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

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52 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems -1 B/H S 0 1 0.0 -0.5 ∆σ ” -1.0 “ 2e 2 2π 2 4 6 Q SO W 8 - 10 1.5 Figure3.11: Thequantumconductivity correction inunitsof 2e 2 /2π asfunctionofmagnetic field B (scaled with bulk relaxation field H s ), and the wire width W scaled with 1/Q SO for pure Rashba coupling and cutoffs 1/D e Q 2 SOτ ϕ = 0.08, 1/D e Q 2 SOτ = 4: Comparison of the zero-mode calculation (grid without shading) to the exact diagonalization where the lowest 21 triplet branches and seven singlet branches were taken into account. singlet and triplet terms of the Cooperon. A significant change should arise if 1 (W = W c ) = 1 . (3.83) τ s τ ϕ Assuming that this occurs for small wire widths, Q SO W < 1, as confirmed for the parameters we used, we apply Eq.(3.81) to Eq.(3.83) and conclude that W c ∼ 1 √ τϕ . (3.84) If we calculate the crossover numerically in the 0-mode approximation we get the relation plotted in Fig.3.13 which coincides with Eq.(3.84). We note that the change from WAL to WL may occur at a different width W c than the change of sign in the correction to the electrical conductivity ∆σ(B = 0) occurs, W WL . However, we find that the ratio W c /W WL is independent of the dephasing rate and the spin-orbit coupling strength Q SO . Furthermore while there is quantitative agreement with the 0-mode approximation in the magnitude of the magnetoconductivity for all magnetic fields for small wire widths W
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 53 L SO , there is only qualitative agreement at larger wire widths. In particular, the total magnitude of the conductivity is reduced considerably in comparison with the 0-mode approximation. We can attribute this to the reduction of the energy of the lowest Cooperon triplet modes due to the emergence of edge modes, which is not taken into account when neglecting transversal spatial variations, as is done in the 0-mode approximation. Therefore, the 0-mode approximation overestimates the suppression of the triple modes, resulting in an overestimate of the conductivity. Similarly, the magnetic field at which the magnetoconductivity changes its sign from negative to positive is already at a smaller magnetic field, as seen by the shift in the minimum of the conductivity towards smaller magnetic fields (Fig.3.12), in comparison to the 0-mode approximation (unshaded) in Fig.3.11. This is in accordance with experimental observations, which showed clear deviations from the 0-mode approximation for larger wire widths, with a stronger magnetic field dependence than obtained in 0-mode approximation.[DLS + 05, LSK + 07, SGP + 06, WGZ + 06] Note that the nonmonotonous behavior of the triplet modes as function of the wire width, seen in Fig.3.7, cannot be resolved in the width dependence of the conductivity. ∆σ−∆σ0 2e 2 /2π 0.05 0.00 1 5 Q SO W c 10 1 0 B/H S Figure 3.12: The relative magnetoconductivity ∆σ(B) − ∆σ(B = 0) in units of 2e 2 /2π, with the same parameters and number of modes as in Fig.3.11. 3.4.4 Other Types of Boundary Conditions Adiabatic Boundary Conditions When the lateral confinement potential V is smooth compared to the SO splitting, that is, if λ F ∂ y V ≪ ∆ α = 2k F α 2 , where λ F is the Fermi wavelength, the boundaries do not preserve the spin, s in ≠ s out , Eq.(3.70), since the spin may adiabatically evolve as the
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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
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
Page 61: Chapter 3: WL/WAL Crossover and Spi
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
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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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