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

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64 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 1.4 1.6 ∆σ/( 2e2 2π ) 1.8 2.0 2.2 0.0 0.5 1.0 1.5 2.0 B/H S Figure 3.18: The magnetoconductance ∆σ(B) in a magnetic field perpendicular to the quantumwell, whereitscouplingviatheZeemantermisconsideredbyexact diagonalization while its effect on the orbital motion is considered effectively by the magnetic phase-shifting rate 1/τ B (W) for wire width Q SO W = 1, dephasing rate 1/τ ϕ = 0.06D e Q 2 SO , and elasticscattering rate 1/τ = 4D e Q 2 SO . The strength of the contribution of the Zeeman term is varied by the material-dependent factor ˜g = gµ B H s /D e Q 2 SO in the range ˜g = 0...1.5 in steps of 0.5: The system changes from positive magnetoconductivity (˜g = 0, green) to negative one (˜g = 0.5...1.5, continuous decrease of the absolute minimum). 3.6 Conclusions In conclusion, in wires with spin-conserving boundaries and a width W smaller than bulk spin precession length L SO , the spin relaxation due to linear Rashba SO coupling is suppressed according to the spin relaxation rate (1/τ s )(W) = (π 2 /3)(W/L SO ) 2 (1/τ s ), where 1/τ s = 2p 2 F α2 2 τ. The enhancement of spin relaxation length L s = √ D e τ s (W) can be understood as follows: The area an electron covers by diffusion in time τ s is WL s . This spin relaxation occurs if that area is equal L 2 SO,[Fal03] which yields 1/L 2 s ∼ 1/D e τ s ∼ (W/L SO ) 2 /L 2 SO, in agreement with Eq.(3.81). For larger wire widths, the exact diagonalization reveals a nonmonotonic behavior of the spin relaxation as function of the wire width of the long-living eigenstates. The spin relaxation rate is first enhanced before it is suppressed as the widths W is decreased. The longest living modes are found to exist at the boundary of wide wires. Since we identified a direct transformation from the spin-diffusion equation to the Cooperon equation, we could show that these edge modes affect the conductivity: the 0-mode approximation overestimates the conductivity for larger wire widths Q SO W > 1 since it does not take into account these edge modes. They add a larger contribution to the negative triplet term of the quantum
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 65 1.2 Q SO Wc 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 g/8m e D e (a) 3.5 3.0 Q SO Wc 2.5 2.0 1.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 B Z /H s (b) Figure 3.19: (a) Change of crossover width W c with g factor: The magnetic field is included as an effective field 1/τ B and in the Zeeman term. The strength of the contribution of the Zeeman term is varied by the material dependent factor ˜g = gµ B H s /D e Q 2 SO. (b) Change of crossover width W c with Zeeman field: To calculate W c , we fix the Zeeman field to a certain value, horizontal axis, while we vary the effective field independently and calculate if negative or positive magnetoconductivity is present. For different Zeeman fields B Z /H s , we find thereby a different width W c . Here, we set g/8m e D e = 1. In (a) and (b), the cutoff due to dephasing is varied: 1/D e Q 2 SOτ ϕ = 0.04,0.06,0.08 (lowest first).
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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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Page 23 and 24: 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 73: 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
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
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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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