Itinerant Spin Dynamics in Structures of ... - Jacobs University

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Contents Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Citations to Previously Published Work . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i iii iv vii viii ix 1 Introduction 1 2 Spin Dynamics: Overview and Analysis of 2D Systems 6 2.1 Short Reminder on the Origin of Spin Orbit Coupling . . . . . . . . . . . . 6 2.2 Dynamics of a Localized Spin . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Spin Dynamics of Itinerant Electrons . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Ballistic Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.2 Spin Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.3 Spin Orbit Interaction in Semiconductors . . . . . . . . . . . . . . . 11 2.3.4 Spin Diffusion in the Presence of Spin-Orbit Interaction . . . . . . . 15 2.4 Spin Relaxation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 D’yakonov-Perel’ Spin Relaxation . . . . . . . . . . . . . . . . . . . 19 2.4.2 DP Spin Relaxation with Electron-Electron and Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.3 Elliott-Yafet Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . 21 2.4.4 Spin Relaxation due to Spin-Orbit Interaction with Impurities . . . 21 2.4.5 Bir-Aronov-Pikus Spin Relaxation . . . . . . . . . . . . . . . . . . . 22 2.4.6 Magnetic Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.7 Nuclear Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.8 Magnetic Field Dependence of Spin Relaxation . . . . . . . . . . . . 23 3 WL/WAL Crossover and Spin Relaxation in Confined Systems 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 One-Dimensional Wires . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 Wires with W > λ F . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Quantum Transport Corrections . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Diagrammatic Approach . . . . . . . . . . . . . . . . . . . . . . . . . 27 iv
Contents v 3.2.2 Weak Localization in Quantum Wires . . . . . . . . . . . . . . . . . 34 3.3 The Cooperon and Spin Diffusion in 2D . . . . . . . . . . . . . . . . . . . . 37 3.4 Solution of the Cooperon Equation in Quantum Wires . . . . . . . . . . . . 42 3.4.1 Quantum Wires with Spin-Conserving Boundaries . . . . . . . . . . 42 3.4.2 Zero-Mode Approximation . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4.3 Exact Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4.4 Other Types of Boundary Conditions . . . . . . . . . . . . . . . . . 53 3.5 Magnetoconductivity with Zeeman splitting . . . . . . . . . . . . . . . . . . 57 3.5.1 2DEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.2 Quantum Wire with Spin-Conserving Boundary Conditions . . . . . 60 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Spin Relaxation anisotropy in the (001) system . . . . . . . . . . . . . . . . 70 4.2.1 2D system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.2 Quasi-1D wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Spin relaxation in quasi-1D wire with [110] growth direction . . . . . . . . . 75 4.3.1 Special case: without cubic Dresselhaus SOC . . . . . . . . . . . . . 76 4.3.2 With cubic Dresselhaus SOC . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Weak Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.5 Diffusive-Ballistic Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.1 Spin Relaxation at Q SO W ≪ 1 . . . . . . . . . . . . . . . . . . . . . 80 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Spin Hall Effect 83 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.1 About the Definition of Spin Current . . . . . . . . . . . . . . . . . . 84 5.2 SHE without Impurities: Exact Calculation . . . . . . . . . . . . . . . . . . 85 5.3 Numerical Analysis of SHE . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3.1 Exact Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3.2 Kernel Polynomial Method . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.3 SHC calculation using KPM . . . . . . . . . . . . . . . . . . . . . . . 101 6 Critical Discussion and Future Perspective 106 List of Symbols 108 List of Figures 110 List of Tables 115 Bibliography 116 A SOC Strength in the Experiment 129
Page 1 and 2: Itinerant Spin Dynamics in Structur
Page 3: Itinerant Spin Dynamics in Structur
Page 7 and 8: Citations to Previously Published W
Page 9 and 10: Dedicated to Linda, my parents and
Page 11 and 12: Chapter 1 Introduction Structure of
Page 13 and 14: Chapter 1: Introduction 3 the coupl
Page 15 and 16: Chapter 1: Introduction 5 Throughou
Page 17 and 18: Chapter 2: Spin Dynamics: Overview
Page 35 and 36: Chapter 3 WL/WAL Crossover and Spin
Page 37 and 38: Chapter 3: WL/WAL Crossover and Spi
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Chapter 3: WL/WAL Crossover and Spi
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Chapter 4 Direction Dependence of S
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Chapter 4: Direction Dependence of
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Chapter 5 Spin Hall Effect 5.1 Intr
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Chapter 5: Spin Hall Effect 85 curr
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Chapter 5: Spin Hall Effect 87 with
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Chapter 5: Spin Hall Effect 89 1.0
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Chapter 5: Spin Hall Effect 91 as s
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Chapter 5: Spin Hall Effect 93 V⩵
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Chapter 5: Spin Hall Effect 95 0.4
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Chapter 5: Spin Hall Effect 97 oper
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Chapter 5: Spin Hall Effect 99 the
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Chapter 5: Spin Hall Effect 101 str
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Chapter 5: Spin Hall Effect 103 Com
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Chapter 5: Spin Hall Effect 105 Fin
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Chapter 6: Critical Discussion and
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Chapter 6: Critical Discussion and
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List of Figures 111 3.3 Exemplifica
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List of Figures 113 4.2 The spin de
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List of Tables 3.1 Singlet and trip
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Bibliography 117 [BB04] [BBF + 88]
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Bibliography 119 [DP71b] [DP71c] [D
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Bibliography 121 [HSM + 06] [HSM +
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Bibliography 123 [MACR06] T. Mickli
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Bibliography 125 [PMT88] [PP95] [PT
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Bibliography 127 [Tor56] H. C. Torr
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Appendix A SOC Strength in the Expe
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Appendix A: SOC Strength in the Exp
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Appendix B: Linear Response 133 in
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Appendix B: Linear Response 135 Pro
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Appendix C Cooperon and Spin Relaxa
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Appendix C: Cooperon and Spin Relax
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Appendix D: Hamiltonian in [110] gr
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Appendix E: Summation over the Ferm
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Appendix F: KPM 151 integer running
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