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80 Chapter 4: Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover free path l e is comparable to the wire width W, we cannot perform the step from to ∑ q E,p + q E ′ ,p ′ −q ∫ 2π 0 = 1 2πντ ∑ GE,σ R (p+q)G E A ′ ,σ ′(p′ −q) (4.47) q dϕ 2π (1+iτve ϕ(Q+2m e a(ϕ).S)) −1 (4.48) and integrate in Eq.(4.3) over the Fermi surface in a continuous way. Instead, we assume k F /W to be finite and sum over the number of discrete channels N = [k F W/π], where [...] is the integer part. Because H γD ∼ E 2 F this constant term due to cubic Dresselhaus should reduce if we reduce the number of channels. If we expand the Cooperon to second order in (Q+2eA+2m e âS) before averaging over the Fermi surface, 〈...〉, and use the Matsubara trick, we get C −1 ( ( ) ) = 2f 1 Q y +2α 2 S x +2 α 1 −γ D v 2f 2 3 S y D e f 1 ( ) +2f 2 (Q x −2α 2 S y −2 α 1 −γ D v 2f 2 3 S x ) f 2 [( ) ( ) +8γDv 2 4 f 4 − f2 3 Sx 2 + f 5 − f2 3 Sy 2 f 2 f 1 ] , (4.49) with m e = 1 and functions f i (ϕ) (App.E) which depend on the number of transverse modes N. In the diffusive case we can perform the continuous sum over the angle ϕ in Eq.(E.3)- (E.7), and we receive the old result with f 1 = f 2 = 1/2, f 3 = 1/8 and f 4 = f 5 = 1/16: H c = (Q y +2α 2 S x +2 (α 1 − 1 ) 2 γ DE F S y ) 2 +(Q x −2α 2 S y −2 (α 1 − 1 ) 2 γ DE F S x ) 2 +(γ D E F ) 2 (S 2 x +S 2 y). (4.50) 4.5.1 Spin Relaxation at Q SO W ≪ 1 In the first section we analyzed the lowest spin relaxation in wires of different direction ina(001) system. We have shown, that forevery direction thereis still afinitespin relaxation at wire width which fulfill the condition Q SO W ≪ 1 due to cubic Dresselhaus SOC. It is clear that this finite spin relaxation vanishes when the width is equal to the Fermi wave length λ F . In the following we show how this finite spin relaxation depends on the number of transverse channels N. We show in Ref.[WK] that the findings are consistent with calculations going beyond the perturbative ansatz. This is possible in a
Chapter 4: Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover 81 similar manner as has been done previously in Ref.[KM02] for wires without SOC, which showed the crossover of the magnetic phase shifting rate, which had been known before in the diffusive and ballistic limit, only. TofindthespectrumoftheCooperonHamiltonianwithboundaryconditionsasinSec.4.2.2, we stay in the 0-mode approximation in the Q space and proceed as before: According to Eq.(4.49), the non-Abelian gauge transformation for the transversal direction y is given by ( [ ( ) ) U = exp −i 2α 2 S x +2 α 1 −γ D v 2f 3 S y ]y . (4.51) f 1 To concentrate on the constant width independent part of the spectrum we extract the absolute minimum at Q = 0, Fig.(4.4) and Fig.(4.5). A clear reduction of the absolute minimum is visible. Due to the factor f 3 /f 1 in the transformation U, the decrease of the minimal spin relaxation depends also on the ratio of Rashba and linear Dresselhaus SOC. E min /ǫ 2 F γ2 D 1.0 0.8 0.6 0.4 0. 0.5 1. 1.5 2. 2.5 3. 0.2 0.0 5 10 15 20 25 30 N Figure 4.4: The lowest eigenvalues of the confined Cooperon Hamiltonian Eq.(4.49), equivalent to the lowest spin relaxation rate, are shown for Q = 0 for different number of modes N = k F W/π. Different curves correspond to different values of α 2 /q s . From Eq.(4.49) it is clear, that not only the H γD is affected by the reduction of the number of channels N but also the shift of the lin. Dresselhaus SOC, α 1 , in the orbital part. A model to extract the ratio of Rashba and lin. Dresselhaus SOC developed in Ref.[SANR09] by Scheid et al. did not show much difference between the strict 1D case and the non-diffusive case with wire of finite width. The results presented here should allow for extending the model to finite cubic Dresselhaus SOC. Deducing from our theory, the direction of the SO field should change with the number of channels due to the mentioned N dependent shift.
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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 77 and 78: Chapter 4 Direction Dependence of S
Page 79 and 80: Chapter 4: Direction Dependence of
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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 and 124: List of Figures 113 4.2 The spin de
Page 125 and 126: List of Tables 3.1 Singlet and trip
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
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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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