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

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54 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems Q SO Wc 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1/ ( ) D e Q 2 SOτ ϕ Figure 3.13: Width of wire WQ SO at which there is a crossover from negative to positive magnetoconductivity as function of the lower cutoff 1/D e Q 2 SOτ ϕ . electron is scattered from such a smooth boundary.[GKD04] If this applies, the potential is adiabatic and the spin of the scattered electron stays parallel to the field B SO as its momentum is changed. This leads to the boundary condition for the spin-density[SDGR06] s x | y=±W/2 = 0, (3.85) s y | y=±W/2 = 0, (3.86) ∂ y s z | y=±W/2 = 0. (3.87) We can transform this boundary condition to the one of the triplet Cooperon by using the unitary rotation between the spin density in the s i representation and the triplet representation of the Cooperon, ˜s i , Eq.(3.62), which leads to the boundary condition 1 √ 2 (−s x +is y )| y=±W/2 = ˜s ⇈ | y=±W/2 = 0, (3.88) ∂ y s z | y=±W/2 = ∂ y˜s ⇉ | y=±W/2 = 0, (3.89) 1 √ (s x +is y )| y=±W/2 = ˜s | y=±W/2 = 0. 2 (3.90) Now, if werequirevanishingmagnetization for the1Dcase, then thediagonalization is done straightforwardly, plotted in Fig.3.14 (see also Ref. [SDGR06]). We use a basis which satisfies the boundary conditions and therefore consists of ∼sin(qy)(1,0,0) T , ∼cos(qy)(0,1,0) T , and ∼sin(qy)(0,0,1) T , with q = nπ/W, n ∈ N ∗ . However, looking at the spin-diffusion operator [Eq.(3.57)], we see immediately that if we set k to zero and use the fact that s x,y must vanish at the boundary and s z has to be constant for the chosen k, we receive
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 55 1.0 0.8 EDQ 2 SO 0.6 0.4 0.2 0.0 0 5 10 15 20 WQ SO Figure 3.14: Spectrum in the case of adiabatic boundaries. Except for states which are polarized in z direction, the relaxation rates for all states diverge at small wire widths, Q SO W ≪ 1. a polarized mode. Although this mode is a trivial solution, it differs from all other due to the fact that it has a finite spin relaxation time as Q SO W vanishes. For the choice of basis for diagonalization, this means: We set q i = n i π/W for respective ˜s i therewith one state is described by {n 1 ,n 2 ,n 3 } = {n,n+p 2 ,n +p 3 }, n ∈ N ∗ , p 2 ∈ {−1,0,1,···}, p 3 ∈ N. In the case {n 1 ,n 2 ,n 3 } = {n,n,n} all branches diverge with reference to the eigenvalues in the limes of Q SO W → 0, so that the spin relaxation time goes to zero for small wires.[SDGR06] In contrast, there is an additional branch in the case of p 2 = −1,p 3 = 0 which has a finite eigenvalue and therefore finite spin relaxation time for small wire widths E D e Q 2 SO = 2+Kx (1− 2 32(Q SOW) 2 ) π 4 +O[(Q SO W) 4 ]. (3.91) The smallest spin relaxation rate for vanishing Q SO W, 1/τ s,1D , which is given for k x = 0, is foundto bean eigenstate polarized in z direction which relaxes with the rate 1/τ s,1D = 2/τ s . It shows compared with the other modes a monotonous behavior as function of Q SO W. If we allow magnetization for the 1D case, then the combination p 1 = −1,p 2 = 0 leads to a valid solution. For wide wires the smallest absolute minimum is the 2D minimum E/D e = (7/16)Q 2 ; there are no edge modes. But already at a width of Q SO SOW = π/ √ 5 ≈ 1.4 all modes except the z-polarized exceed the rate 1/τ s,1D .
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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
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 63: 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
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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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