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

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56 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems Tubular Wires In tubular wires, such as carbon nanotubes, and InN nanowires in which only surface electrons conduct,[PHC + 09] and radial core-shell InO nanowires,[JLS + 08] the tubular topology of the electron system can be taken into account by periodic boundary conditions. Inthefollowing, wefocusonwireswherethedominant SOcouplingisofRashbatype. Ifone requires furthermore that this SO-coupling strength is uniform and the wire curvature can be neglected,[PHC + 09] the spectrum of the Cooperon propagator can be obtained by substituting in Eq.(3.52) the transverse momentum Q y by the quantized values Q y = n2π/W, n is an integer, when W is the circumference of the tubular wire. Thus, the spin relaxation rate remains unchanged, 1/τ s = (7/16)D e Q 2 SO. If then a magnetic field perpendicular to the cylinder axis is applied as done in Ref. [PHC + 09], there remains a negative magnetoconductivity due to the WAL, which is enhanced due to the dimensional crossover from the 2D correction to the conductivity Eq.(C.39) to the quasi-one-dimensional behavior of the quantum correction to the conductivity [Eq.(3.79)]. In tubular wires in which the circumference fulfills the quasi-one-dimensional condition W < L ϕ , the WL correction can then be written as ∆σ = √ √ HW √ Hϕ +B ∗ (W)/4 − HW √ Hϕ +B ∗ (W)/4+H s (W) √ HW −2√ Hϕ +B ∗ (W)/4+7H s (W)/16 (3.92) in units of e 2 /2π. As in Eq.(3.79), we defined H W = 1/4eW 2 , but the effective external magnetic field differs due to the different geometry: Assuming that W < l B , we have[Ket07] 1 = D e (2e) 2 B 2 〈y 2 〉 (3.93) τ B = D e (2e) 21 ( ) BW 2 (3.94) 2 2π and the effective external magnetic field yields ( ) BW 2 B ∗ (W) = (2e) (3.95) 2π = (2e)(Br tube ) 2 , (3.96)
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 57 with the tube radius r tube . The spin relaxation field H s is H s = 1/4eD e τ s , with 1/τ s = 2p 2 F α2 2 τ, or in terms of the effective Zeeman field B SO, H s = gγ g 16 B SO (ǫ F ) 2 ǫ F . (3.97) Thus the geometrical aspect, 〈y 2 〉 tube /〈y 2 〉 planar ≈ 6.6, might resolve the difference between measured and calculated SO coupling strength in Ref. [PHC + 09] where a planar geometry has been assumed to fit the data. This assumption leads in a tubular geometry to an underestimation of H s (W). The flux cancellation effect is as long as we are in the diffusive regime, l e ≪ W, negligible. 3.5 Magnetoconductivity with Zeeman splitting In the following, we want to study if the Zeeman term, Eq.(3.42), is modifying the magnetoconductivity. Accordingly, we assume that the magnetic field is perpendicular to the 2DES. Taking into account the Zeeman term to first order in the external magnetic field B = (0,0,B) T , the Cooperon is according to Eq.(3.43) given by Ĉ(Q) = 1 D e (Q+2eA+2eA S ) 2 +i 1 2 γ g(σ ′ −σ)B . (3.98) This is valid for magnetic fields γ g B ≪ 1/τ. Due to the term proportional to (σ ′ −σ), the singletsectoroftheCooperonmixeswiththetripletone. Wecanfindtheeigenstates ofC −1 , |i〉 with the eigenvalues 1/λ i . Thus, the sum over all spin up and down combinations αβ,βα in Eq.(3.35) for the conductance correction simplifies in the singlet-triplet representation to (AppendixC.1) ∑ C αββα = ∑ i αβ (−〈⇄ |i〉〈i|⇄〉+〈⇈ |i〉〈i|⇈〉 +〈⇉ |i〉〈i|⇉〉+〈 |i〉〈i|〉)λ i . (3.99) 3.5.1 2DEG The coupling of the singlet to the triplet sector lifts the energy level crossings at K = ±1/ √ 2 of the singlet E S and the triplet branch E T− as can be seen in Fig.3.15 for a nonvanishing Zeeman coupling. The spectrum, which is not positive definite anymore for
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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
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
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Page 77 and 78: Chapter 4 Direction Dependence of S
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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
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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
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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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