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60 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems or dephasing rate as the Zeeman coupling mixes all the spin states, breaking time-reversal invariance. Strong Field If we expand the spectrum in 1/˜B ≪ 1, we find that all modes have the same gap proportional to the strength of the SO coupling, D e Q 2 SO , while two modes attain a finite imaginary part with opposite sign E B,2D,1 /D e Q 2 SO = 1+K 2 x, (3.106) E B,2D,2 /D e Q 2 SO = 1+K2 x +O(1/˜B), (3.107) E B,2D,3/4 /D e Q 2 SO = 1+K 2 x ∓i ˜B +O(1/˜B). (3.108) Thus, a strong Zeeman field polarizes the spins and leads to their precession. The SO interaction, which is too weak to flip the spins, merely results in a relaxation of all modes, corresponding to a dephasing of the spin precession. 3.5.2 Quantum Wire with Spin-Conserving Boundary Conditions In the following, we want to study if a Zeeman field modifies the magnetoconductivity and can shift the crossover from positive to negative Magnetoconductivity as function of wire width W. We have seen that for appropriate parameters the critical width W c is small compared with L SO . Therefore we stay in the 0-mode approximation to get a better overview of the physics. To do so, we first analyze the spectrum. The modes with low decay rates are situated at K x = 0 and K x ≈ ±1 for small widths and small enough Zeeman field, ˜B 1, as can be seen in Fig.3.17. For Kx = 0, we have ( E min,0 /D e Q 2 = ˜B 2 SO + ˜B 4 1+ (Q SOW) 2 ) (3.109) 12 and for K x = ±1, E min,±1 /D e Q 2 SO = (Q SOW) 2 ( + 24 ˜B 1 2 2 − (Q SOW) 2 ) 96 ( + ˜B 3 4 16 − (Q SOW) 2 384 ) . (3.110) As in the 2D case, we have a mode which is independent of the Zeeman field and the spectrum is equal to E t0 with the eigenvector (0,1,0,1) T . Using this spectrum, we estimate
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 61 4 3 R[E]/DeQ 2 SO 2 1 0 2 1 0 1 2 (a) K x 1.5 1.0 I[E]/DeQ 2 SO 0.5 0.0 0.5 1.0 1.5 2 1 0 1 2 K x (b) ˜B 0 1 2 Figure 3.16: (a) Real and (b) imaginary parts of the spectrum of the 2D Cooperon with Zeeman term of the strength gµ B B/D e Q 2 SO = 0...2 in steps of 0.25 (w.l.o.g K y = 0). The B independent mode E T0 is not shown.
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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
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Chapter 1: Introduction 5 Throughou
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Chapter 2: Spin Dynamics: Overview
Page 19 and 20: 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 69: 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
Page 115 and 116: Chapter 5: Spin Hall Effect 105 Fin
Page 117 and 118: Chapter 6: Critical Discussion and
Page 119 and 120: 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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