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Quantum Fluctuations Across the Superconductor-Insulator Transition

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Quantum Fluctuations Across the Superconductor-Insulator Transition Quantum Fluctuations Across the Superconductor-Insulator Transition Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Hasan Khan, B.S. Graduate Program in Physics The Ohio State University 2019 Dissertation Committee: Nandini Trivedi, Advisor Yuan-Ming Lu Rolando Vald´esAguilar Richard J. Furnstahl c Copyright by Hasan Khan 2019 Abstract Superconductivity has been at the heart of research into quantum phenomena since its discovery over a century ago. More recently efforts have been made to un- derstand the nature of the quantum phase transition (QPT) that separates the super- conducting and insulating phases in certain 2D materials at zero temperature. This superconductor-insulator transition (SIT) has been theoretically and experimentally proven to be driven by quantum fluctuations of the superconducting phase instead of the breakup of Cooper pairs. In this thesis we present a study of quantum fluctua- tions across the SIT and how they can be imaged in both theoretical simulations and experimental measurements. We begin with an overview of the field from a historical perspective, describing the development of the theory of SITs driven by experiments on thin films. We present the Josephson junction array (JJA) model as a paradigm to investigate the quantum phase fluctuation-driven SIT using quantum Monte Carlo (QMC) techniques. We explore the manifestation of quantum fluctuations across the SIT in three different local measurements: the diamagnetic susceptibility χ(r), two-particle den- sity of states P (r;!), and compressibility κ(r), revealed through their local maps and calculated via QMC. χ(r) probes the system's ability to generate diamagnetic currents and its local map displays growing fluctuations upon increasing both tem- perature the quantum tuning parameter g. Remarkably, however, these fluctuations ii persist well below Tc as the SIT is approached, indicating the quantum nature of these fluctuations. We compare our results to SQUID susceptometry measurements performed on thin-film NbTiN and find good qualitative agreement. The map of κ(r) paints a similar picture when tuned via g, but in contrast to χ(r), we find a funda- mental difference in its evolution with temperature, providing a complementary local probe to χ(r). P (r;!), obtained using Maximum Entropy analytic continuation of imaginary time QMC data, shows strongly diminished zero-energy spectral weight in nearly-insulating islands, correlating with regions of suppressed κ(r). We discuss the experimental implications of our results for scanning Josephson spectroscopy, com- pressibility, and scanning SQUID measurements, the first time these quantities have been discussed together in the context of quantum fluctuations. iii To my loving parents, who have always been there for me. iv Acknowledgments The time I spent pursuing my Ph.D. in the Department of Physics at OSU would not have been fruitful without the help of countless individuals. First, I would like to thank my advisor Nandini Trivedi for being so supportive and helping to guide me through this program the past six years. Her insight has been invaluable in pushing the direction of my research projects and helping them come to completion. The majority of the lessons I have learned in my time here have come from her, and for that I am eternally grateful. I would also like to thank her for providing me financial support via grants during my time in her group. There are a number of other individuals I would like to thank for helping me grow from a young graduate student to the competent researcher I am today. Mason Swan- son was the most patient mentor a student could ever ask for. He helped transition me into the group when I was new and never turned me down when I needed help with my first project. Tim McCormick coached me on the fundamentals of being a strong, independent researcher and became a great friend for the remainder of my time here. I also want to thank Yen Lee Loh for helping to guide me through some of my first research projects and for being a great source of discussion during my research career. I want to thank all of my colleagues for their support and help. Many of the people I have met in the physics department have become good friends over the years, v including James Rowland, Tamaghna Hazra, Chris Svoboda, David Nozadze, Mehdi Kargarian, Kyungmin Lee, Joe Szabo, and Ian Osborne. Outside the department, I want to express my gratitude to the friends who kept me company during my time at OSU, including Dan Litwak, Alex Perhac, Jess Tran, Paul Justice, Tim Gao, Ian Froning, and Mark Meng. I could not have gotten through the program without the support of my family who was there for me despite being hundreds of miles away. I am incredibly thankful to my parents for their love and support these past six years and for allowing me to pursue my passion. I want to thank my sister Hina Khan for being the best sibling in the world and my fianc´eeSanah Choudry for her emotional support in my final year. Finally, I would like to thank the US-Israel Binational Science Foundation for funding the majority of my research in my time at OSU. vi Vita September 15, 1991 . .Born|Baltimore, MD May 2013 . .B.S. Physics and Astronomy| University of Rochester, Rochester, NY August 2013 - present . Graduate Research Assistant|The Ohio State University, Columbus, OH Publications Research Publications Local spectroscopies across the superconductor-insulator transition, H. Khan and N. Trivedi, arXiv:1812.06954. Imaging quantum fluctuations near criticality, A. Kremen, H. Khan, Y.L. Loh, T.I. Baturina, N. Trivedi, A. Frydman, and B. Kalisky, Nature Physics 14, 1205 (2018). A machine learning approach to the Berezinskii-Kosterlitz-Thouless transition in clas- sical and quantum models, M. Richter-Laskowska, H. Khan, N. Trivedi, and M.M. Ma´ska, Condens. Matter Phys. 21, 33602 (2018). Fields of Study Major Field: Physics vii Table of Contents Page Abstract . ii Dedication . iv Acknowledgments . .v Vita......................................... vii List of Figures . .x 1. Introduction . .1 2. The Josephson Junction Array Model of the Superconductor-Insulator Transition . .5 2.1 The Superconductor-Insulator Transition . .6 2.1.1 Early Experiments . .6 2.1.2 Persistence of the Single-Particle Gap . .8 2.2 The Josephson Junction Array Model . 10 2.2.1 The Model . 10 2.2.2 The Quantum-Classical Mapping . 12 2.2.3 Superfluid Stiffness . 14 3. Local Diamagnetic Susceptibility . 18 3.1 The Diamagnetic Susceptibility . 18 3.1.1 Global Diamagnetic Susceptibility . 20 3.1.2 Local Diamagnetic Susceptibility . 22 3.2 Monte Carlo Results . 23 3.3 Experimental Comparisons . 26 viii 4. Local Density of States and Local Compressibility . 28 4.1 Global Quantities . 29 4.1.1 Bosonic Spectral Function . 30 4.1.2 Compressibility . 34 4.2 Local DOS and Compressibility . 35 4.3 Discussion and Outlook . 38 Bibliography . 41 Appendices 46 A. Quantum-Classical Mapping . 46 A.1 Mapping to Classical XY Model . 46 A.2 Mapping to Integer Current Model . 50 B. Linear Response and Kubo Formulas . 52 B.1 Superfluid Stiffness and Compressibility in the (2+1)D XY Model . 52 B.1.1 Current and the Electromagnetic Response Tensor . 53 B.1.2 Superfluid Stiffness . 55 B.1.3 Compressibility . 57 B.2 Diamagnetic Susceptibility in the (2+1)D Integer Current Model . 58 C. Analytic Continuation . 61 C.1 The Maximum Entropy Method . 62 C.2 Sum Rules . 64 ix List of Figures Figure Page 2.1 (a) Measurements of sheet resistance R on a sample of thin-film Bi as a function of temperature for varying thicknesses [1]. The film exhibits superconducting behavior at low temperatures for larger thicknesses and rapidly becomes insulating for smaller thicknesses. The film’s superconducting Tc decreases with decreasing thickness, suggesting a thickness-tuned SIT at T = 0. (b) Schematic phase diagram of an SIT tuned by a non-thermal parameter g. As the SIT is approached, Tc decreases and eventually vanishes. .7 2.2 (a) BdG calculations on a disordered 2D system show a spectral gap that persists and remains the same order of magnitude for all disorder strengths, while the superfluid stiffness decreases significantly [2]. (b) Comparison of DQMC calcuations (theory [3]) and STS measurements (experiment [4, 5]) on disordered 2D superconductors. Both show the persistence of a hard gap across the SIT and the granular structure of superconducting islands. .9 2.3 (a) The granular structure of thin film superconductors (left) allows us to study the SIT in terms of an effective model describing supercon- ducting islands coupled via the Josephson effect (right) [6]. (b) When the Josephson coupling is strong, the phases are able to align and carry Cooper pairs across the system (superconductor). When the charging energy is strong, tunneling is prohibitive and the Cooper pairs become localized (insulator). 11 x 2.4 (a) The 2D JJA can be simulated in Monte Carlo by mapping to either of two classical actions: the (2+1)D XY model or the (2+1)D integer current model. (b) Examples of Monte Carlo configurations (projected onto the xy-plane) in the XY and ICM models. In the XY model we see coherent phases clustering together and the formation of vortices and anti-vortices (marked by blue and red squares respectively). In the ICM we see the system forming closed current loops that carry magnetization (shaded regions, discussed in the next chapter). 14 2.5 (a) Monte Carlo results for ρs as a function of temperature performed on a 64 × 64 lattice. ρs rapidly jumps to zero at Tc, following the universal 2T/π jump of the KT transition.
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