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

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28 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems (a) (b) Figure 3.1: Two different experimental approaches to extract the wire width dependence of spin relaxation rate. (a) Measurement by Kerr rotation (extracted from [HSM + 06]) and (b) using magnetoconductivity experiments (extracted from [LSK + 07]). all classical paths α with their corresponding actions S α P(r,r ′ ) ≈ ∑ α A α e iSα . (3.1) It is intuitively clear that only this interference processes will survive self-averaging which are independent of the impurity position. Disadvantageous for conductivity is clearly, if an electron returns to the point he started from. This are paths where the relative phase is independent of the position of the impurities: |P(r,r)| 2 ≈ ∑ α,β A α A β e i(Sα−S β) . (3.2) There are two possibilities, which cancel the phase factor: The first one is pure classic, namely representing a scattering that causes the electron to traverse the way α backwards. The reason for the second one is with the time-reversal-symmetry of the system, which allows to be β the time-reversed path of α, more specific |P(r,r)| 2 ≈ ∑ α 〈|A α| 2 +|A α| 2 〉+2R ∑ α 〈A αA ∗ α 〉 (3.3) = |P class (r,r)| 2 +|P WL (r,r)| 2 . (3.4)
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 29 Figure 3.2: Measured WL corrections on a thin PdAu film, Ref.[DO79]. The resistivity increases logarithmically as the temperature decreases. Summing over all possible paths will generally yield only the classical and the pure quantum-mechanical term. Other path have in general a large phase difference and will average out. Thus, the quantum-mechanical return-probability is twice the classical one. This effect is called WL. This effect can be destroyed if we think about the Aharonov- Bohm-Effect[AB59]: Switching on a magnetic field, Fig.(3.3), we can add an additional phase to the propagating electron which destroys the constructive interference, leading to an enhanced conductivity, |P(r,r)| 2 ≈ ∑ α = ∑ α 〈(A αe i2πϕ +A αe −i2πϕ )(A ∗ α e −i2πϕ +A ∗ α e i2πϕ )〉 (3.5) 〈2|A α| 2 (1+cos(4πϕ))〉 (3.6) with ϕ = φ/φ 0 ,φ 0 = h e . Because of 〈(1 +cos(4πϕ))〉 = 1 we have no quantum corrections on average in this case. The quantum correction to the conductivity appears therefore in the difference between the classical diffusion with the corresponding propagator (Diffuson) and the propagation which includes quantum interference between two possibilities for a particle to traverse a closed loop in opposite -time reversed- directions described by the Cooperon. To calculate this quantum correction at zero frequency ω, one applies Kubo formula (Appendix B.1) which yields σ xx (ω = 0) ≡ σ = e2 2π m 2 e Vol ∫ ∞ 0 dE ( − ∂f(E) ) 〈Tr[δ(E −H 0 )p x δ(E −H 0 )p x ]〉 ∂E imp . (3.7)
Page 1 and 2: Itinerant Spin Dynamics in Structur
Page 3 and 4: Itinerant Spin Dynamics in Structur
Page 5 and 6: Contents v 3.2.2 Weak Localization
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: Chapter 3: WL/WAL Crossover and Spi
Page 41 and 42: 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
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Chapter 4: Direction Dependence of
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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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