Transport and Superconductivity in Spin-Orbit Coupled Electron Systems
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Transport and Superconductivity in Spin-Orbit Coupled Electron Systems by Joel Gordon Hutchinson A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics University of Alberta c Joel Gordon Hutchinson, 2019 Abstract This thesis provides a theoretical discussion of several new phenomena associated with spin-orbit coupling in systems that lack inversion symmetry. Chapter 1 gives an in- troduction to the context of spin-orbit coupling in condensed matter physics and the role of inversion symmetry breaking. The remainder of the thesis is divided into two parts. Part 1 explores the effects of spin-orbit coupling on low-energy electron-impurity scattering. First, we study the single-particle scattering problem (Chapters 3 and 4) and find a host of unusual properties at ultra-low energies, including a quantized cross section. Chapter 5 extends these results to transport in a many-body system, where the quantized cross section manifests itself as a quantized conductivity. Part 2 explores the effect of spin-orbit coupling on superconductivity within BCS theory. In Chapter 7, we find that the critical temperature can be tuned by this coupling. In Chapter 8 we discuss the symmetry of the superconducting order parameter in the presence of at- tractive interactions between nearest-neighbour electrons. There we find that the gap function symmetry can change as a function of all the material parameters, including the temperature and spin-orbit coupling. ii Preface Chapters 3, 4, 5, and 7 of this dissertation are an elaboration and summary, with permis- sion, of references [1{5], of which I am the first author. Many results are taken directly from these papers. All figures and tables are my original work. Joseph Maciejko super- vised the projects outlined in Part 1. Frank Marsiglio supervised the projects outlined in Part 2. The setup and motivation for Chapter 7 was provided by our collaborator Jorge Hirsch. iii \The task is, not so much to see what no one has yet seen; but to think what nobody has yet thought, about that which everybody sees." Erwin Schr¨odinger iv Acknowledgements I would like to start by thanking both of my supervisors Joseph Maciejko and Frank Marsiglio. Their support and encouragement over the last four years has been truly invaluable. Their passion for physics is infectious, and their ability to think through difficult research challenges is inspiring. It is a rare privilege to have a mentor that puts their students first, and communicates freely and effectively with them. It is even more rare to have two such mentors. I count myself lucky because of this. I would also like to thank my committee member Lindsay LeBlanc. Over the course of my PhD she has injected some crucial experimental questions and insights into this otherwise theoretical work. I wish to thank my collaborator Jorge Hirsch as well. He is a genuine person and a brilliant physicist. His perseverance in following his own ideas has led to some very interesting concepts and paved the way for Chapter 7 of this thesis. A special thanks goes to my friends and colleagues in the department for their support and discussion, both academic and otherwise. In particular, I would like to acknowledge David Purschke, whose insatiable curiosity, and long discussions with me led to the pump-probe proposal described in Section 5.1.2. Several funding agencies made this work possible, including the National Science and Engineering Research Council (NSERC) and Alberta Innovates - Technology Futures. Their support allowed me to dedicate my efforts to research without worrying about finances. v I am eternally grateful to my partner Farah Elawar. Her countless hours of support, from helping me prepare talks to editing this thesis, got me through my degree. I have never met such a loving person and I am humbled to be the recipient of her relentless encouragement and support. I only hope I can return the favour when she writes her PhD thesis. I owe a huge thanks to my entire family. In particular, I would like to thank my parents, not only for their support and compassion throughout this degree, but also for recognizing my dreams early in life and standing behind them. My mother's generosity and work ethic is truly inspirational and has given me the confidence needed to complete this work. My father is proof that curiosity is one of the most important assets for parenting. Little actions, like leaving a popular physics book on the coffee table, or asking me whether light was a particle or a wave when I was too young to understand either, set me on this path. For that reason, I dedicate this dissertation to him. vi Table of Contents Abstract . ii Preface . iii Acknowledgements . v Table of Contents . vii List of Tables . x List of Figures . xi Abbreviations . xiii Physical Constants . xiv Symbols . xv 1 Introduction . 1 1.1 Spin-orbit coupling . 2 1.2 Rashba spin-orbit coupling . 2 1.3 Group theory and chronology of spin-orbit coupling . 4 1.4 Basic properties of the continuum Rashba model . 10 1.5 Basic properties of the Rashba model on a square lattice . 13 1.6 How to read this thesis . 16 I Rashba scattering and transport 17 2 Introduction . 18 2.1 Motivation: spintronics . 18 2.2 Motivation: interacting systems . 21 2.3 Single-particle scattering formalism in 2D . 22 3 Single-particle Rashba scattering at low energy: simple potentials . 28 3.1 Low-energy setup . 28 3.2 Hard-disk scattering . 30 3.2.1 S-matrix . 33 3.2.2 Differential cross section . 35 3.2.3 Total cross section . 39 3.3 Delta-shell scattering . 40 3.4 Discussion . 42 4 Universality of low-energy Rashba scattering . 46 4.1 Scattering quantities . 47 4.1.1 Cross section and optical theorem . 49 4.2 Universal Rashba T -matrix . 50 4.3 Example potentials . 57 4.3.1 Delta function potential . 57 4.3.2 Circular barrier potential . 58 vii 4.3.3 Delta-shell potential . 59 4.4 Discussion . 62 5 Unconventional transport in low-density Rashba systems . 64 5.1 Semiclassical Boltzmann transport . 65 5.1.1 Zero temperature . 66 5.1.1.1 DC Conductivity . 66 5.1.1.2 AC Conductivity . 71 5.1.2 Finite temperature . 72 5.2 Self-consistent full Born approximation . 76 5.2.1 Single-particle properties . 76 5.2.2 Kubo conductivity . 80 5.3 Discussion . 88 II BCS superconductivity 91 6 Introduction . 92 6.1 BCS theory and tight-binding . 92 6.2 Spin-orbit coupling and superconductivity . 96 6.3 Symmetry of the gap function . 98 7 Enhancement of superconducting Tc due to the spin-orbit interaction . 101 7.1 Correlated hopping . 101 7.2 Gap equations . 105 7.3 Results . 107 7.4 Discussion . 112 8 Superconducting order parameter symmetry . 114 8.1 Extended Hubbard model critical temperature . 114 8.2 Free energy . 117 8.3 Extended Hubbard model at finite temperature . 119 8.4 Extended Rashba-Hubbard model critical temperature . 123 8.5 Discussion . 128 9 Conclusion . 130 Bibliography . 133 Appendix A Character tables of some relevant point groups . 147 Appendix B Spin-degenerate scattering . 149 Appendix C Symmetries of the Rashba S and T matrices . 152 C.1 Symmetry of the S-matrix . 152 C.2 Symmetry of the T -matrix . 154 Appendix D Rashba Green's function . 157 Appendix E Derivation of the Rashba S and T matrix relation . 160 E.1 First derivation . 160 E.2 Second derivation . 162 Appendix F Derivations of important integrals . 164 − F.1 The full Born integral Il . 164 l F.2 The self-consistent full Born integral I− . 168 viii F.3 Density of states . 170 F.4 Advanced-retarded integrals . 171 Appendix G Mean-field theory for superconductivity in tight-binding models . 172 G.1 Extended Hubbard model without spin-orbit coupling . 172 G.2 Rashba model with correlated hopping . 176 G.3 Rashba superconducting density of states . 178 Appendix H Critical points of the free energy . 181 H.1 Corollaries of the Morse condition . 183 ix List of Tables 1.1 Spin-splitting energies . 4 1.2 E representations of some point group elements . 8 6.1 Nearest-neighbour tight-binding models . 95 A.1 C2v point group character table . 147 A.2 C4v point group character table . 147 A.3 C6v point group character table . ..