAs [107]. 5.2.1 States measured In order to pioneer this technique on Sr2RuO4 we measured the SO splitting at the \u00CE\u0093 point, or ~k = (0, 0, 0), as indicated in Fig. 5.3. This allows us to use both the experimental geometry and initial-state wave-functions that are the most straightforward to understand, leading to a clear-cut proof of the SO entanglement in these bands. At this point in k-space non-relativistic band structure calculations predict two degenerate bands of dxz and dyz orbital character. Here the other bands are far enough away in energy that they can be ignored in the following discussion, which we will do by considering only the Ising part of the SO interaction. This breaks the degeneracy by hybridizing these bands and the electron spin into two states, with a pre- dicted \u00CE\u00B6eff \u00E2\u0088\u00BC 90 meV splitting between them: a higher-energy state with the z components of orbital and spin angular momentum parallel [d\u00E2\u0086\u0091z+1z , d \u00E2\u0086\u0093z \u00E2\u0088\u00921z , illustrated in panel (b)], and a low-energy state where they are antiparallel [d\u00E2\u0086\u0091z\u00E2\u0088\u00921z , d \u00E2\u0086\u0093z +1z , illustrated in panel (c)]. Here d+1z \u00E2\u0089\u00A1 \u00E2\u0088\u009A 1/2(\u00E2\u0088\u0092dxz \u00E2\u0088\u0092 idyz) has m`z = 1, while d\u00E2\u0088\u00921z \u00E2\u0089\u00A1 \u00E2\u0088\u009A 1/2(dxz \u00E2\u0088\u0092 idyz) has m`z = \u00E2\u0088\u00921, and \u00E2\u0086\u0091z represents 21The Kramers degeneracy states that a state at +k with (pseudo) spin up is equivalent to the state at \u00E2\u0088\u0092k with (pseudo) spin down. The Kramers degeneracy, together with the inversion symmetry in the crystal structure of Sr2RuO4 (which equates +k with \u00E2\u0088\u0092k), results in the existence of degenerate states with opposite (pseudo) spin at every k-point. 104 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement SzSySx (b) (c) Z \u00CE\u0093 M X \u00CE\u0093 En er gy (e V ) 0 1 -1 (a) -.5 .50 1 Figure 5.3: Illustration of the states measured by CPS-ARPES. (a) the band structure of Sr2RuO4, as in Fig. 5.1, with the eigenstates measured by CPS-ARPES illustrated in real space in panels (b) and (c). These illus- trations show, for both halves of the Kramers degenerate pair, from left to right, the phase of both spin parts of the wave-function (\u00CF\u0095\u00E2\u0086\u0093/\u00CF\u0095\u00E2\u0086\u0091, quantized in the z direction) and spin expectation values in the x, y, and z directions as calculated by LMTO/NMTO+SO. 105 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement spin. Optical selection rules dictate that both \u00E2\u0088\u0086\u00CC\u0080 = \u00C2\u00B11 and \u00E2\u0088\u0086ml = \u00C2\u00B11. For d orbitals the change in ` will strongly favour the d \u00E2\u0086\u0092 p transition over d\u00E2\u0086\u0092 f owing to cross-section at the used photon energies, while the change in ml will depend on polarization. A circularly polarized photon travelling along the z direction could be either circular plus (\u00E2\u008A\u0095) or minus (\t). When a \u00E2\u008A\u0095 (\t) photon is absorbed by the high-energy parallel d\u00E2\u0086\u0091z+1z , d\u00E2\u0086\u0093z\u00E2\u0088\u00921z state mlz must increase (decrease) by one; but as mlz = 2 is forbidden in the strongly favoured p transition, electrons from the d\u00E2\u0086\u0093z\u00E2\u0088\u00921z (d \u00E2\u0086\u0091z +1z ) will dominate, resulting in \u00E2\u0086\u0093z (\u00E2\u0086\u0091z) spin polarization. Similarly, using \u00E2\u008A\u0095 (\t) light will result in photoemission with the opposite, \u00E2\u0086\u0091z (\u00E2\u0086\u0093z), spin polarization from the low energy antiparallel state d\u00E2\u0086\u0091z\u00E2\u0088\u00921z , d \u00E2\u0086\u0093z +1z . This is illustrated in Appendix C, Fig. C.3. 5.2.2 ARPES results When observed with regular, spin-integrated, angle-resolved photoemission spectroscopy (ARPES) the energy distribution curve (EDC) from these two states at the \u00CE\u0093 point shows both predicted features as a broad hump with width \u00E2\u0088\u00BC 400 meV, as shown in Fig 5.4 using data taken at the University of British Columbia (UBC). That the individual states cannot be resolved despite an energy resolution five times greater than the predicted splitting indicates that the states have a broad fundamental lineshape. However, as we will demonstrate, it is still possible to measure a splitting between these two states because of their fundamentally different character using CPS-ARPES. 5.2.3 CPS-ARPES results SARPES data was taken at the Paul Scherrer Institute using the Swiss Light Source COPHEE endstation, with \u00E2\u0088\u00BC1\u00E2\u0097\u00A6 angular resolution22 and energy res- olution \u00E2\u0088\u00BC100 meV. Samples were cleaved in situ with base pressures in the 10\u00E2\u0088\u00929 mbar range at low temperature (40 K). Owing to the relatively high 22Equivalent to \u00E2\u0088\u00BC 0.1 pi a at 24 eV and \u00E2\u0088\u00BC 0.2 pi a at 56 eV photon energy. 106 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement 0 0.25 0.50 0.75 1 -1000 -800 -600 -400 -200 0 200 1234 \u00CE\u00B2\u00CE\u00B1' \u00CE\u00B3 kx (\u00CF\u0080/a) 0 En er gy (m eV ) Intensity (a.u.) Intensitymin max \u00CE\u0093 \u00CE\u009C \u00CE\u0093 \u00CE\u009C (a) (b) Figure 5.4: Spin-integrated ARPES on 5.2 K Sr2RuO4 at the \u00CE\u0093 point, per- formed at UBC using 21.22 eV He I spectral line, with energy resolution of \u00E2\u0088\u00BC 17 meV and angular resolution equivalent to \u00E2\u0088\u00BC 0.01 pia . (a) shows an EDC integrated over \u00E2\u0088\u00BC 0.03 pia around the \u00CE\u0093 point, as marked in the band map (b). In (b) the location of the band map is marked in the Fermi surface map in the inset. Note that this sample, cleaved at low temperature and ultra-high vacuum conditions, shows clear evidence of the reconstructed sur- face (as seen by the folded \u00CE\u00B1\u00E2\u0080\u00B2 band), while those on which CPS-ARPES was performed did not. In either case no additional folded features will appear in an EDC at the \u00CE\u0093 point, owing to the nature of the folding, which maps the X point (which has no features in that energy range) to the \u00CE\u0093 point. 107 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement temperature and pressure, no features associated with the reconstructed surface [6, 56] were observed. In order to compensate for the observed strong circular dichroism on this material, all measurements were repeated using both circular plus and minus light, and the cross-polarization P\u00CC\u0083\u00E2\u008A\u0097 was calculated using the geometric formula P\u00CC\u0083\u00E2\u008A\u0097SMotts = \u00E2\u0088\u009A I\u00E2\u008A\u0095L I R \u00E2\u0088\u0092 \u00E2\u0088\u009A I\u00E2\u008A\u0095R I L\u00E2\u0088\u009A I\u00E2\u008A\u0095L I R + \u00E2\u0088\u009A I\u00E2\u008A\u0095R I L (5.1) for each of the Mott polarimeter pairs. (Here SMotts is the Sherman function, an empirically determined measure of the efficiency of a Mott pair, while I is the measured intensity for the left, L, and right, R, halves of the Mott pair using circular plus, \u00E2\u008A\u0095, and minus, \t, light.) With the effects of dichroism eliminated the cross polarization will be equivalent to the true incoming electron polarization P given by P = I\u00E2\u0086\u0091 \u00E2\u0088\u0092 I\u00E2\u0086\u0093 I\u00E2\u0086\u0091 + I\u00E2\u0086\u0093 ; (5.2) however, we will continue to use P\u00CC\u0083\u00E2\u008A\u0097 everywhere the measured cross polariza- tion was used. The total intensity from all detectors and light polarizations (IT ) was used with the measured cross polarizations to reconstruct the spin intensities according to I\u00E2\u0086\u0091 = IT (1 + P\u00CC\u0083\u00E2\u008A\u0097)/2, I\u00E2\u0086\u0093 = IT (1\u00E2\u0088\u0092 P\u00CC\u0083\u00E2\u008A\u0097)/2 for all three spatial dimensions and transformed into crystal coordinate directions ac- cording to the experimental geometry. Aspects of raw data analysis are presented in more detail in Appendix C, \u00C2\u00A7C.1. When the two states discussed in \u00C2\u00A7 5.2.1 are observed with CPS-ARPES two distinct features, split in the z direction only, become apparent. This splitting was observed with both 24 eV (shown in Fig. 5.5) and 56 eV photons over two energy ranges (shown in Figs. 5.6 and 5.7). Estimating the split- ting directly from the z direction curves [panel (e)] in each data set yields estimates of 83 meV, 101 meV, and 110 meV (respectively) for an average of 98 \u00C2\u00B1 14 meV. However owing to the experimental geometry (light incident at 45\u00E2\u0097\u00A6 relative to the analyzer), photons cannot be incident directly along 108 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement -0.1 0.0 0.1 C ro ss -P ol ar iz at io n Energy (eV) 0 4 8 12 C ou nt s ( 10 6 ) x y z 0.0 1.00.5 Binding Energy (eV) 0.0 1.00.5 Integrated spin(a) (b) Energy (eV) 0 2 4 6 x y z 0.0 1.00.5 Energy (eV) 0.0 1.00.5 Energy (eV) 0.0 1.00.5 , , (c) (d) (e) C ou nt s ( 10 6 ) 0 2 4 6 Energy (eV) 0.0 1.00.5 z Parallel Antiparallel (f) C or re ct ed C ou nt s ( 10 6 ) Figure 5.5: CPS-ARPES on Sr2RuO4 at the \u00CE\u0093 point, performed at COPHEE with 24 eV photon energy, measuring the states as highlighted in Fig. 5.3. (a) total intensity, (b) cross-polarization, and spin intensities (c\u00E2\u0080\u0093e) relative to the crystal coordinate directions. (f) shows the state intensity required to give the results in (e), corrected for photons incident at 45\u00E2\u0097\u00A6 and assuming no background. Error bars represent statistical uncertainty plotted at 95% confidence, with locally weighted [108] fits shown. 109 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement -0.1 0.0 0.1 C ro ss -P ol ar iz at io n Energy (eV) C ou nt s ( 10 6 ) x y z Binding Energy (eV) Integrated spin(a) (b) 0.0 0.5 1.0 0.0 0.5 1.0 0 2 4 6 8 Energy (eV) C ou nt s ( 10 6 ) x y z Energy (eV) Energy (eV) , , (c) (d) (e) 0 1 2 3 4 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 C or re ct ed C ou nt s ( 10 6 ) Energy (eV) z Parallel Antiparallel (f) 0.0 0.5 1.0 0 1 2 3 4 Figure 5.6: CPS-ARPES on Sr2RuO4 at the \u00CE\u0093 point, performed at COPHEE with 56 eV photon energy, as in Fig. 5.5 except for photon energy. 110 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement -0.1 0.0 0.1 C ro ss -P ol ar iz at io n Energy (eV) 0 4 8 12 C ou nt s ( 10 6 ) x y z Binding Energy (eV) Integrated spin(a) (b) 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 Energy (eV) 0 2 4 6 x y z Energy (eV) Energy (eV) , , (c) (d) (e) 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 C ou nt s ( 10 6 ) 0 2 4 6 Energy (eV) z Parallel Antiparallel (f) 0.0 1.0 2.0 3.0 C or re ct ed C ou nt s ( 10 6 ) Figure 5.7: CPS-ARPES on Sr2RuO4 at the \u00CE\u0093 point, performed at COPHEE with 56 eV photon energy over an extended energy range. As in Figs. 5.5 and 5.6 but using 56 eV photon energy and over a greater energy range than that shown in Fig 5.6 (this represents a second experiment). 111 5.2. Measuring spin\u00E2\u0080\u0093orbital entanglement the z direction (additionally there is a node in the final state preventing the experiment from being performed in the second Brillouin zone (BZ) with normal incident light and 45\u00E2\u0097\u00A6 emission). The result is that with equal prob- ability the photons will interact either with mlz , resulting in spin-polarized photoemission as described above, or with mlx , resulting in non-polarized photoemission (which will be measured as either \u00E2\u0086\u0091 or \u00E2\u0086\u0093 with equal probabil- ity from both parallel and antiparallel states). Therefore, in the z direction, the spin intensity (I) measured for the (\u00E2\u008A\u0095\u00E2\u0086\u0093,\t\u00E2\u0086\u0091) configuration will be given by I\u00E2\u008A\u0095\u00E2\u0086\u0093,\t\u00E2\u0086\u0091 = Iparallel 2 + Iparallel 4 + Iantiparallel 4 (5.3) (and opposite for I\u00E2\u008A\u0095\u00E2\u0086\u0091,\t\u00E2\u0086\u0093). Here Iparallel and Iantiparallel represent the total photoemission intensity given off by the states discussed in \u00C2\u00A7 5.2.1; the single term on the left represents the spin-polarized photoemission described above (divided by two due to half photons interacting with mlz), while the two terms on the right represent the non-spin polarized photoemission (divided by four due to half photons interacting with mlx , and half of those being measured as either up or down). This dilution causes a decrease in the observed splitting, which can be corrected for by taking Iparallel = (3I\u00E2\u008A\u0095\u00E2\u0086\u0093,\t\u00E2\u0086\u0091 \u00E2\u0088\u0092 1I\u00E2\u008A\u0095\u00E2\u0086\u0091,\t\u00E2\u0086\u0093)/2 = IT (1\u00E2\u0088\u0092 2P\u00CC\u0083\u00E2\u008A\u0097)/2 (5.4) (and opposite for Iantiparallel) to recover the intensities from each state; this assumes that all observed intensity is from only these two states with no background. Such a correction is shown in panel (f) in each figure and results in splitting estimates of 125 meV, 184 meV, and 172 meV (respectively) for an average of 160\u00C2\u00B130 meV. These limits represent an upper (160\u00C2\u00B130 meV) and lower bound (98 \u00C2\u00B1 14 meV) on the splitting between these two states, which shows good agreement with the predicted splitting of \u00CE\u00B6eff \u00E2\u0088\u00BC 90 meV. The possible enhancement of the energy splitting is another reminder of the importance of electronic correlations in this material. Most impor- tantly, however, the existence of these two states, from which spin-polarized 112 5.3. Implications for pairing photoemission can be generated using circularly polarized light in the z direction only, is a clear confirmation of the theoretical prediction of SO coupling in Sr2RuO4 and its consequences for the entangled nature of the wave-functions. 5.3 Implications for pairing To understand the impact of this entanglement on the description of super- conductivity in this material, we must look at the basic building blocks of the Cooper pair. Cooper assumed the two-particle wave-function describing a Cooper pair to be of the form \u00CE\u00A8(r1, \u00CF\u00831, r2, \u00CF\u00832) = \u00CF\u0095(r1\u00E2\u0088\u0092r2)\u00CF\u0086spin\u00CF\u00831,\u00CF\u00832 , assuming the momentum of the Cooper pair to be zero and where the spin part can be either singlet: \u00CF\u0086spin\u00CF\u00831,\u00CF\u00832 = \u00E2\u0088\u009A 1 2 (| \u00E2\u0086\u0091\u00E2\u0086\u0093\u00E3\u0080\u0089 \u00E2\u0088\u0092 | \u00E2\u0086\u0093\u00E2\u0086\u0091\u00E3\u0080\u0089) (5.5) (total spin S = 0) or triplet: \u00CF\u0086spin\u00CF\u00831,\u00CF\u00832 = \u00EF\u00A3\u00B1\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B2\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B3 | \u00E2\u0086\u0091\u00E2\u0086\u0093\u00E3\u0080\u0089\u00E2\u0088\u009A 1 2 (| \u00E2\u0086\u0091\u00E2\u0086\u0093\u00E3\u0080\u0089+ | \u00E2\u0086\u0093\u00E2\u0086\u0091\u00E3\u0080\u0089) | \u00E2\u0086\u0091\u00E2\u0086\u0093\u00E3\u0080\u0089 (5.6) (S = 1) [109, 110]. This allows one to classify superconductors in singlet or triplet paired states. However, a fundamental assumption to this description is that one can write the wave-function of each electron as a simple product of spatial and spin parts: \u00CF\u0088(k, \u00CF\u0083) = \u00CF\u0095(k) \u00CF\u0086spin\u00CF\u0083 . (5.7) However, this separation into a product of independent charge and spin func- tions cannot be used when SO coupling is important; instead, the general equation becomes \u00CF\u0088(k, \u00CF\u0083) = c\u00E2\u0086\u0091\u00CF\u0095\u00E2\u0086\u0091(k)\u00CF\u0086 spin \u00E2\u0086\u0091 + c\u00E2\u0086\u0093\u00CF\u0095\u00E2\u0086\u0093(k)\u00CF\u0086 spin \u00E2\u0086\u0093 , (5.8) 113 5.3. Implications for pairing \u00CE\u00B2 \u00CE\u00B3 \u00CE\u00B1kz in \u00CF\u0080/c 0\u00CE\u0093, 1/8 1/4 3/4 7/8 1Z, 3/8 1/2 5/8 Z M X \u00CE\u0093 M X Spin Character Figure 5.8: Orbital and spin character around the Fermi surface sheets of Sr2RuO4 as calculated by LMTO/NMTO+SO. Spin character is shown by colouring the orbitals according to spin expectation value in the z direction for one half of the Kramers degenerate pair \u00E2\u0080\u0093 all orbitals are degenerate with a second state that has opposite character. More detail is presented in Fig. 5.9 for the orbitals along the diagonal at kz = 3/4 pi c \u00E2\u0080\u0093 those highlighted here in green. 114 5.3. Implications for pairing SzSySx \u00CE\u00B2 kx = ky kz = 3/4 \u00CF\u0080/c \u00CE\u00B3 kx = ky kz = 3/4 \u00CF\u0080/c \u00CE\u00B1 kx = ky kz = 3/4 \u00CF\u0080/c (a) (b) (c) Figure 5.9: Orbitals illustrated in greater detail, for selected k-points in Sr2RuO4 along the zone diagonal at kz = 3/4 pi c , as highlighted in Fig. 5.8; (a\u00E2\u0080\u0093c) show the \u00CE\u00B2, \u00CE\u00B3, and \u00CE\u00B1 bands, respectively. Here both halves of the Kramers degenerate pair are illustrated, showing, from left to right, the phase of both spin parts of the wave-function (\u00CF\u0095\u00E2\u0086\u0093/\u00CF\u0095\u00E2\u0086\u0091, quantized in the z direction) as well as spin expectation values in the x, y, and z directions. Note that the illustrations of spin character should not be interpreted to mean that these states could alternatively be represented by a single wave- function, as the phase information in \u00CF\u0095\u00E2\u0086\u0093 and \u00CF\u0095\u00E2\u0086\u0091 has been lost. 115 5.3. Implications for pairing with \u00E2\u0086\u0091 and \u00E2\u0086\u0093 specifying the spin, c\u00E2\u0086\u0091 and c\u00E2\u0086\u0093 eigenstate-dependent pre-factors, and \u00CF\u0095\u00E2\u0086\u0091 and \u00CF\u0095\u00E2\u0086\u0093 two independent wave-functions. In locations with strong mixing between \u00CF\u0095\u00E2\u0086\u0091 and \u00CF\u0095\u00E2\u0086\u0093, it is not possible to use a description in terms of spin eigenstates. In Fig. 5.8 we illustrate orbitals in real space showing spin character for a variety of momentum space locations on the Fermi surface sheets of Sr2RuO4 for one quarter of the BZ. Along the edges of the zone, e.g., near the intersections with \u00CE\u0093\u00E2\u0086\u0092M or Z\u00E2\u0086\u0092M, we find that the orbitals do not show strong mixing, and we notice also that in these areas the \u00CE\u00B2 and \u00CE\u00B1 bands are of dxz/dyz orbital character, while the \u00CE\u00B3 band is of dxy character, as expected. At these locations the wave-function could be well approximated by a simple product of spatial and spin parts, as in Eq. 5.7. However, close to the zone diagonal, e.g., near the intersections of the sheets with \u00CE\u0093\u00E2\u0086\u0092X or Z\u00E2\u0086\u0092X, this is not the case; here we find strong mixing in orbital character for all bands and, in the \u00CE\u00B3 and \u00CE\u00B2 bands, also mixing in spin character. Orbitals from a selection of these locations with strong mixing, those directly along the zone diagonal at kz = 3/4 pi c , are chosen for a more detailed illustration in Fig. 5.9. In Fig. 5.9 we explicitly see how, for the \u00CE\u00B3 and \u00CE\u00B2 that show strong spin mixing, this is a direct consequence of the fact that the wave-function cannot be split into independent charge and spin parts and can only be represented by Eq. 5.8. As discussed earlier, Cooper took the momentum of a pair to be zero and the spin part to be either singlet or triplet. Even in cases where the wave- function cannot be separated into spin and spatial components as in Eq. 5.7, we can still work out the interaction of the spin between the canonical two states at k and \u00E2\u0088\u0092k \u00E2\u0080\u0093 and therefore whether the spin states are singlet/triplet or something else. We will follow this interaction, written as Hk,\u00E2\u0088\u0092kint \u00E2\u0088\u009D ~sk \u00C2\u00B7 ~s\u00E2\u0088\u0092k (5.9) where ~s is the spin operator, between these two states around the Fermi surface sheets for all three bands in Fig. 5.10. Regardless of the pairing glue in Sr2RuO4, this total spin of the pair should be conserved. This is the case 116 5.3. Implications for pairing \u00CE\u00B1 s k . s -k \u00CE\u00980 \u00CF\u0080 \u00CE\u00980 \u00CF\u0080 \u00CE\u00980 \u00CF\u0080 (a) (d) (g) (b) (e) (h) (c) (f) (i) 0.0 -0.2 0.2 0.4 0.6 0.0 -0.2 0.2 0.4 0.6 0.0 -0.2 0.2 0.4 0.6 s k . s -k s k . s -k \u00CE\u00B2kz \u00CE\u00B3 0 \u00CF\u0080/2 \u00CF\u0080 Figure 5.10: \u00E3\u0080\u0088~sk \u00C2\u00B7 ~s\u00E2\u0088\u0092k \u00E3\u0080\u0089 around the Fermi surface sheets in Sr2RuO4, as calculated by LMTO/NMTO+SO, for \u00CE\u00B2 (a,d,g), \u00CE\u00B3 (b,e,h), and \u00CE\u00B1 (c,f,i) bands at kz = 0 (a\u00E2\u0080\u0093c), pi/2 (d\u00E2\u0080\u0093f), and pi (g\u00E2\u0080\u0093i). The value of k in the xy- plane is defined using \u00CE\u0098 as in Fig. 5.1. 117 5.4. Conclusion whether it be phonons or even a magnetic interaction, which fundamentally results from the Coulomb interaction. In Fig. 5.10 we see that, between states at k and \u00E2\u0088\u0092k, spin does occasion- ally form a singlet/triplet system \u00E2\u0080\u0093 for example, in the \u00CE\u00B1 band at \u00CE\u0098 = 0, pi/2, and pi. However, it does not do so elsewhere. For the \u00CE\u00B2 and \u00CE\u00B3 bands we find that occasionally there exists two doublets, but most often two singlets and a doublet. Overall, in all bands, we find that there exists much k-dependent mixing between singlet/triplet, doublet/doublet, and other spin configura- tions rendering a description in terms of spin eigenstates meaningless. 5.4 Conclusion In conclusion we have shown that Sr2RuO4, a known unconventional su- perconductor, may well be much more unconventional than has been as- sumed so far. The theoretically predicted entanglement of spin and orbital descriptions has been experimentally confirmed with SARPES using circu- larly polarized light. This means that the states at the Fermi energy cannot be written as a simple product of charge and spin functions, resulting in a situation where one cannot build normal Cooper pairs in terms of pure spin singlet or triplet states. In the future it would be highly desirable to calcu- late the superconducting gap function using the relativistic LDA eigenstates and possible realistic superconducting pairing interactions. Such a descrip- tion, by allowing mixed states, could reconcile the experimental results that suggest that the superconducting state in Sr2RuO4 cannot be purely triplet in character. These mixed states necessarily have interesting implications beyond spin, as the Pauli principle guarantees the total state, including spin degrees of freedom, to be of odd parity under particle exchange. 118 Chapter 6 Conclusion This thesis covered many details over a range of topics, here we will reca- pitulate the main conclusions and their context. The ARPES system at UBC A state of the art angle-resolved photoemission spectroscopy (ARPES) sys- tem has been constructed and characterized here at the University of British Columbia (UBC). It has already begun producing publication-quality data on a wide variety of systems [6, 84, 111\u00E2\u0080\u0093113], and will continue to do so into the future. Spectral Function Analysis ARPES, by measuring the spectral function, provides access to an extremely rich data source. Not only can fundamental elements like the electronic band structure be resolved, but it is also possible to learn about the many-body electronic interactions in a system. We have shown that none of the renormalizations measured in the quasi- particle dispersion necessarily scale linearly with the dimensionless cou- pling found in the Hamiltonian, as is often assumed in the interpretation of ARPES results. It should therefore be expected that in many real-world systems \u00CE\u00BBeff 6= \u00CE\u00BB. This helps explain some of the exceedingly large renor- malization values sometimes reported in ARPES literature [9\u00E2\u0080\u009312], which, for example, have been as high as 30 (in Ref. 10). Understanding the mecha- nisms by which such renormalizations do not necessarily imply unphysically large couplings is a key step toward the correct interpretation of these re- sults. 119 Chapter 6. Conclusion Additionally, we have developed and thoroughly tested a variety of meth- ods that can be applied to determine the self-energy of the underlying in- teractions from the spectral function. A key difference between this and previous works along a similar vein, is the assessment of how our methods perform in the presence of a momentum-dependent self-energy. This is an important distinction as, while momentum independence is often assumed, it is unlikely to occur in real-world materials; at best it will be a question of how much momentum dependence exists. Further, it is not possible to know with certainty that a self-energy is momentum independent from sim- ple lineshape alone \u00E2\u0080\u0093 a Lorentzian form is a necessary, but not sufficient, condition. We show that by comparing quasiparticle renormalizations it is possible to determine the degree of momentum dependence in the vicinity of the quasi-particle peak. Additionally, we developed and presented two inde- pendent methods of determining the self-energy: Kramers\u00E2\u0080\u0093Kronig bare-band fitting (KKBF), which can find the self-energy if the momentum dependence is not too strong; and a lineshape ratio-based method, which will correctly produce the form of the self-energy regardless of its momentum-dependence anywhere the lineshape is Lorentzian. These methods show promise for future ARPES studies, such as those led by B. Ludbrook on MgB2, which is currently in preparation for publica- tion. In this material, a multi-gap superconductor [114], it is thought that a strongly momentum-dependent electron-phonon coupling may be responsi- ble for the differing gap sizes measured for each of the two bulk bands. Such a coupling would surely cause momentum-dependence in the self-energy of the quasiparticle excitations as well. This makes the use of our newly developed methods, which explicitly account for possible momentum-dependence, par- ticularly valuable. Confidently determining the self-energy of the quasipar- ticle excitations in MgB2 may help unlock the mystery of how the otherwise- conventional superconductivity in this material could be different between its two bands. 120 Chapter 6. Conclusion ARPES on Sr2RuO4 With the advancement of the ARPES technique a new set of features was recently discovered in data on Sr2RuO4 [6, 16, 115]; these features could not be explained by the existing model of surface reconstruction, and appeared to show a splitting that could be indicative of a wide variety of phenom- ena. Through extensive ARPES measurements on these newly discovered features, and a comparison against both bulk and slab local-density approx- imation (LDA) calculations with the inclusion of spin\u00E2\u0080\u0093orbit (SO) coupling, we have shown that the structural reconstruction observed on the surface must progressively relax into the bulk; this causes a corresponding progres- sion in the electronic band structure, which explains all features and their observed characteristics. In addition to the implications for the study of Sr2RuO4 it is likely that, in other materials with a surface reconstruction, a similar progression of states may exist. In much the same way that ARPES results on Sr2RuO4 were originally resolved as being due to a surface recon- struction [56], this work demonstrates that it is now necessary to consider also the structure of the sub-surface layers. Such considerations will be- come ever more important as advancements continue in both the resolution of ARPES systems as well as the level of detail with which materials are studied. In addition to demonstrating the existence of a progression of electronic structure from the surface to the bulk, this work also provides the most detailed information yet presented on Fermi surface volumes and veloci- ties for both the surface and sub-surface layers, as well as the many body renormalization of these velocities as compared to LDA. The progression of renormalizations, in particular, demonstrates a strong enhancement of electronic correlations in the surface layer. This may be due to reduced dimensionality, and similar enhancements have recently observed in other materials as well \u00E2\u0080\u0093 for example the underdoped cuprates [84, 90]. The morals of this story also show strong ties to the rapidly advancing field of interface-physics, by demonstrating that the effects of the surface interface propagate over multiple layers and that its reduced dimensionality 121 Chapter 6. Conclusion strongly enhances the electronic correlations. By exploiting and combining such effects through engineered interfaces and layers, the field of interface- physics shows promise as an avenue for the future development of next- generation electronic devices in much the same way that semiconductors revolutionized and brought about the electronics we enjoy today. CPS-ARPES, Spin\u00E2\u0080\u0093orbit coupling, and superconductivity in Sr2RuO4 Problems exist in the current struggle to classify the superconductivity in Sr2RuO4, specifically the spin-configuration of the paired electrons. While some experiments show that Sr2RuO4 is a chiral p-wave spin-triplet super- conductor [92, 93, 95], others cannot be fully explained by such a description [97\u00E2\u0080\u009399] \u00E2\u0080\u0093 specifically regarding the spin-triplet nature of the pairing interac- tion. The SO interaction, expected to be significant for a heavy transition metal like ruthenium, offers a possible resolution to this conflict by coupling the spin and motion of electrons in the lattice. While the band-structure predictions of SO coupling have previously been confirmed for Sr2RuO4 [17], the actual coupling of spin and orbital descriptions has not. It is this en- tanglement between spin and orbital motion that we probed directly and confirmed with circularly polarized photon spin- and angle-resolved photoe- mission spectroscopy (CPS-ARPES). By confirming this coupling between the spin and orbital motion of the electrons we directly show that the usual method of writing the electronic wave-function as product of independent spin and spatial parts is invalid. This interferes with the first steps of writing a Cooper pair under the normal description. Additionally, we calculate the spin-eigenstates for the canoni- cal pair of electrons around the Fermi surface, for all bands, using the full SO description and show that the available spin-eigenstates vary contin- uously between singlet\u00E2\u0080\u0093triplet, doublet\u00E2\u0080\u0093doublet, and other spin configura- tions. This shows that attempting to classify the pairing in Sr2RuO4 as \u00E2\u0080\u009Cspin-triplet\u00E2\u0080\u009D is not actually meaningful, and that the pairing must instead be even more unconventional than is generally assumed. Moving forward, 122 Chapter 6. Conclusion it would be highly desirable to calculate superconducting properties using possible realistic pairing interactions and the full relativistic LDA+SO eigen- states. By allowing mixed and transitory states, such a description provides more spin configurations than the traditional singlet\u00E2\u0080\u0093triplet eigenstates, as well as a means for the apparent spin-state of the pair to change; neither of these things are possible using the regular formulation. 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A.1 Calculating renormalizations from perturbation theory Starting from the Hamiltonian, Eq. 2.2, H = \u00E2\u0088\u0091 k \u00CE\u00B5bkc \u00E2\u0080\u00A0 kck+\u00E2\u0084\u00A6 \u00E2\u0088\u0091 Q b\u00E2\u0080\u00A0QbQ+ \u00E2\u0088\u0091 K,Q gQ\u00E2\u0088\u009A N c\u00E2\u0080\u00A0K\u00E2\u0088\u0092QcK(b \u00E2\u0080\u00A0 Q+b\u00E2\u0088\u0092Q), (A.1) which for momentum k has a zeroth-order ground state \u00E3\u0080\u0088k|0 consisting of a single electron c\u00E2\u0080\u00A0k|0\u00E3\u0080\u0089 with energy \u00CE\u00B5bk, we find that the first-order perturbation energy is 0 owing to a lack of phonons: E1(k) = \u00E3\u0080\u00880|ck \u00E2\u0088\u0091 K,Q gQ\u00E2\u0088\u009A N c\u00E2\u0080\u00A0K\u00E2\u0088\u0092QcK(b \u00E2\u0080\u00A0 Q+b\u00E2\u0088\u0092Q) c \u00E2\u0080\u00A0 k|0\u00E3\u0080\u0089 = 0. (A.2) The first-order ground state adds a single phonon: C1\u00E2\u0084\u00A6 = \u00E2\u0088\u0091 L \u00E3\u0080\u00880|ck\u00E2\u0088\u0092LbL \u00E2\u0088\u0091 K,Q gQ\u00E2\u0088\u009A N c\u00E2\u0080\u00A0K\u00E2\u0088\u0092QcK(b \u00E2\u0080\u00A0 Q+b\u00E2\u0088\u0092Q) \u00CE\u00B5bk \u00E2\u0088\u0092 \u00CE\u00B5bk\u00E2\u0088\u0092L \u00E2\u0088\u0092 \u00E2\u0084\u00A6 c\u00E2\u0080\u00A0k|0\u00E3\u0080\u0089 (A.3) C1\u00E2\u0084\u00A6 = \u00E2\u0088\u0091 L gL\u00E2\u0088\u009A N(\u00CE\u00B5bk \u00E2\u0088\u0092 \u00CE\u00B5bk\u00E2\u0088\u0092L \u00E2\u0088\u0092 \u00E2\u0084\u00A6) (A.4) |k\u00E3\u0080\u00891 = c\u00E2\u0080\u00A0k|0\u00E3\u0080\u0089 + \u00E2\u0088\u0091 L gL\u00E2\u0088\u009A N(\u00CE\u00B5bk \u00E2\u0088\u0092 \u00CE\u00B5bk\u00E2\u0088\u0092L \u00E2\u0088\u0092 \u00E2\u0084\u00A6) c\u00E2\u0080\u00A0k\u00E2\u0088\u0092Lb \u00E2\u0080\u00A0 L|0\u00E3\u0080\u0089, (A.5) 139 A.2. Kramers\u00E2\u0080\u0093Kronig bare-band fitting which gives the second-order correction to the ground-state energy: E2(k) = \u00E3\u0080\u0088k|0 \u00E2\u0088\u0091 L,M gM\u00E2\u0088\u009A N c\u00E2\u0080\u00A0L\u00E2\u0088\u0092McL(b \u00E2\u0080\u00A0 M+b\u00E2\u0088\u0092M) |k\u00E3\u0080\u00891 (A.6) E2(k) = 1 N \u00E2\u0088\u0091 L gL\u00E2\u0088\u0092kgk\u00E2\u0088\u0092L \u00CE\u00B5bk \u00E2\u0088\u0092 \u00CE\u00B5bk\u00E2\u0088\u0092L \u00E2\u0088\u0092 \u00E2\u0084\u00A6 . (A.7) This, changing the sum to an integral over all possible phonon momenta, becomes (for example, in 3D): E2(k) = 1 (2pi)3 pi\u00E2\u0088\u00AB \u00E2\u0088\u0092pi pi\u00E2\u0088\u00AB \u00E2\u0088\u0092pi pi\u00E2\u0088\u00AB \u00E2\u0088\u0092pi gL\u00E2\u0088\u0092kgk\u00E2\u0088\u0092L \u00CE\u00B5bk \u00E2\u0088\u0092 \u00CE\u00B5bk\u00E2\u0088\u0092L \u00E2\u0088\u0092 \u00E2\u0084\u00A6 \u00E2\u0088\u0082kx \u00E2\u0088\u0082ky \u00E2\u0088\u0082kz. (A.8) In 1D this integral can be solved exactly and was presented in a simpli- fied form to show the band for both models as predicted by perturbation theory in Eqs. 2.8 and 2.9. However, for higher dimensions it was evaluated numerically. In multiple dimensions, deterministic algorithms for numerical integra- tion can require large calculation times and often provide no estimate for the error associated with the calculation. For these reasons Monte-Carlo integration was chosen to integrate Eq. A.1. Code to evaluate the integral found in Eq. A.1 and also provide an estimate of the error using the GNU scientific library [51] implementation of the VEGAS Monte Carlo integration algorithm [49, 50] 23 is provided in \u00C2\u00A7D.1. A.2 Kramers\u00E2\u0080\u0093Kronig bare-band fitting The method outlined here varies slightly from techniques previously de- scribed in the literature, which generally reduce the functional form for \u00CE\u00B5bk substantially in order to facilitate an exact solution for A(k, \u00CF\u0089) as they often deal with data very close to the Fermi energy over a small range [23, 33\u00E2\u0080\u009335]. 23This was a relatively new feature in version 1.15, released only two months before submission of the manuscript [2]. 140 A.2. Kramers\u00E2\u0080\u0093Kronig bare-band fitting In our analysis we have instead expanded everything about km, essentially using a new linear approximation for \u00CE\u00B5bk on each momentum distribution curve (MDC) slice. Although our method has the disadvantage that it does not work as well near the zone boundary where the band velocity goes to zero (other methods that make a second-order approximation can successfully predict and fit the non-Lorentzian shape in this region and may continue to work in this regime), ours has the distinct advantage that it works over a much larger energy range and allows fitting based on an infinite variety of bare-band models (so long as they are differentiable). Most importantly, by its form it also explicitly shows that the self-energies are evaluated along the km path in the case where there is global momentum dependence in the self- energy. One might imagine that for the analysis of a particular experiment one may have reasons to choose one method over another, or perhaps even a hybrid of the two. Here we will describe the idea of a Kramers\u00E2\u0080\u0093Kronig bare-band fitting (KKBF) as implemented for our method; its application to other methods is similar. KKBF is a technique whereby a Lorentzian fit is first performed on every slice of constant energy, \u00CF\u0089\u00CC\u0083, according to Eq. 2.13. The values of km and \u00E2\u0088\u0086km from the fits can provide the self-energies for every (\u00CF\u0089\u00CC\u0083, km) point, within the limitations above, if the bare-band \u00CE\u00B5bk is known. As an analytic complex function the real and imaginary parts of the self-energy are Kramers\u00E2\u0080\u0093Kronig (KK) related: \u00CE\u00A3\u00E2\u0080\u00B2 , \u00E2\u0080\u00B2\u00E2\u0080\u00B2KK(k, \u00CF\u0089) = \u00C2\u00B1 1 pi P \u00E2\u0088\u00AB \u00E2\u0088\u009E \u00E2\u0088\u0092\u00E2\u0088\u009E \u00E2\u0088\u0082\u00CE\u00BE \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2 , \u00E2\u0080\u00B2MDC(k, \u00CE\u00BE) \u00CE\u00BE \u00E2\u0088\u0092 \u00CF\u0089 . (A.9) It is possible to \u00E2\u0080\u009Cfit\u00E2\u0080\u009D the bare-band parameters by choosing them such that \u00CE\u00A3\u00E2\u0080\u00B2MDC \u00E2\u0089\u00A1 \u00CF\u0089\u00CC\u0083\u00E2\u0088\u0092\u00CE\u00B5bkm and \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2MDC \u00E2\u0089\u00A1\u00E2\u0088\u0092vbkm\u00E2\u0088\u0086km are self-consistent with \u00CE\u00A3\u00E2\u0080\u00B2KK and \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2KK. Since neither the Kramers\u00E2\u0080\u0093Kronig relationships (Eq. A.9) nor the MDC relationships (Eq. 2.15) are sensitive to a constant offset in both the real self-energy and bare-band, this is unconstrained by the method and both \u00CE\u00A3\u00E2\u0080\u00B2 and \u00CE\u00B5bk are free. In our study we have simply made the calculation of differences between \u00CE\u00A3KK and \u00CE\u00A3MDC insensitive to a constant offset and set the final offset to zero by hand at the end to allow comparison. 141 A.2. Kramers\u00E2\u0080\u0093Kronig bare-band fitting -100 -50 0 50 \u00CE\u00A3' ( m eV ) -150 -100 -50 0 \u00CE\u00A3' ' (m eV ) \u00CE\u00A3'MDC \u00CE\u00A3'tail \u00CE\u00A3'KK \u00CE\u00A3''MDC \u00CE\u00A3''tail \u00CE\u00A3''KK -100 -50 0 50 \u00CE\u00A3' ( m eV ) -150 -100 -50 0 \u00CE\u00A3' ' (m eV ) \u00CE\u00A3'MDC \u00CE\u00A3'tail \u00CE\u00A3'KK \u00CE\u00A3''MDC \u00CE\u00A3''tail \u00CE\u00A3''KK Energy (meV) 0 50 100 150 -100 -50 0 50 \u00CE\u00A3' ( m eV ) -150 -100 -50 0 \u00CE\u00A3' ' (m eV ) \u00CE\u00A3'MDC \u00CE\u00A3'tail \u00CE\u00A3'KK \u00CE\u00A3''MDC \u00CE\u00A3''tail \u00CE\u00A3''KK (a) (b) (c) (d) (e) (f) Figure A.1: Tail extrapolation for use in KK transformations, with self- energies as defined in Fig. 2.10 for the MA(1) Holstein problem with \u00E2\u0084\u00A6 = 50 meV and \u00CE\u00BB= 0.15. Panels (a) and (b) have the bias used in fitting the tails set too small, (e) and (f) have the bias set too large, and (c) and (d) have it set just right. 142 A.2. Kramers\u00E2\u0080\u0093Kronig bare-band fitting In our implementation of this idea, a simple third-order polynomial was used to fit the bare-band with an initial guess found by fitting MDC peak maxima. We then used the Levenberg\u00E2\u0080\u0093Marquardt algorithm [116] as imple- mented in the mpfit package for IDL [53] to vary band parameters. We found that the standard sum-of-squares minimization on the differences between \u00CE\u00A3KK and \u00CE\u00A3MDC did not perform as well as a concave-down function, as it placed too much weight on outlying points far away. In order to evaluate the integrals in Eq. A.9 with a finite region of data, tails were extrapolated before a Fourier-based transform was performed (the tails were then dis- carded, leaving the analysis of MDC and KK curves only within the data region). These tails were extrapolated by fitting an inverse polynomials to each side of the data, weighing the fit for each side\u00E2\u0080\u0099s tail with an exponen- tially decaying bias parameter. A bias parameter of zero would weigh the entire curve equally, while a large bias parameter would concentrate only near that data edge. It is possible for problems such as tails, overweighted outliers, and uncon- strained offsets to compound each other. An unconstrained constant offset in \u00CE\u00A3\u00E2\u0080\u00B2MDC and \u00CE\u00B5 b k leads toward a tendency for a small linear offset in both, which, when Kramers\u00E2\u0080\u0093Kronig transformed, will distort \u00CE\u00A3\u00E2\u0080\u00B2\u00E2\u0080\u00B2KK most visibly near the edges of the data, where it can interfere with a good fit of the tails. This in turn can lead to inaccuracies at these edges, which, if overweighted, can distort the bare-band fit itself. This runaway condition results in a fit that gets progressively worse through iterations and will never find the cor- rect bare-band. In practice we found that the tail bias parameter as well as the concavity of the function used to process errors must be carefully ad- justed by hand in order to prevent this problem, which can be accomplished simply by looking at whether or not the tail approximation continues to appear reasonable through successive iterations. In Fig. A.1 we show some typical examples of how the tail bias parameter can affect the fitting; each pair of panels represents the final \u00E2\u0080\u009Csolution\u00E2\u0080\u009D of the entire band minimization problem using a given tail bias. Plots like these form the guide to be used when adjusting the bias parameters by hand while looking for the best solution. In panels (a) and (b) the tail bias is too small, 143 A.2. Kramers\u00E2\u0080\u0093Kronig bare-band fitting and so the found tail is the best approximation that fits the whole curve. In panel (b) this causes a discontinuity for the low-energy tail right at the boundary, which in turn causes a cusp in the KK transform visible in (a). Despite this, the overall fit is not too bad, with reasonable general agreement between MDC and KK self-energies, meaning the found bare-band is likely close to the real solution. In panels (c) and (d) the tail bias is good, which results in a realistic fit at all boundaries and a good agreement between MDC and KK self-energies, giving confidence that the found bare-band is accurate. In panels (e) and (f) the tail bias is too strong, which results in a tail fit depending too much on the data right at the edges. This results in a KK transform poor enough to throw off the band fitting entirely resulting in a found bare-band that is likely not close to the true band, shown by generally poor agreement between MDC and KK self-energies. 144 Appendix B ARPES data preparation In order to analyze angle-resolved photoemission spectroscopy (ARPES) data, our group wrote a full object-oriented software suite in the Interactive Data Language (IDL), with many command-driven tools but with a graphical user interface (GUI) for most steps. This effort was undertaken largely by Giorgio Levy, Jonathan Rosen, and me. Each of us developed much of our own code regarding data management and workflow, with many core rou- tines (or pieces of them) shared and often co-authored. In order to transform the raw detector image into an intensity map over angle and energy, we generally followed a workflow as illustrated in Fig. B.1 to prepare ARPES data24. It is this data-preparation workflow that will be discussed here. Along with the data of interest, reference gold and slit-array images are also taken. The slit-array image is used to correct distortions in and characterize the angular direction of the image (described in \u00C2\u00A7B.1), while the gold image is used to define the Fermi energy and correct distortions in the energy direction (described in \u00C2\u00A7B.2). Finally, in order to transform the angular data into momentum space, a Fermi surface map is created from a series of images (all themselves slit and gold corrected) taken for varying sample angles. This Fermi surface map is used to characterize any offset angles from sample mounting as well as the sample work function to allow emission angle and kinetic energy to be accurately related to the electron\u00E2\u0080\u0099s origin in energy\u00E2\u0080\u0093momentum space, described in \u00C2\u00A7B.3. 145 Appendix B. ARPES data preparation Gold data Data of interest Slit-array data Multiple images at different angles, all slit/gold corrected angle (slit) correction pixels pi xe ls pixels pi xe ls pixels pi xe ls angle pi xe ls angle pi xe ls angle pi xe ls energy (gold) correction angle to k-space mapping angle en er gy angle en er gy angle en er gy momentum en er gy momentum en er gy momentum en er gy as acquired Fermi surface map angle an gl e momentum m om en tu m integrate at EF Figure B.1: An illustration of ARPES data preparation workflow to bring the raw image into energy\u00E2\u0080\u0093momentum space. 146 Appendix B. ARPES data preparation Figure B.2: A screen shot of IDL while defining peak positions in preparation for correcting angular distortions. 147 B.1. Angle (slit) correction B.1 Angle (slit) correction By inserting a slit-array between the sample and the analyzer (see Fig. 3.1) an image with lines evenly spaced in tan(\u00CE\u00B8) can be taken. The first step to transform the raw detector image into energy\u00E2\u0080\u0093momentum space is to use this slit-array image to characterize any distortions in the angular direction and warp the image to correct for them. Peak locations are first determined at a variety of energy slices along the image25, which are then fit to determine a smooth form for each line. These lines, along with the image data to be corrected, are then forced to have the correct spacing by individually warping each 1D energy slice. The use of a 1D transform prevents mixing in the direction the transform does not define and also allows attention to be paid to making sure the total intensity of each slice remains constant, as no readily available 2D warping routines have this feature26. B.2 Energy (gold) correction After correcting the angular distortions and establishing the angular axis, a similar procedure is applied in the energy direction using data taken on polycrystalline gold. By fitting a Fermi edge to each constant angle slice and performing a polynomial fit to the resultant Fermi edge locations, the form of the Fermi edge constant-energy contour can be found. The chemical potential is then set to this location and, since a warping cannot be defined 24This workflow is most relevant for quasi-2D systems, such as Sr2RuO4, where the location in kz is not of paramount importance owing to the small kz dispersion. 25Depending on data quality, defining the peak positions in an efficient manner can be non-trivial. While the peak location may be \u00E2\u0080\u009Cobvious\u00E2\u0080\u009D to a human, writing an algorithm that can quickly and reliably locate each peak with minimal human intervention in noisy data is non-trivial. Instead of a standard least-squares fit to a fence of many peaks, which would have many parameters requiring good initialization guesses, data is smoothed over a region and primary peak locations selected based on topographic prominence. 262D image warping routines are, in general, designed explicitly not to conserve total intensity, instead keeping local intensity the same. This is because if you warp a pho- tograph, you expect the colours not to change \u00E2\u0080\u0093 but in our case we do want the local intensity to change to reflect the conservation of intensity overall. 148 B.3. Angle to k-space mapping by a single constant-energy contour27, each momentum distribution curve (MDC) slice in the image is shifted so that the Fermi edge is straight. The scale of the energy axis is taken from the analyzer pass energy. B.3 Angle to k-space mapping After transforming all data into energy\u00E2\u0080\u0093angle space, the final step is to convert emission angle to the crystal momentum of the electron initial state. If the crystal were aligned perfectly, this would be straightforward, as the momentum in vacuum, determined from emission angles and kinetic energy, can be transformed from detector space to crystal space according to the known sample-holder geometry28. However, the small offset angles acquired when mounting the sample must be taken into account. To solve this problem and determine the offset angles, a large number of scans are taken at a variety of sample geometries, usually by incrementing the cryostat angle across its full range of travel in the direction perpen- dicular to the analyzer entrance slit. This data set can be integrated in a window near the Fermi energy, and a Fermi surface map generated using as- sumed offset angles and work function. These assumed parameters can then be varied until the Fermi surface map generated has the correct size and symmetries around kx = ky = 0. The GUI developed to perform and test these transforms for different conditions is shown in Fig. B.3. In addition to drawing grid lines whose spacing can be chosen based on the known recip- rocal lattice size, it is possible to overlay images on top of the transformed data \u00E2\u0080\u0093 in this case the results of a band structure Fermi surface calculation. 27The Fermi edge could be swept across the detector, by changing the kinetic energy of the analyzer, to generate constant energy contours at any location \u00E2\u0080\u0093 but there is no reason to assume that any warping present would be the same for different analyzer settings. 28Straightforward for the in-plane kx and ky directions, where momentum is conserved during photoemission owing to symmetry. The kz value of the initial state in crystal space is much harder to define, as this direction is not conserved, and calculating it requires knowledge of the inner potential. At a synchrotron the kz dispersion can be measured by varying photon energy continuously, and the inner potential inferred. However, using spectral lines as a light source does not provide enough data points to do this. Generally the kz location is either inferred from a comparison with band structure calculations, or a quasi-2D material is selected in which case the exact kz location is not important. 149 B.3. Angle to k-space mapping Figure B.3: A screen shot of the GUI from which offset angles are chosen; this GUI drives the angle-to\u00E2\u0080\u0093momentum space transformation. Note the \u00E2\u0080\u009Cphoton energy\u00E2\u0080\u009D label is a misnomer and should more properly be \u00E2\u0080\u009Cphoton energy minus work function\u00E2\u0080\u009D. 150 B.3. Angle to k-space mapping It is also possible to display composite images folded along high-symmetry directions (not shown). With careful consideration, angles defined in this manner are likely accurate to better than 0.75\u00E2\u0097\u00A6. Once the offset angles for a given sample are defined, the angular direction can be transformed into momentum space and is ready for more meaningful analysis. 151 Appendix C CPS-ARPES data preparation Here we will go over some details of circularly polarized photon spin- and angle-resolved photoemission spectroscopy (CPS-ARPES) omitted from Chap- ter 5 for the sake of readability. A discussion on the experimental setup and how the relative spin intensities in each direction are produced from the raw detector data is found in \u00C2\u00A7C.1. \u00C2\u00A7C.2 explores how the shape of the \u00E2\u0080\u009Cwiggle\u00E2\u0080\u009D measured in polarization is related to the shape and splitting of the under- lying peaks and why direct estimates based on the width of the wiggle alone will fail. For further information about spin- and angle-resolved photoemis- sion spectroscopy (SARPES) as an experiment, as well as a discussion of some results relating to Rashba splitting and topological insulators, both the review by Dil [117] as well as that by Meier, Dil, and Osterwalder [118] are highly recommended. C.1 How polarization is measured The analyzer/detector at the Complete Photoemission Experiment (COPHEE) endstation is roughly similar to ours at the University of British Columbia (UBC), with the exception that instead of a 2D detector, there is a hole that permits energy- and angle-filtered electrons to pass through and be focused and accelerated into a Mott detector array to discriminate spin29. 29Alternatively, the electrons can also be directed to a series of channeltron detectors, in cases where spin resolution is not required, such as data taken for sample alignment purposes. It should also be noted that the Mott detector array coordinate system is rotated with respect to that of the sample. 152 C.1. How polarization is measured Chopper Gold foil Detectors Gold foil Detectors x y z Polarimeter II Px, Pz Polarimeter I Py, Pz Incoming electrons Figure C.1: Illustration of the Mott detector setup at COPHEE. Electrons are discriminated in angle and energy by a hemispherical analyzer before being split between two pairs of Mott polarimeters at right angles, for a total of eight detectors. 153 C.1. How polarization is measured The Mott detector array consists of two sets of Mott detector pairs at 90\u00E2\u0097\u00A6 to each other; electrons are directed toward one or the other by a chopper operating at 2 Hz, as illustrated in Fig. C.1 [117, 118]. This results in a total of eight detectors, with the polarization in one direction (z) theoretically overdetermined, although during the measurements presented here one pair of z detectors was malfunctioning. The electron polarization P measured in any given direction, is simply the normalized difference in intensity between electrons measured as spin \u00E2\u0080\u009Cup\u00E2\u0080\u009D (I\u00E2\u0086\u0091) and \u00E2\u0080\u009Cdown\u00E2\u0080\u009D (I\u00E2\u0086\u0093) and is given by P = I\u00E2\u0086\u0091 \u00E2\u0088\u0092 I\u00E2\u0086\u0093 I\u00E2\u0086\u0091 + I\u00E2\u0086\u0093 , (C.1) i.e., a stream of pure \u00E2\u0080\u009Cup\u00E2\u0080\u009D electrons would have P = 1 and pure \u00E2\u0080\u009Cdown\u00E2\u0080\u009D would have P = \u00E2\u0088\u00921 regardless of the overall intensity. Mott scattering is inherently not very efficient at sorting spin30. The scattering asymmetry, A, between the intensities IL and IR in any given detector pair is given by A = IL \u00E2\u0088\u0092 IR IL + IR = PS, (C.2) where P is the electron polarization and where S, the Sherman function, is 0.068 at COPHEE31 \u00E2\u0080\u0093 i.e., each Mott pair is 6.8% efficient at sorting (detected) electrons left or right based on their incoming spin polarization. Additionally, we observe a strong circular dichroism \u00E2\u0080\u0093 the overall inten- sity varies depending on light polarization, with similar magnitude to the already small variations from electron polarization. For this reason a com- bination of measurements with opposite light polarizations is needed. Here 30Although inefficient, using Mott scattering is the best available method of sorting electron spin \u00E2\u0080\u0093 an inhomogeneous magnetic field, i.e. a Stern Gerlach experiment, does not work for charged particles. 31The Sherman \u00E2\u0080\u009Cfunction\u00E2\u0080\u009D is a function of the scattering energy, quality of the foil, and detector geometry; for our purposes it is an empirically determined constant. 154 C.1. How polarization is measured Energy (eV) -0.1 0.0 0.1 C ro ss -P ol ar iz at io n -0.1 0.0 0.1 C ro ss -P ol ar iz at io n -0.1 0.0 0.1 C ro ss -P ol ar iz at io n x` y` z` C ou nt s ( 10 6 ) 0.0 0.5 1.0 Energy (eV) 0.0 0.5 1.0 0.0 0.4 0.8 1.2 C ou nt s ( 10 6 ) 0.0 0.4 0.8 1.2 C ou nt s ( 10 6 ) 0.0 0.4 0.8 1.2 IL IR IR IL IL IR IR IL IL IR IR IL (a) (b) (c) (d) (e) (f) Figure C.2: Raw detector data from CPS-ARPES measurements on Sr2RuO4 at the \u00CE\u0093 point using 24 eV photon energy. (a,c,e) show the in- tensity at each left (L) / right (R) half of the (x, y, z) detector pairs for both circular plus (\u00E2\u008A\u0095) and minus (\t) light, which is transformed using the cross-asymmetry to find the electron polarization for each pair in detector space (b,d,f). These twelve data sets are used to produce the polarization and spin intensities in Fig. 5.5; note that, in this geometry, changing from detector to sample coordinate systems mixes x and y only. Error bars rep- resent statistical uncertainty plotted at 95% confidence. 155 C.1. How polarization is measured we use the geometric cross-asymmetry, A\u00CC\u0083\u00E2\u008A\u0097, defined as A\u00CC\u0083\u00E2\u008A\u0097 = \u00E2\u0088\u009A I\u00E2\u008A\u0095L I R \u00E2\u0088\u0092 \u00E2\u0088\u009A I\u00E2\u008A\u0095R I L\u00E2\u0088\u009A I\u00E2\u008A\u0095L I R + \u00E2\u0088\u009A I\u00E2\u008A\u0095R I L . (C.3) Using the geometric cross-asymmetry A\u00CC\u0083\u00E2\u008A\u0097 has the advantage that it can correct for both circular dichroism as well as detector efficiencies, assuming the intensity at each detector can be written in the following form: I\u00E2\u008A\u0095L = D \u00E2\u008A\u0095EL \u00E2\u0086\u0091, I\u00E2\u008A\u0095R = D\u00E2\u008A\u0095ER \u00E2\u0086\u0093, I\tL = D \tEL \u00E2\u0086\u0093, and I\tR = D\tER \u00E2\u0086\u0091 . (C.4) Here D\u00E2\u008A\u0095 and D\t are multiplicative factors representing the circular dichro- ism, EL and ER represent possibly different linear detector efficiencies, while \u00E2\u0086\u0091 and \u00E2\u0086\u0093 represent the photoemitted states in which we are interested \u00E2\u0080\u0093 those that (may) change scattering probability with light polarization. It should be noted that the sometimes-used arithmetic cross-asymmetry, given by ( (I\u00E2\u008A\u0095L + I R )\u00E2\u0088\u0092 (I\u00E2\u008A\u0095R + I\tL ) ) / ( (I\u00E2\u008A\u0095L + I R ) + (I \u00E2\u008A\u0095 R + I L ) ) , does not have this property \u00E2\u0080\u0093 it will correct for either D\u00E2\u008A\u0095 6= D\t or EL 6= ER but not both at the same time. The 12 raw energy distribution curves (EDCs) (3 directions \u00C3\u0097 2 detectors \u00C3\u0097 2 light polarizations) gathered at the \u00CE\u0093 point by sweeping the analyzer kinetic energy and used to generated the polarization, total intensity, and spin intensities in Fig. 5.5 are presented in Fig. C.2. Here the strong circular dichroism can be seen as the difference in average value between the data taken with \u00E2\u008A\u0095 and \t, with \t showing greater intensity overall32. Possibly differing detector efficiencies can be seen as well, with the L, L, and R detectors returning a consistently higher number of counts in the x, y, and 32Note that this scaled difference in intensity between \u00E2\u008A\u0095 and \t was also seen to exist on polycrystalline copper, so some of the intensity difference could also be from the in- tensity of light produced by the beamline undulator for different polarizations, but this will be corrected for in the cross-asymmetry in both cases. Possible accidental photon energy shifts also associated with changing light polarization were evaluated on the same polycrystalline copper sample, and this was not found to occur. 156 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? z directions, respectively. Finally, it can be readily seen that all curves (by eye) appear to be almost identical rescalings of the same curve \u00E2\u0080\u0093 it is for this reason that such high count rates are necessary in order to properly determine the slight difference between them and why, despite having \u00E2\u0088\u00BC106 counts, the error in polarization still ends up being substantial. Finally, once the cross polarizations in each direction are determined, the associated spin intensities can be worked out by rearranging Eq. C.1 with IT = I\u00E2\u0086\u0091 + I\u00E2\u0086\u0093 to get I\u00E2\u0086\u0091 = IT (1 + P\u00CC\u0083\u00E2\u008A\u0097)/2, I\u00E2\u0086\u0093 = IT (1\u00E2\u0088\u0092 P\u00CC\u0083\u00E2\u008A\u0097)/2. (C.5) C.2 Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? Looking at the cross-polarization in Fig. C.2(f), one might estimate the width of the \u00E2\u0080\u009Cwiggle\u00E2\u0080\u009D to be \u00E2\u0088\u00BC 500 meV. That the estimate of splitting be- tween these two states ends of being a factor of five or so lower than that may then initially come as a surprise. For this reason we will consider here, for a toy model where all parameters are controlled, the appearance of the \u00E2\u0080\u009Cwiggle\u00E2\u0080\u009D that would be measured in polarization as a function of peak width, splitting, and background intensity for both Lorentzian and Gaussian lineshapes. Our model, as the states measured in Chapter 5, consists of a pair of spin\u00E2\u0080\u0093 orbit (SO) split states, where the splitting between the two states is (usually) not sufficiently large compared with peak width to resolve each individually with regular photoemission33, as illustrated in Fig. C.3. In Fig. C.3 we see that the width of the \u00E2\u0080\u009Cwiggle\u00E2\u0080\u009D is actually comparable to the width of the splitting, but we will soon see that this is only a coincidence for the particular parameters used. Varying the shape of the measured polarization as a function of peak 33Why you cannot resolve each peak, whether because of a fundamental broadening or simply instrumental resolution, is not important. 157 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? Total signal, as with regular photoemission momentum parallel momentum antiparallel ml = +1, s = -1/2 ml = -1, s = +1/2 ml = +1, s = +1/2 ml = -1, s = -1/2 (b) Circ+ light (c) Circ- light Po lar iza tio n Po lar iza tio n ml = +2, s = -1/2 ml = 0, s = +1/2 ml = +2, s = +1/2 ml = 0, s = -1/2 ml = 0, s = -1/2 ml = -2, s = +1/2 ml = 0, s = +1/2 ml = -2, s = -1/2 not allowed allowed allowed not allowed Energy Energy Int en sit y (a) Unpolarized light (ml ml + 1) (ml ml - 1) background signal Energy Figure C.3: Spin-polarized photoemission from a model SO-split system showing (a) total intensity from all sources and polarization measured using circular plus (b) or minus (c) light. 158 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0 0.5 1 1.5 2 -0.8 -0.4 0 0.4 0.8 -0.6 -0.2 0.2 0.6 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 splitting -10 -5 0 5 10 -10 -5 0 5 10 Int en sit y ( a.u .) Di ffe ren ce (a .u. ) Position (a.u.) Position (a.u.) (a) (c) (e) (b) (d) (f) Po lar iza tio n Figure C.4: Peak splitting simulation example, varying the splitting in a background-free environment, for Lorentzian (a,c,e) and Gaussian (b,d,f) lineshapes of constant unit width. (a,b) show the original peaks (thin lines) and sum (thick lines), while (c,d) show the difference of the two peaks. (e,f) show the polarization \u00E2\u0080\u0093 i.e., the normalized difference (difference divided by sum). 159 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? 0 0.5 1 1.5 2 2.5 -1.5 -1 -0.5 0 0.5 1 1.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 0.5 1 1.5 2 -2 -1 0 1 2 -1 -0.5 0 0.5 1 width 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 -10 -5 0 5 10 -10 -5 0 5 10 Int en sit y ( a.u .) Di ffe ren ce (a .u. ) Position (a.u.) Position (a.u.) (a) (c) (e) (b) (d) (f) Po lar iza tio n Figure C.5: Peak splitting simulation example, varying the peak width in a background-free environment, for Lorentzian (a,c,e) and Gaussian (b,d,f) lineshapes of constant unit splitting. (a,b) show the original peaks (thin lines) and sum (thick lines), while (c,d) show the difference of the two peaks. (e,f) show the polarization \u00E2\u0080\u0093 i.e., the normalized difference (difference di- vided by sum). 160 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? 0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000 background 0 1 2 3 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -10 -5 0 5 10 -0.4 -0.2 0 0.2 0.4 0 1 2 3 -0.8 -0.4 0 0.4 0.8 -10 -5 0 5 10 -1 -0.5 0 0.5 1 Int en sit y ( a.u .) Di ffe ren ce (a .u. ) Position (a.u.) Position (a.u.) (a) (c) (e) (b) (d) (f) Po lar iza tio n Figure C.6: Peak splitting simulation example, varying the background, for Lorentzian (a,c,e) and Gaussian (b,d,f) lineshapes of constant unit width and splitting. (a,b) show the original peaks (thin lines) and sum (thick lines), while (c,d) show the difference of the two peaks. (e,f) show the polarization \u00E2\u0080\u0093 i.e., the normalized difference (difference divided by sum). 161 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? 0.5 1 1.5 2 2.5 3 -0.8 -0.4 0 0.4 0.8 -0.4 -0.2 0 0.2 0.4 0.5 1 1.5 2 2.5 3 -1 -0.5 0 0.5 1 -0.4 -0.2 0 0.2 0.4 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 splitting -10 -5 0 5 10 -10 -5 0 5 10 Int en sit y ( a.u .) Di ffe ren ce (a .u. ) Position (a.u.) Position (a.u.) (a) (c) (e) (b) (d) (f) Po lar iza tio n Figure C.7: Peak splitting simulation example, varying the splitting with a background of 0.5, for Lorentzian (a,c,e) and Gaussian (b,d,f) lineshapes of constant unit width. (a,b) show the original peaks (thin lines) and sum (thick lines), while (c,d) show the difference of the two peaks. (e,f) show the polarization \u00E2\u0080\u0093 i.e., the normalized difference (difference divided by sum). 162 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? width 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.5 1 1.5 2 2.5 3 3.5 -1.5 -1 -0.5 0 0.5 1 1.5 -0.4 -0.2 0 0.2 0.4 0.5 1 1.5 2 2.5 3 -2 -1 0 1 2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -10 -5 0 5 10 -10 -5 0 5 10 Int en sit y ( a.u .) Di ffe ren ce (a .u. ) Po lar iza tio n Position (a.u.) Position (a.u.) (a) (c) (e) (b) (d) (f) Figure C.8: Peak splitting simulation example, varying the width with a background of 0.5, for Lorentzian (a,c,e) and Gaussian (b,d,f) lineshapes of constant unit splitting. (a,b) show the original peaks (thin lines) and sum (thick lines), while (c,d) show the difference of the two peaks. (e,f) show the polarization \u00E2\u0080\u0093 i.e., the normalized difference (difference divided by sum). 163 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? shape, background levels, and width is shown in Figs. C.4\u00E2\u0080\u0093C.8. In the first cases, as we vary both splitting (Fig. C.4) and width (Fig. C.5), the polar- ization behaves vastly differently between peak shapes. In the Lorentzian case, where the intensity drops off slowly as 1/x2, the peak shape takes on a wiggle; conversely in the Gaussian case, where the peak shape drops off quickly as e\u00E2\u0088\u0092x2 , the polarization asymptotically approaches 1 and \u00E2\u0088\u00921 at the extremes. This difference, due to the tails and in limits that would not be possible to measure experimentally owing to zero intensity, changes quickly in the presence of a background intensity, as shown in Fig. C.6, where a background is slowly added. This causes the polarization to quickly become a wiggle in both cases (and, in fact, any case where the intensity in the peaks goes to zero). Figs. C.7 and C.8 show the same progressively changing split- ting and width as Fig. C.4 and C.5 only with a background intensity, closer to what might be expected in a real experiment. Fig. C.7 shows that (in the parameter regime shown where the peaks overlap strongly), as splitting is increased, most of the modulation is not in the shape or width of the wiggle but in its amplitude, while Fig. C.8 shows that changing peak shape (rather than splitting) can strongly change both the amplitude as well as the shape and width of the wiggle. In fact, for the case of two Lorentzians with equal area and width with no background, it is possible to solve exactly for the width (defined as the distance from maxima to minima), amplitude (max to min), and the slope in the middle of the wiggle in the polarization: Width = \u00E2\u0088\u009A FWHM2 + Splitting2 (C.6) Amplitude = 2 Splitting\u00E2\u0088\u009A FWHM2 + Splitting2 . (C.7) Slope0 = 4 Splitting FWHM2 + Splitting2 . (C.8) In this case, and where the splitting is small compared with the peak widths, the amplitude of the wiggle and slope in the middle scale linearly with split- ting, while the width is a constant. Given these simple forms, a natural temptation would be to combine these easily measured quantities to re- 164 C.2. Polarization from a toy model \u00E2\u0080\u0093 what\u00E2\u0080\u0099s in a wiggle? trieve the splitting (for example, as Width \u00C2\u00B7 Amplitude/2); however, this approximation fails quickly34 for non-zero background. While real-life cases of asymmetric lineshapes and non-constant back- grounds will complicate matters, in general it appears that \u00E2\u0080\u0093 as a rough rule of thumb for peaks very broad compared with the splitting \u00E2\u0080\u0093 the width of the splitting is related more strongly to the amplitude of the \u00E2\u0080\u009Cwiggle\u00E2\u0080\u009D than to its width, although in a way that depends strongly on the background, peak form, and peak width. In general, making quantitative estimates of the magnitude peak splitting directly from the shape of the wiggle in polar- ization is not recommended; it must be combined with the measured total intensity. 34This approximation fails faster than linearly with increasing background and would (for example) underestimate the splitting by a factor of 2 for peaks with a splitting of 0.1, full width at half maximum (FWHM) 2 and area 1 once the background had reached 0.1. 165 Appendix D Software routines A large quantity of software was written for the analysis of both angle- resolved photoemission spectroscopy (ARPES) (138 000 lines of code35), and circularly polarized photon spin- and angle-resolved photoemission spec- troscopy (CPS-ARPES) data (2600 lines of code36) in the Interactive Data Language (IDL). However, none of that code is reproduced here; the level of interdependency between routines and the fact that IDL is a proprietary language does not make that practical. Here we include one routine \u00E2\u0080\u0093 that written in C++ and used to perform the Monte Carlo integration using the GNU scientific library [51] VEGAS [49, 50] algorithm. This is included because it is relatively short, is self- contained, requires only open-source libraries, and would be straightforward to modify in order to calculate the same integral based on a different coupling or bare-band. Evaluating such an integral with a deterministic integration engines, especially in three dimensions, is also liable to take a very long time. D.1 Monte Carlo integration using the GNU VEGAS algorithm The code works by evaluating both the bare-band and the band as predicted by perturbation theory for \u00CE\u00BB = \u00E2\u0088\u0086\u00CE\u00BB at k = 0 and k = \u00E2\u0088\u0086k, where \u00E2\u0088\u0086k was set 35As counted by CLOC 1.56 and roughly divided as 20k managed by J. A. Rosen, 86k managed by G. Levy, 2k contributed by summer students, and 30k that I managed \u00E2\u0080\u0093 although there has been a lot of intermingling, particularly between Levy and me, and Levy\u00E2\u0080\u0099s code base also includes some libraries modified to suit our own purposes. This total also includes some software for low-energy electron diffraction (LEED) and x-ray photoelectron spectroscopy (XPS) analysis. 36Managed solely by me. 166 D.1. Monte Carlo integration using the GNU VEGAS algorithm to 0.005 and \u00E2\u0088\u0086\u00CE\u00BB was set to 0.005. Setting either of these values too high could result in an error because the band or renormalization can no longer be approximated as linear, and too low could result in an error because the renormalization becomes difficult to detect. The main body of the code is Integrator.cpp, which relies on a def- inition for the coupling as provided by either Holstein.cpp or Breath- ing.cpp, using shared parameters found in shared.h. For speed, these are linked at compile time \u00E2\u0080\u0093 an example makefile is included, which cre- ates two binaries named Holstein and Breathing, as well as a bash script do them all to run all the calculations necessary to produce Figs. 2.7 and 2.8. D.1.1 shared.h //special variable types used by Integrator, Holstein and Breathing struct model_params { double t; double Omega; double g2; size_t dimension; char cut; size_t monte_calls; size_t monte_warmups; double delta_k;}; struct integral_params { struct model_params params; double *k; }; D.1.2 Holstein.cpp #include #include #include #include #include \"shared.h\" using namespace std; string modelname(){ return \"Holstein Polaron\"; } 167 D.1. Monte Carlo integration using the GNU VEGAS algorithm double numerator_1d (double l[1], double k[1], model_params params){ return 1; } double numerator_2d (double l[2], double k[2], model_params params){ return 1; } double numerator_3d (double l[3], double k[3], model_params params){ return 1; } D.1.3 Breathing.cpp #include #include #include #include #include #include \"shared.h\" using namespace std; string modelname(){ return \"Breathing Mode Coupling\"; } double numerator_1d (double l[1], double k[1], model_params params){ return double(2) * gsl_pow_int( sin( (l[0] - k[0])/double(2) ) , 2); } double numerator_2d (double l[2], double k[2], model_params params){ return double(2) * gsl_pow_int( sin( (l[0] - k[0])/double(2) ) 168 D.1. Monte Carlo integration using the GNU VEGAS algorithm + sin( (l[1] - k[1])/double(2) ) , 2); } double numerator_3d (double l[3], double k[3], model_params params){ return double(2) * gsl_pow_int( sin( (l[0] - k[0])/double(2) ) + sin( (l[1] - k[1])/double(2) ) + sin( (l[2] - k[2])/double(2) ) , 2); } D.1.4 Integrator.cpp #include #include #include #include #include #include #include #include \"shared.h\" using namespace std; //function prototypes for what\u00E2\u0080\u0099s in either Holstein or Breathing, //depending on what compiled library is linked string modelname(); double numerator_1d (double l[1], double k[1], model_params params); double numerator_2d (double l[2], double k[2], model_params params); 169 D.1. Monte Carlo integration using the GNU VEGAS algorithm double numerator_3d (double l[3], double k[3], model_params params); //Inline functions for readability and changability but retaining speed inline double bareband_1d (double k[1], model_params params){ return -double(2)*(params.t)*( cos(k[0]) ); } inline double bareband_2d (double k[2], model_params params){ return -double(2)*(params.t)*( cos(k[0]) +cos(k[1]) ); } inline double bareband_3d (double k[3], model_params params){ return -double(2)*(params.t)*( cos(k[0]) + cos(k[1]) + cos(k[2]) ); } //but use the above when concerned about speed double bare_band (double k[], model_params params){ double bareband_energy; switch (params.dimension) { case 1: bareband_energy = bareband_1d(k,params); break; case 2: bareband_energy = bareband_2d(k,params); break; case 3: bareband_energy = bareband_3d(k,params); break; } 170 D.1. Monte Carlo integration using the GNU VEGAS algorithm return bareband_energy; } inline double denominator_1d (double l[1], double k[1], model_params params){ return (params.Omega) - bareband_1d(k,params) + bareband_1d(l,params); } inline double denominator_2d (double l[2], double k[2], model_params params){ return (params.Omega) - bareband_2d(k,params) + bareband_2d(l,params); } inline double denominator_3d (double l[3], double k[3], model_params params){ return (params.Omega) - bareband_3d(k,params) + bareband_3d(l,params); } //The meat of things double integrate_this_1d (double l[1], size_t dim, void * p) { struct integral_params * i_params = (struct integral_params *) p; return numerator_1d( l, i_params->k, i_params->params ) / denominator_1d( l, i_params->k, i_params->params ); } double integrate_this_2d (double l[2], size_t dim, void * p) { struct integral_params * i_params = (struct integral_params *) p; return numerator_2d( l, i_params->k, i_params->params ) / denominator_2d( l, i_params->k, i_params->params ); 171 D.1. Monte Carlo integration using the GNU VEGAS algorithm } double integrate_this_3d (double l[3], size_t dim, void * p) { struct integral_params * i_params = (struct integral_params *) p; return numerator_3d( l, i_params->k, i_params->params ) / denominator_3d( l, i_params->k, i_params->params ); } void display_results(double result, double error){ cout << \"integral value : \" << result; cout << \"; estimated error : \" << error << \"\n\"; } double perturbation_energy (double k[], model_params params, double *error, gsl_rng *r){ //my variables double prefactor = gsl_pow_int( double(2) * M_PI , -params.dimension); //int monte_iterations_counter = 0; //gsl monte integral setup double res, err; double lb_1d[] = {-M_PI}; double lb_2d[] = {-M_PI,-M_PI}; double lb_3d[] = {-M_PI,-M_PI,-M_PI}; double ub_1d[] = {M_PI}; double ub_2d[] = {M_PI,M_PI}; double ub_3d[] = {M_PI,M_PI,M_PI}; double *lb; double *ub; 172 D.1. Monte Carlo integration using the GNU VEGAS algorithm double chisq; //cout << \"\u00E2\u0080\u0099\" << gsl_rng_get(r) << \"\u00E2\u0080\u0099\"; gsl_monte_vegas_state *s = gsl_monte_vegas_alloc(params.dimension); integral_params int_params = {params,k}; gsl_monte_function integral; integral.params= &int_params; integral.dim=params.dimension; // setup the integral differently in different dimensions... switch (params.dimension) { case 1: integral.f = &integrate_this_1d; //cout << \"calculating perturbed band for k = \" // << k[0] << endl; lb=lb_1d; ub=ub_1d; break; case 2: integral.f = &integrate_this_2d; //cout << \"calculating perturbed band for k = \" // << k[0] << \",\" << k[1] << endl; lb=lb_2d; ub=ub_2d; break; case 3: integral.f = &integrate_this_3d; //cout << \"calculating perturbed band for k = \" // << k[0] << \",\" << k[1] << \",\" << k[2] << endl; lb=lb_3d; ub=ub_3d; 173 D.1. Monte Carlo integration using the GNU VEGAS algorithm break; } //cout << \"warming-up integration grid\" << endl; gsl_monte_vegas_integrate (&integral, lb, ub, params.dimension, params.monte_warmups, r, s, &res, &err); //cout << \" warm-up complete, \"; //display_results(res,err); //cout << \"converging...\" << flush; do { // if (monte_iterations_counter == 1 ) // { // cout << \" err est off: chisq/dof= \" << chisq << flush; // } gsl_monte_vegas_integrate (&integral, lb, ub, params.dimension, params.monte_calls/5, r, s, &res, &err); chisq = gsl_monte_vegas_chisq(s); //chisq per DOF. Should be close to 1 or error is underestimated // if (monte_iterations_counter > 0) // { // cout << \" ... \" << chisq << flush; // } // monte_iterations_counter=monte_iterations_counter+1; } while (fabs (chisq - 1.0) > 0.4); //cout << \"converged, \"; //display_results (res, err); 174 D.1. Monte Carlo integration using the GNU VEGAS algorithm gsl_monte_vegas_free(s); *error = - prefactor * params.g2 *err; return - prefactor * params.g2 * res; } double perturbed_band (double k[], model_params params, double *error, gsl_rng *r){ return bare_band(k,params) + perturbation_energy(k,params,error, r); } inline double d2_from_points (double f_zero, double f_delta, double delta){ // assumes it\u00E2\u0080\u0099s even about zero (ie - 1st deriv is zero) return double(2)*(f_delta - f_zero)/gsl_pow_2(delta); } double renorm_at_0 (model_params params, double *error_in_renorm, gsl_rng *r){ 175 D.1. Monte Carlo integration using the GNU VEGAS algorithm double pert_0, pert_delta, pert_d2; double bare_0, bare_delta, bare_d2; double integral_error_0, integral_error_delta; double renorm, error_in_d2, perror_in_d2; double k0[params.dimension]; double kd[params.dimension]; //find second deriv bb and setup the k vectors //depending on what slice you pick //note that d2 is not strictly correct on the diagonals, //but that\u00E2\u0080\u0099s ok because the sqrt(2) or sqrt(3) cancels out in the ratio. switch (params.dimension) { case 1: k0[0]=double(0); kd[0]=params.delta_k; break; case 2: k0[0]=double(0); k0[1]=double(0); kd[0]=params.delta_k; switch (params.cut) { case \u00E2\u0080\u0099s\u00E2\u0080\u0099: kd[1]=double(0); break; case \u00E2\u0080\u0099d\u00E2\u0080\u0099: kd[1]=params.delta_k; break; } break; case 3: k0[0]=double(0); k0[1]=double(0); k0[2]=double(0); 176 D.1. Monte Carlo integration using the GNU VEGAS algorithm kd[0]=params.delta_k; switch (params.cut) { case \u00E2\u0080\u0099s\u00E2\u0080\u0099: kd[1]=double(0); kd[2]=double(0); break; case \u00E2\u0080\u0099d\u00E2\u0080\u0099: kd[1]=params.delta_k; kd[2]=params.delta_k; break; } break; } bare_0 = bare_band(k0, params); bare_delta = bare_band(kd, params); bare_d2 = d2_from_points(bare_0, bare_delta, params.delta_k); pert_0 = perturbation_energy(k0, params, &integral_error_0 , r); pert_delta = perturbation_energy(kd, params, &integral_error_delta, r); pert_d2 = d2_from_points(pert_0, pert_delta, params.delta_k); renorm = bare_d2/(pert_d2+bare_d2); //simple multiplication was done to get d2 from delta // - so the error scales perror_in_d2 = fabs( (integral_error_0+integral_error_delta) /(pert_delta-pert_0) ); error_in_d2 = fabs( pert_d2*perror_in_d2 ); //likewise for renormalization (but now the error is on dispersion) *error_in_renorm = fabs( renorm * error_in_d2/(pert_d2+bare_d2) ); return renorm; } 177 D.1. Monte Carlo integration using the GNU VEGAS algorithm void cmd_error_message(char *argv[], double monte_calls, double delta_k, double lambda, double delta_omrat, double omrat_min, double omrat_max) { cout << \"You must supply at least 3 arguments:\" << \"dimension (1,2,3) ; cut (to side or diag) ; filename\" << endl; cout << \" ie: $ \" << argv[0] << \" 2 side output.dat -option value\" << endl; cout << \"optional flags must go at the end and are:\" << endl; cout << \" -monte_calls (currently \" << monte_calls << \" )\" << endl; cout << \" -delta_k (currently \" << delta_k << \" )\" << endl; cout << \" -delta_lambda (currently \" << lambda << \" )\" << endl; cout << \" -delta_omrat (currently \" << delta_omrat << \" )\" << endl; cout << \" -omrat_min (currently \" << omrat_min << \" )\" << endl; cout << \" -omrat_max (currently \" << omrat_max << \" )\" << endl; } //main actually makes \"figure 3\" - slope of renormalization at k=0 lambda->0 int main (int argc, char *argv[]) { double t=1.0; double lambda=0.005; //the value to check renormalization at //double omrat=1.0; size_t dimension; 178 D.1. Monte Carlo integration using the GNU VEGAS algorithm char cut;// = \u00E2\u0080\u0099s\u00E2\u0080\u0099; //\"s\" for side or \"d\" for diagonal string outputfilename; size_t monte_calls=500000; size_t monte_warmups=10000; double delta_k = 0.005; double delta_omrat=0.02; double omrat_min=0.02; double omrat_max=2.001; //read from the command line if (argc < 4) { cmd_error_message(argv,monte_calls,delta_k,lambda, delta_omrat,omrat_min,omrat_max); return 1; } bool validarg; validarg = ( sscanf(argv[1],\"%zd\",&dimension) == 1 ); cut = argv[2][0]; outputfilename = (string)argv[3]; if ( dimension > 3 || ! validarg || (cut != \u00E2\u0080\u0099s\u00E2\u0080\u0099 && cut != \u00E2\u0080\u0099d\u00E2\u0080\u0099) ) { cmd_error_message(argv,monte_calls,delta_k,lambda, delta_omrat,omrat_min,omrat_max); 179 D.1. Monte Carlo integration using the GNU VEGAS algorithm return 1; } for (int i=4; i < argc; i=i+2) { validarg=0; if (strcmp(argv[i], \"-monte_calls\") == 0 && (i+1) < argc ) { validarg = ( sscanf(argv[i+1],\"%zd\",&monte_calls) == 1 ); } if (strcmp(argv[i], \"-delta_k\") == 0 && (i+1) < argc ) { validarg = ( sscanf(argv[i+1],\"%lf\",&delta_k) == 1 ); } if (strcmp(argv[i], \"-delta_lambda\") == 0 && (i+1) < argc ) { validarg = ( sscanf(argv[i],\"%lf\",&lambda) == 1 ); } if (strcmp(argv[i], \"-delta_omrat\") == 0 && (i+1) < argc ) { validarg = ( sscanf(argv[i+1],\"%lf\",&delta_omrat) == 1 ); } if (strcmp(argv[i], \"-omrat_min\") == 0 && (i+1) < argc ) { validarg = ( sscanf(argv[i+1],\"%lf\",&omrat_min) == 1 ); } if (strcmp(argv[i], \"-omrat_max\") == 0 && (i+1) < argc ) { validarg = ( sscanf(argv[i+1],\"%lf\",&omrat_max) == 1 ); } if ( ! validarg ) { cmd_error_message(argv,monte_calls,delta_k,lambda, 180 D.1. Monte Carlo integration using the GNU VEGAS algorithm delta_omrat,omrat_min,omrat_max); cout << \"Your flag: \" << argv[i] << endl; return 1; } } //monte_calls=monte_calls * dimension; //done initializing command line arguments //make a plot of slope of renormalization double omrat, renorm, slope, r_error, s_error; omrat=omrat_min; cout << modelname() << \" pertubation renormalization\" << endl; cout << \" Dimension : \" << dimension << endl; cout << \" Cut : \" << cut << endl; cout << \" t : \" << t << endl; cout << \" delta_lambda : \" << lambda << endl; cout << \" delta_k : \" << delta_k << endl; cout << \" monte_warmups : \" << monte_warmups << endl; cout << \" monte_calls : \" << monte_calls << endl; cout << \" delta_omrat : \" << delta_omrat << endl; cout << \" omrat_min : \" << omrat_min << endl; cout << \" omrat_max : \" << omrat_max << endl << endl; cout << \"Outputting to file: \" << outputfilename << endl; //open output file ofstream output(outputfilename.c_str(),ios::ate); output.precision(6); 181 D.1. Monte Carlo integration using the GNU VEGAS algorithm output << \"# \" << modelname() << \" in Dimension : \" << dimension << endl; output << \"# Cut : \" << cut << endl; output << \"# t : \" << t << endl; output << \"# delta_lambda : \" << lambda << endl; output << \"# delta_k : \" << delta_k << endl; output << \"# monte_warmups : \" << monte_warmups << endl; output << \"# monte_calls : \" << monte_calls << endl; output << \"# omrat \t renorm_slope \t abs_error_in\" << endl; //random number generator for the loop const gsl_rng_type *T; gsl_rng *r; gsl_rng_env_setup (); T = gsl_rng_default; r = gsl_rng_alloc (T); //time junk for the loop double ratiodone; time_t started, finished, now, last, difference, future; struct tm * timeinfo; time(&started); timeinfo = localtime(&started); cout << \"Started: \" << asctime(timeinfo) << endl; last=started; //the loop do { double Omega=omrat*t*double(2); // omrat := Omega/2t double g2=lambda*t*Omega*double(dimension*2); // lambda := g^2 / 2 t D Omega 182 D.1. Monte Carlo integration using the GNU VEGAS algorithm model_params params={ t, Omega, g2 , dimension , cut , monte_calls , monte_warmups , delta_k }; renorm = renorm_at_0(params,&r_error,r); slope = (renorm-double(1)) / lambda; s_error = r_error / lambda; output << omrat << \"\t\" << slope << \"\t\" << s_error << endl; ratiodone=(omrat-omrat_min+delta_omrat)/(omrat_max-omrat_min); cout << (int)( 100*ratiodone ) << \"% done; \"; cout << \" omrat= \" << omrat << \", slope= \" << slope; //cout << \", %err= \" << (100*s_error/slope); cout << \", abserr= \" << s_error; time(&now); cout << \", took: \" << difftime(now,last)/60 << \"min\"; last=now; difference = now - started; future = now + difference*(1-ratiodone)/ratiodone; timeinfo = localtime(&future); cout << \", done \" << asctime(timeinfo) << flush; omrat+=delta_omrat; } while (omrat <= omrat_max); //free random number generator gsl_rng_free(r); //time junk time(&finished); 183 D.1. Monte Carlo integration using the GNU VEGAS algorithm timeinfo = localtime(&finished); cout << \"Finished: \" << asctime(timeinfo) << endl; cout << \"Took: \" << difftime(finished,started)/3600 << \" hours\" << endl; output.close(); return 0; } D.1.5 makefile #compiler CPP = g++ #dirs INC_DIR = /opt/local/include LIB_DIR = /opt/local/lib INC_DIR2 = /usr/local/include LIB_DIR2 = /usr/local/lib # flags COMP_FLAGS = -Wall -Wno-unknown-pragmas -c -I$(INC_DIR) -I$(INC_DIR2) LINK_FLAGS = -lgsl -lgslcblas -lm -L$(LIB_DIR) -L$(LIB_DIR2) #sources HOLST_OBJECT = Holstein.o HOLST_SOURCE = Holstein.cpp BREAT_OBJECT = Breathing.o BREAT_SOURCE = Breathing.cpp SHARED_OBJECT = Integrator.o 184 D.1. Monte Carlo integration using the GNU VEGAS algorithm SHARED_SOURCE = Integrator.cpp OBJECTS = $(HOLST_OBJECT) $(BREAT_OBJECT) $(SHARED_OBJECT) Holstein: $(HOLST_OBJECT) $(SHARED_OBJECT) $(CPP) $(LINK_FLAGS) $(HOLST_OBJECT) $(SHARED_OBJECT) -o Holstein Breathing: $(BREAT_OBJECT) $(SHARED_OBJECT) $(CPP) $(LINK_FLAGS) $(BREAT_OBJECT) $(SHARED_OBJECT) -o Breathing Holstein.o: $(HOLST_SOURCE) $(CPP) $(COMP_FLAGS) $(HOLST_SOURCE) Breathing.o: $(BREAT_SOURCE) $(CPP) $(COMP_FLAGS) $(BREAT_SOURCE) Integrator.o: $(SHARED_SOURCE) $(CPP) $(COMP_FLAGS) $(SHARED_SOURCE) .PHONY clean: rm -f *~ $(OBJECTS) Holstein Breathing htest: Holstein ./Holstein 1 s test.dat -monte_calls 50000 btest: Breathing ./Breathing 1 s test.dat -monte_calls 50000 185 D.1. Monte Carlo integration using the GNU VEGAS algorithm D.1.6 do them all ./Holstein 1 s 1Ds_Holstein.dat -monte_calls 5000000 ./Holstein 2 s 2Ds_Holstein.dat -monte_calls 50000000 ./Holstein 2 d 2Dd_Holstein.dat -monte_calls 50000000 ./Holstein 3 s 3Ds_Holstein.dat -monte_calls 500000000 ./Holstein 3 d 3Dd_Holstein.dat -monte_calls 500000000 ./Breathing 1 s 1Ds_Breathing.dat -monte_calls 5000000 ./Breathing 2 s 2Ds_Breathing.dat -monte_calls 50000000 ./Breathing 2 d 2Dd_Breathing.dat -monte_calls 50000000 ./Breathing 3 s 3Ds_Breathing.dat -monte_calls 500000000 ./Breathing 3 d 3Dd_Breathing.dat -monte_calls 500000000 186"@en . "Thesis/Dissertation"@en . "2013-05"@en . "10.14288/1.0073781"@en . "eng"@en . "Physics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "Attribution-NonCommercial-NoDerivs 3.0 Unported"@en . "http://creativecommons.org/licenses/by-nc-nd/3.0/"@en . "Graduate"@en . "Spin- and angle-resolved photoemission spectroscopy on unconventional superconductor strontium ruthenate"@en . "Text"@en . "http://hdl.handle.net/2429/44166"@en .