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44 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems Ã s ∼ −m e α 2 Q SO y/e, like the vector potential of the external magnetic field B. Thus, it follows, that for W < L SO , the spin relaxation rate is 1/τ s ∼ Q 2 SO〈y 2 〉 ∼ Q 2 SOW 2 /12, vanishing for small wire widths. As announced at the beginning, we thus see that the presence of boundaries diminishes the spin relaxation already at wire widths of the order of L SO . If we include only pure Neumann boundaries to the Hamiltonian H c , i.e., using the wrong covariant derivative, this would not affect the absolute spin relaxation minimum and it would be equal to the nonzero one in the 2D case. We give a more precise answer in the following. 3.4.2 Zero-Mode Approximation For W < L ϕ , we can usethe fact that thenthtransversenonzero-modes contribute terms to the conductivity which are by a factor W/nL ϕ smaller than the 0-mode term, with n a nonzero integer number. Therefore, it should be a good approximation to diagonalize the effective quasi-one-dimensional Cooperon propagator, which is the transverse 0-mode expectation value of the transformed inverse Cooperon propagator, Eq.(3.73), ˜H1D = 〈0 | ˜H c | 0〉. It is crucial to note that ˜H 1D contains additional terms, created by the non-Abelian transformation, whichshowsthat takingjustthetransversezero-modeapproximation of the untransformed Eq.(3.48) would yield a different, incorrect result. We can now diagonalize ˜H 1D and finally find the dispersion of quasi-1D triplet modes E t0 = Q 2 x + 1 D e 2 Q2 SOt SO , (3.76) ( √ ) E t± = Q 2 x D + 1 e 4 Q2 SO 4−t SO ± t 2 +64 Q2 x SO Q 2 (1+c SO (c SO −2)) , (3.77) SO where c SO and t SO are functions of the wire width W as given by c SO = 1− 2sin(Q SOW/2) , t SO = 1− sin(Q SOW) Q SO W Q SO W . (3.78) One notices that in the limit of Q SO W → ∞ we do not recover the previous 2D solution. This boundary effect will be clarified later on. Inserting Eq.(3.76) and Eq.(3.77) into the expression for the quantum correction to the conductivity Eq.(3.46), takingintoaccount themagneticfieldbyinsertingthemagneticrate 1/τ B (W) and the finite temperature by inserting the dephasing rate 1/τ ϕ (T), it remains to perform the integral over momentum Q x , as has been done in Ref. [Ket07]. For Q SO W < 1,
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 45 the WL correction can then be written as √ √ HW ∆σ = √ Hϕ +B ∗ (W)/4 − HW √ Hϕ +B ∗ (W)/4+H s (W) √ HW −2√ Hϕ +B ∗ (W)/4+H s (W)/2 (3.79) in units of e 2 /2π. We defined H W = 1/4eW 2 and the effective external magnetic field ( ) −1 ) B ∗ (W) = 1− (1+ W2 B. (3.80) The spin relaxation field H s (W) is for Q SO W < 1, 3l 2 B H s (W) = 1 12 (Q SOW) 2 H s , (3.81) suppressed in proportion to (W/L SO ) 2 similar to B ∗ (W), Eq.(3.80). Here, H s = 1/4eD e τ s , with 1/τ s = 2p 2 F α2 2τ. As mentioned above, the analogy to the suppression of the effective magnetic field, Eq.(3.80), is expected, since the SO coupling enters the transformed Cooperon, Eq.(3.73), like an effective magnetic vector potential.[Fal03] Cubic Dresselhaus coupling, however, would give rise to an additional spin relaxation term, see Eqs. (3.45) and (C.35), which has no analogy to a magnetic field and is therefore not suppressed in diffusive wires. In Chapter4 we will show that this additional term, though it cannot vanishfor Q SO W ≪ 1, is widthdependent, since theterm Eq.(3.45) in theCooperon Hamiltonian H c is also transformed to obtain the modified Neumann boundaries by applying the transformation U A , which is W dependent. When W is larger than SO length L SO , coupling to higher transverse modes may become relevant even if W < L ϕ is still satisfied, since the SO interaction may introduce coupling to highertransversemodes.[Ale06]Wewillstudythesecorrectionsbynumericalexactdiagonalizationinthenextsection. Onecanexpectthatinballisticwires,l e > W,thespinrelaxation rate is suppressed in analogy to the flux cancellation effect, which yields the weaker rate, 1/τ s (W) = (W/Cl e )(D e W 2 /12L 4 S), where C = 10.8.[BvH88a, DK84, KM02] Before we investigate the exact diagonalization in the pure Rashba case, we consider an anisotropic field with linear Rashba and Dresselhaus SO coupling to see which form the long persisting spindiffusion modes have in narrow wires. Also, here, we can take advantage of the equivalence of Cooperon and spin-diffusion equation as far as time-reversal symmetry is not violated. We findthreesolutions whosespinrelaxation ratedecay proportional toW 2 forα 2 ≠ α 1 and
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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 53: 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⩵
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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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