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Spectroscopy and Phenomenology of Unconventional Superconductors

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Spectroscopy and Phenomenology of Unconventional Superconductors Spectroscopy and phenomenology of unconventional superconductors by Nicholas Robert Lee-Hone M.Sc., McGill University, 2013 B.Sc., McGill University, 2011 Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Physics Faculty of Science © Nicholas Robert Lee-Hone 2020 SIMON FRASER UNIVERSITY Fall 2020 Copyright in this work is held by the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Declaration of Committee Name: Nicholas Robert Lee-Hone Degree: Doctor of Philosophy Thesis title: Spectroscopy and phenomenology of unconventional superconductors Committee: Chair: Igor F. Herbut Professor, Physics David M. Broun Co-Supervisor Associate Professor, Physics Erol Girt Co-Supervisor Professor, Physics Malcolm P. Kennett Committee Member Associate Professor, Physics J. Steven Dodge Examiner Associate Professor, Physics William A. Atkinson External Examiner Professor, Physics and Astronomy Trent University ii Abstract The challenge of condensed matter physics is to understand how various states of matter emerge from periodic or amorphous arrangements of atoms. This is typically easier if there are no impurities in the material under investigation. However, many of the properties that make materials technologically useful stem from the very disorder and impurities that complicate their study. In this work we investigate three unconventional superconductors within the dirty BCS framework of superconductivity, with impurities treated in the self- consistent t-matrix approximation. The three materials behave quite differently and allow us to explore and reveal a rich, subtle, and poorly appreciated range of effects that must be considered when studying superconducting materials and probing the limits of applicability of current theories of superconductivity. Microwave spectroscopy measurements of FeSe, a member of the iron-based family of su- perconductors, were performed at frequencies below 1 GHz using a novel helical resonator. The measurements show that the material is particularly clean, with an electronic mean free path that exceeds that of all known binary compounds. They also reveal the presence of a finite gap on both superconducting bands, strongly constraining which pairing states should be considered for FeSe. Nb-doped SrTiO3 has an extremely low carrier concentration and multiple electronic bands at the Fermi sruface. Microwave spectroscopy measurements, this time with a stripline res- onator, reveal a single spectroscopic gap despite clear evidence for multiple superconducting bands. Elastic scattering introduced via Nb dopants is proposed as the mechanism by which the two bands homogenize into a single spectroscopic signature, as per Anderson’s theorem. The impurity hypothesis is tested with single-band fits to the superfluid density and is found to be in good agreement with measured data. Finally La2−xSrxCuO4, a member of the famous cuprate family of superconductors is ex- plored in detail under the assumption that Born scattering with large amounts of impurities can explain a number of puzzling measurements that have caused controversy in the high-Tc community over the past few years. Many features of the experimental measurements can quantitatively be accounted for with very few free parameters and no fine tuning. iii Keywords: unconventional superconductivity, BCS, disorder, microwave spectroscopy, Mat- subara formalism, dirty superconductor, FeSe, Nb-STO, LSCO, La2−xSrxCuO4, helical res- onator iv Acknowledgements During my time at SFU, I had the privelage of working closely on a regular basis with my supervisors Erol Girt and David Broun. Their constant guidance, encouragement, trust, and willingness to actively engage in the day-to-day lab work has been an inspiration. There are no words to describe how much I value their mentorship and friendship. From the very beginning, Colin Truncik, Natalie Murphy, Ricky Chu and Taras Chouinard made me feel at home in David’s lab. Since then many others have made my time in the research group very enjoyable; thank you in particular to Ulas Özdemir, and Steph Labbé. Tommy McKinnon, Paul Omelchenko, Eric Montoya, and Monica Arora, all helped me learn how to operate the instruments in Erol’s lab. Without their guidance I would have been lost. I have also been very fortunate to have had the opprotunity to collaborate with re- searchers from around the world, including but not limited to Doug Bonn, Marc Scheffler, and Dieter Suess. I have fond memories of the collaboration with Markus Thiemann, who visited us from Stuttgart for over a year. I am also grateful for the guidance and work of Peter Hirschfeld and Vivek Mishra, with whom we have fostered a very successful ongoing collaboration. Thank you to my grandparents, Roger and Margaret, and my sister Chloe for supporting me, and to my parents, Susan and Michel for allowing me to spend innumerable hours on the computer learning how to program as a teenager. It really paid off! And finally thank you to Helen, who has supported me, encouraged me, and endured me through all these years. This thesis would not have been possible without all of you. v Table of Contents Declaration of Committee ii Abstract iii Acknowledgements v Table of Contents vi List of Tables ix List of Figures x 1 Introduction 1 1.1 A brief history of superconductivity . 1 1.2 Conventional and unconventional superconductivity . 4 1.3 Impurity studies of superconductivity . 6 2 Theory 8 2.1 BCS theory . 8 2.2 Self-consistent t-matrix approximation . 12 2.2.1 Anderson’s theorem . 22 2.2.2 Gap equation and Tc ........................... 24 2.2.3 Abrikosov-Gor’kov formula . 25 2.2.4 Critical scattering rate . 27 2.3 Experimental observables . 28 2.3.1 Density of states . 28 2.3.2 Superfluid density . 30 2.3.3 Optical conductivity . 34 2.3.4 Heat capacity . 35 3 Electrodynamics of superconductors 39 3.1 London theory . 39 3.2 Electrodynamics of superconductors . 41 vi 3.3 Surface impedance . 44 4 Experimental methods 48 4.1 Cavity perturbation . 48 4.2 Obtaining calibrated surface impedance from cavity perturbation . 50 4.3 Helical resonator . 51 5 Multiband superconductivity 56 5.1 FeSe . 56 5.1.1 Nodes or no nodes, that is the question . 59 5.1.2 Experiment details . 61 5.1.3 Results . 63 5.1.4 Anisotropic two-band model . 66 5.1.5 Superfluid density fits . 71 5.1.6 Outlook . 73 5.1.7 Contributions . 74 5.2 Nb-SrTiO3 .................................... 74 5.2.1 Experiment details . 77 5.2.2 Results . 79 5.2.3 Anderson’s theorem in two-band superconductors . 82 5.2.4 Is Nb-STO really a dirty superconductor? . 85 5.2.5 Contributions . 90 6 Disorder in La2−xSrxCuO4 91 6.1 Dirty d-wave BCS theory on a realistic Fermi surface . 97 6.1.1 Superfluid density . 99 6.1.2 Optical conductivity . 100 6.2 Results . 101 6.2.1 Finding the missing spectral weight . 104 6.3 Outlook . 111 6.4 Contributions . 112 7 Conclusions 113 Bibliography 115 Appendix A Evaluating elliptic integrals 130 A.1 Arithmetic geometric mean . 131 A.2 Reference implementation . 131 Appendix B Scanning tunnelling spectroscopy 134 vii B.1 Temperature dependence of tunneling . 135 viii List of Tables Table 4.1 Details of the 9 attempts at creating a helical coil resonator. 53 Table 5.1 Material properties and sample dimensions. 77 ix List of Figures Figure 1.1 Timeline of superconducting transition temperatures. 5 Figure 2.1 Real space and k-space representation of BCS pairing. 9 Figure 2.2 Schematic of the excitation spectrum of a superconductor. 11 Figure 2.3 Temperature dependence of the gap for dirty superconductors. 26 Figure 2.4 Evaluation of the Abrikosov-Gor’kov formula. 27 Figure 2.5 Density of states for dirty s-wave and dirty d-wave superconductors. 29 Figure 2.6 Superfluid density calculation starting from the density of states. 31 Figure 2.7 Density of states and superfluid density for clean superconductors. 32 Figure 2.8 Superfluid density suppression for an s-wave superconductor. 33 Figure 2.9 Superfluid density suppression for a d-wave superconductor. 34 Figure 2.10 Conductivity for a dirty d-wave superconductor with impurities. 36 Figure 2.11 Heat capacity for s-wave and d-wave superconductors with impurities. 37 Figure 3.1 Electromagnetic fields outside and inside a superconductor. 42 Figure 4.1 Cavity perturbation apparatus and plot of the S21 parameter. 49 Figure 4.2 Schematic of the novel helical coil resonator inside a Cu enclosure. 52 Figure 4.3 Picture of the first 8 helical resonators attempts. 53 Figure 4.4 Comsol simulations of the helical resonator. 55 Figure 5.1 Crystal sturcture of five common types of iron-based superconductors. 57 Figure 5.2 Schematic of the FeAs lattice, Fermi surface, and band structure. 59 Figure 5.3 Tunneling conductance data from several sources. 60 Figure 5.4 Possible gap symmetry scenarios for multiband superconductors. 62 Figure 5.5 Surface resistance of FeSe plotted on linear and logarithmic scales. 64 Figure 5.6 Real part of the quasiparticle conductivity, σ1(T ). 65 Figure 5.7 Quasiparticle relaxation rate from fits to a two-fluid model. 66 Figure 5.8 Superfluid density obtained from fits to a two-fluid model. 67 Figure 5.9 Gap function for various pairing symmetries. 68 Figure 5.10 Fit to the superfluid density using a two-band extended s-wave model. 72 Figure 5.11 Bogoliubov quasiparticle interference measurements of the FeSe gaps. 73 Figure 5.12 Crystal structure, Fermi surface, and band structure of Nb-SrTiO3. 75 x Figure 5.13 Tunneling spectroscopy data for Nb-doped SrTiO3. 76 Figure 5.14 Doping dependence of the electron mobility in Nb-doped SrTiO3. 78 Figure 5.15 Microwave stripline resonator and sample.
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