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

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32 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems where d is the dimension. To get the quantum corrections to the Drude conductivity, we have to include the additional contribution by considering the connection ofG R andG A due to the impurity potential V: 〈Γ〉 imp = + ( + + ) +··· (3.20) This sum can be separated into uncrossed and crossed diagrams. As known from standard literature both can be calculated in an analog way. Summing up only ladder diagrams will lead to the Diffuson ˆD Γ D E,E ′(p,p′ ) = ( ) δ p,p ′ + + +··· = 1 p+q (3.21) 1− ∑ q p ′ +q = G R E(p)G A E ′(p′ ) 1 τ ˆD E,E ′(p,p ′ ), (3.22) It is important to notice that each ladder diagram is of the same order as the Drude diagram 1 . In contrast to this classical contribution, the diagrams where the impurity lines, which connect the advanced and retarded lines, cross are smaller by the factor 1/(p F l) (see e.g. Ref.[Ram82]). Eq.(3.22) can be solved easily for ˆD if we expand the Diffuson in (E ′ −E) and (p ′ −p) (the pole stems from particle conservation): ˆD E,E ′(p,p ′ ) = 1 i(E −E ′ )+D e (p ′ −p) 2, (3.23) with the diffusion constant D e = v 2 F τ/d. If Rσ xx is now calculated not only by using the bubble diagram, we end up with a correction of the momentum relaxation time τ in Eq.(3.19) being replaced by the transport time ∫ τ 0 ∼ dp F |V(p F −p ′ F )|2 (1−p F ·p ′ F ). (3.24) However, we are interested in the calculation which goes beyond this class of diagrams. Time-reversal symmetry helps to sum up the group of crossed diagrams via unknotting 1 bubble diagram
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 33 them: p i p o p i p o Γ = Γ (3.25) p o ′ p i ′ p i ′ p o ′ p i p o p i p o = Γ = Γ (3.26) p i ′ p o ′ −p o ′ −p i ′ Exploiting Eq.(3.25) makes the calculation of the maximally crossed diagrams easier: Γ C E,E ′(p,p′ ) = + + + ··· (3.27) = 1 p+q (3.28) 1− ∑ q p ′ −q = G R E (p)G A E ′(p′ ) 1 τĈE,E ′(p,p′ ), (3.29) with the Cooperon 2 propagator Ĉ for E Fτ ≫ 1 (E F , Fermi energy) given by Ĉ ω=E−E ′(Q = p+p ′ ) = τ ⎛ ⎜ ⎝1− ∑ q E,p + q E ′ ,p ′ −q ⎞ ⎟ ⎠ −1 . (3.30) In contrast to the Diffuson, the infrared divergence is now at p = −p ′ , i.e. the correction to the conductivity for ω = 0, ∆σ = 2 e2 π 1 ∑ m 2 (−p 2 x )G R (p)G A (p)G R (Q−p)G A (Q−p) 1 τĈω=0(Q) (3.31) p,Q is due to the factor (−p 2 x), in the case without magnetic field and SOC, negative. The divergent nature explains post hoc the choice of the maximally crossed diagrams. Notice that one obtains the Cooperon using time-reversal symmetry from the Diffuson. We will use this later to map the Cooperon equation onto the spin diffusion equation. 2 The name stems from the singularity at total momentum being zero, as in the case of a Cooper pair where the consequence is superconductivity.
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 and 38: Chapter 3: WL/WAL Crossover and Spi
Page 41: 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 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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