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

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50 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 0. 6 E {t−,n>0} E t0 E {t−,0} 0. 5 E {t+,0} E/DeQ 2 SO 0. 4 0. 3 0. 2 E {t0,0} 0.1 1 2 3 Q SO W/π 4 5 6 Figure 3.9: Absolute minima of the lowest eigenmodes E {t0,0} , E {t−,0} , and E {t+,0} plotted as function of Q SO W/π = 2W/L SO . We note that the minimum of E {t−,0} is located at ±K x ≠ 0. For comparison, the solution of the zero-mode approximation E t0 is shown. values as function of W, obtained in the 0-mode approximation,[Ket07] is diminished according to the exact diagonalization. However, there remains a sharp maximum of E t0 at Q SO W/π ≈ 1.2 and a shallow maximum of E t− at Q SO W/π ≈ 2.5. As noted above, the values of the energy minima of E t0 and E t− at larger widths W are furthermore diminished as a result of the edge mode character of these modes. Comparison to Solution of Spin Diffusion Equation in Quantum Wires As shown above, the spin-diffusion operator and the triplet Cooperon propagator have the same eigenvalue spectrum as soon as time-symmetry is not broken. Therefore, the minima of the spin-diffusion modes, which yield information on the spin relaxation rate, must be the same as the one of the triplet Cooperon propagator as plotted in Fig.3.9. In Ref. [SDGR06], the value at K x = 0, with K i = Q i /Q SO , has been plotted, as shown in Fig.3.10. We note, however, that this does not correspond to the global minimum plotted in Fig.3.9. The two lowest states exhibit two minima as can be seen in Fig.3.7: one local at K x = 0 and one global, which is for large Q SO W at K x ≈ 0.88. The first one is equal to the results given by Ref. [SDGR06]. For the WL correction to the conductivity, however, it is important to retain the global minimum, which is dominant in the integral over the longitudinal momenta.
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 51 1 0.8 F 1 E/DeQ 2 SO 0.6 0.4 F 2 F 3 0.2 0 0 1 2 3 4 5 Q SO W/π Figure 3.10: Lowest eigenvalues at K x = 0 plotted against Q SO W/π. For comparison, the global minimum of the Cooperon spectrum for Q SO W 9 is plotted, F 3 . Curves F 1 [n] are given by 7/16 + ( (n/(Q SO W/π)) √ 15/4 ) 2 , n ∈ N. F2 shows the energy minimum of the 2D case, F 2 ≡ F 1 [n = 0]. Vertical dotted lines indicate the widths at which the lowest two branches degenerate at K x = 0. They are given by n/( √ 15/4); consider that the wave vector for the minimum of the E T− mode is ( √ 15/4)Q SO . Magnetoconductivity Now, we can proceed to calculate the quantum corrections to the conductivity using the exact diagonalization of the Cooperon propagator. In Fig.3.11, we show the resulting conductivity as function of magnetic field and as function of the wire width W. Here, we have included for all wire widths the lowest seven singlet modes and the lowest 21 triplet modes. We choose this number of modes so that we included sufficient modes to describe correctly the widest wires considered with Q SO W = 10. Thus, for the considered low-energy cutoff, due to electron dephasing rate 1/τ ϕ of 1/D e Q 2 τ SO ϕ = 0.08 and the high energy cutoff 1/D e Q 2 SOτ = 4 due to the elastic scattering rate, we estimate that seven singlet modes fall in this energy range. Since for every transverse mode there are one singlet and three triplet modes, we therefore have to include 21 triplet modes, accordingly. We note a change from positive to negative magnetoconductivity as the wire width becomes smaller than the spin precession length L SO , in agreement with the results obtained within the 0-mode approximation, as reported earlier,[Ket07] plotted for comparison in Fig.3.11 (without shading). At the width, where the crossover occurs, there is a very weak magnetoconductance. This crossover width W c does depend on the lower cutoff, provided by the temperature-dependent dephasing rate 1/τ ϕ . To estimate the dependence of W c on the dephasing rate, we have to analyze the contribution of each term in the denominator of
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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
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 59: 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
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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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