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

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114 List of Figures 5.6 (a) DOS of a system of size L 2 = 40 2 with Rashba SOC, α 2 = 0.8t calculated with exact diagonalization with cutoff η = 0.0215 (blue) and KPM with M = 500 moments. (b) Relation between number of moments M and cutoff η. 98 5.7 (a) Averaged DOS ρ avr calculated with KPM (30 impurity configurations) for system size of L 2 = 280 2 with Rashba SOC α 2 = 0.5t, for different impurity strengths: V = 1t(black), V = 4t(red), V = 6t(green), V = 8t(blue) (b) Monotoneous reduction of typical DOS ρ typ with system size L = 70, 140, 200, 280, for impurity strength V = 8t. . . . . . . . . . . . . . 100 5.8 (a) Log-plot of typical DOS ρ typ at V = 8t for different system sizes L with Rashba SOC α 2 = 0.5t at half filling, calculated with KPM. The dashed line isalinearfittothelog-data whichyieldsalocalization lengthofξ ≈ 100a. (b) Localization length ξ at E F = 0, plotted as a function of impurity strength V.101 5.9 Finite size analysis of the typical DOS in relation to the averaged one, ρ typ /ρ avr , here plotted for E F = 0: The system size has been changed with L = 70, 140, 200, 280. The impurity strength for the different curves is given by V/t = 1,3,4,5,6,8 (monotone from top to bottom). . . . . . . . . . . . . 102 5.10 a)+b) Matrix element density function j(x,y) for a clean system with 70×70 sites and pure Rashba SOC of strength α 2 = 1t using the analytical solution Eq.(5.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.11 SHC σ SH (E F ,M) calculated with KPM for a system with 70×70 sites, pure Rashba SOC of strength α 2 = 1t and M = 200 moments (blue curve). For comparison the analytical solution is plotted (red curve), corresponding to Fig.5.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 C.1 Weak localization correction in 2D in units of (2e 2 /2π). The parameters are c 1 = 1/D e Q 2 SOτ ϕ and c 2 = 1/D e Q 2 SOτ. Thick line indicates ∆σ = 0. . . . . . 143
List of Tables 3.1 Singlet and triplet states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.1 Values of Rashba parameter α 2 measured in experiments [QW=quantum wire]. (List extracted from[FMAE + 07].) . . . . . . . . . . . . . . . . . . . . 130 A.2 Experimental values of m e [VMRM01] . . . . . . . . . . . . . . . . . . . . . 130 A.3 Values of Dresselhaus parameter γ D measured in experiments. (List extracted from[FMAE + 07].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 A.4 Measured α 2 /α 1 ratios with spin-galvanic effect (SGE) and Circular photogalvaniceffect(CPGE)withtheaccordingparametersat4.2K [QW=quantum wire].[GGB + 07] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 115
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Itinerant Spin Dynamics in Structur
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Itinerant Spin Dynamics in Structur
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Contents v 3.2.2 Weak Localization
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Citations to Previously Published W
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Dedicated to Linda, my parents and
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Chapter 1 Introduction Structure of
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Chapter 1: Introduction 3 the coupl
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Chapter 1: Introduction 5 Throughou
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Chapter 2: Spin Dynamics: Overview
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Chapter 3 WL/WAL Crossover and Spin
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Chapter 3: WL/WAL Crossover and Spi
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Page 73 and 74: Chapter 3: WL/WAL Crossover and Spi
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Page 77 and 78: Chapter 4 Direction Dependence of S
Page 79 and 80: Chapter 4: Direction Dependence of
Page 93 and 94: Chapter 5 Spin Hall Effect 5.1 Intr
Page 95 and 96: Chapter 5: Spin Hall Effect 85 curr
Page 97 and 98: Chapter 5: Spin Hall Effect 87 with
Page 99 and 100: Chapter 5: Spin Hall Effect 89 1.0
Page 101 and 102: Chapter 5: Spin Hall Effect 91 as s
Page 103 and 104: Chapter 5: Spin Hall Effect 93 V⩵
Page 105 and 106: Chapter 5: Spin Hall Effect 95 0.4
Page 107 and 108: Chapter 5: Spin Hall Effect 97 oper
Page 109 and 110: Chapter 5: Spin Hall Effect 99 the
Page 111 and 112: Chapter 5: Spin Hall Effect 101 str
Page 113 and 114: Chapter 5: Spin Hall Effect 103 Com
Page 115 and 116: Chapter 5: Spin Hall Effect 105 Fin
Page 117 and 118: Chapter 6: Critical Discussion and
Page 119 and 120: Chapter 6: Critical Discussion and
Page 121 and 122: List of Figures 111 3.3 Exemplifica
Page 123: List of Figures 113 4.2 The spin de
Page 127 and 128: Bibliography 117 [BB04] [BBF + 88]
Page 129 and 130: Bibliography 119 [DP71b] [DP71c] [D
Page 131 and 132: Bibliography 121 [HSM + 06] [HSM +
Page 133 and 134: Bibliography 123 [MACR06] T. Mickli
Page 135 and 136: Bibliography 125 [PMT88] [PP95] [PT
Page 137 and 138: Bibliography 127 [Tor56] H. C. Torr
Page 139 and 140: Appendix A SOC Strength in the Expe
Page 141 and 142: Appendix A: SOC Strength in the Exp
Page 143 and 144: Appendix B: Linear Response 133 in
Page 145 and 146: Appendix B: Linear Response 135 Pro
Page 147 and 148: Appendix C Cooperon and Spin Relaxa
Page 149 and 150: Appendix C: Cooperon and Spin Relax
Page 157 and 158: Appendix D: Hamiltonian in [110] gr
Page 159 and 160: Appendix E: Summation over the Ferm
Page 161: Appendix F: KPM 151 integer running
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