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

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42 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems x 0 L SO 2 L SO S 〈S〉 0 y 0 −S 0 √ 2S0 0 〈S〉 z − √ 2S 0 Figure 3.5: Persistent spin helix solution of the spin-diffusion equation for equal magnitude of linear Rashba and linear Dresselhaus coupling, Eq.(3.68). The condition for persistence is thus rather ˜α 1 = α 2 . This has been confirmed in a recent measurement (Ref. [KWO + 09]). The existence of such long-living modes has an effect on the quantum corrections to the conductivity. In this case, ˜α 1 = α 2 = α 0 , there is only WL in 2D.[PP95, SKK + 08] In the next sections we will make use of the equivalence of the triplet sector of the Cooperon propagator and the spin-diffusion propagator in quantum wires with appropriate boundary conditions and show how long-living modes may change the quantum corrections to the conductivity. 3.4 Solution of the Cooperon Equation in Quantum Wires 3.4.1 Quantum Wires with Spin-Conserving Boundaries The conductivity of quantum wires with width W < L ϕ = √ D e τ ϕ is without SO interaction dominated by the transverse zero-mode Q y = 0. This yields the quasi-1D WL correction.[KCCC92] However, in the presence of SO interaction, setting simply Q y = 0 is not correct. If we consider spin-conserving boundaries, rather one has to solve the Cooperon equation with the following modifiedboundaryconditions as derived in AppendixC.2(Refs. [AF01, MFA02]): (− τ D e n·〈v F [γ g B SO (k)·S]〉−i∂ n ) C| ∂S = 0, (3.70)
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 43 where 〈...〉 denotes the average over the direction of v F and k which we rewrite using Eq.(C.28) for the given geometry as (−i∂ y +2e(A S ) y )C ( x,y = ± W ) = 0, ∀x, (3.71) 2 where n is the unit vector normal to the boundary ∂S and x is the coordinate along the wire. The transverse zero-mode Q y = 0 does not satisfy this condition. Therefore, it is convenient to perform a non-Abelian gauge transformation,[AF01, MC00] so that the transformed problem has Neumann boundary conditions, and the transformed Cooperon Hamiltonian can therefore be diagonalized in zero-mode approximation for quantum wires. Since in quantum wires these boundary conditions apply only in the transverse direction, a transformation acting in the transverse direction is needed: Ĉ → ˜Ĉ = U A ĈU † A , with U A = exp(i2e(A S ) y y). Then, the boundary condition simplifies to −i∂ y ˜C(x,y = ±W/2) = 0, ∀x, and the Hamiltonian changes to ˜H c = Q 2 −2Q SO Q x [cos(Q SO y)S y −sin(Q SO y)S z ] +Q 2 SO [cos2 (Q SO y)Sy 2 +sin2 (Q SO y)Sz 2 −sin(Q SO y)cos(Q SO y)(S y S z +S z S y )] (3.72) = (Q+2eÃ s ) 2 . (3.73) where the effective vector potential A S , as introduced in Eq.(3.43), A S = m e e ˆαS = m e e ⎛ ⎝ 0 −α 2 0 α 2 0 0 ⎛ ⎞ ⎞ S x ⎠⎜ ⎝ S y S z ⎟ ⎠ , (3.74) is transformed to the effective vector potential Ã s after the transformation U A has been applied to the Hamiltonian Ã s ≡ m e e ˜ˆα(y)S ⎛ = m e e ⎝ 0 −α 2cos(Q SO y) −α 2 sin(Q SO y) 0 0 0 ⎛ ⎞ ⎠⎜ ⎝ ⎞ S x S y ⎟ ⎠ , (3.75) S z which varies with the transverse coordinate y on the length scale of L SO . Now, we can see already that for narrow wires W < L SO , this vector potential varies linearly with y,
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 51: 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
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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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