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

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12 Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems precess around the x-axis as it moves along. For narrow quantum wells, where 〈kz 2〉 ∼ 1/a 2 z ≥ kF 2 the linear term exceeds the cubic Dresselhaus terms. For a typical value of 〈kz 2〉 = 0.036/nm2 , one gets accordingly[Win04b], α 1 =0.99 meVnm (GaAs), 0.98 meVnm (InAs), 27.4 meVnm (InSb). A special situation arises for quantum wells grown in the [110]- direction, where it turns out that the spin-orbit field is pointing normal to the quantum well, as shown in Fig.2.2, so that an electron whose spin is initially polarized along the normal of the plane, remains polarized as it moves in the quantum well. 0 SIA 0 BIA111 0 0 010 0 BIA001 110 0 BIA110 0 001 0 1 10 0 100 Figure 2.2: The spin-orbit vector fields for linear structure inversion asymmetry (Rashba) coupling, and for linear bulk inversion asymmetry (BIA) spin-orbit coupling for quantum wells grown in [111], [001] and [110] direction, respectively. In quantum wells with asymmetric electrical confinement the inversion symmetry is broken as well. The resulting spin-orbit coupling, the structural inversion asymmetry coupling
Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 13 (SIA), also called Rashba-spin-orbit interaction[Ras60] is given by H R = α 2 (σ x k y −σ y k x ), (2.15) whereα 2 dependsontheasymmetryoftheconfinementpotentialV(z)inthedirectionz, the growth direction of the quantum well, and can thus be deliberately changed by application of a gate potential. This dependence allows one, in principle, to control the electron spin with a gate potential, which can therefore be used as the basis of a spin transistor.[DD90] One should stress that the expectation value of the electrical field E c = −∂ z V(z) in the conduction band state vanishes if the effective mass m e is not position dependent. However it can be shown that the parameter α 2 is modulated by the electric field in the valence band and the z-dependent Pauli splitting ∆ 0 .[FMAE + 07] Several measured values of α 2 are listed in Tab.A and ratios of α 2 and α 1 in Tab.A. We can combine all spin-orbit couplings by introducing the spin-orbit field such that the Hamiltonian has the form of a Zeeman term: H SO = −sB SO (k), (2.16) where the spin vector is s = σ/2. But we stress again that since B SO (k) → B SO (−k) = −B SO (k)underthetimereversaloperation, spin-orbitcouplingdoesnotbreaktimereversal symmetry, since the time reversal operation also changes the sign of the spin, s → −s. Only an external magnetic field B breaks the time reversal symmetry. Thus, the electron spin operator ŝ is for fixed electron momentum k governed by the Bloch equations with the spin-orbit field, ∂ŝ ∂t = ŝ×(B+B SO(k))− 1ˆτ s ŝ. (2.17) The spin relaxation tensor is no longer necessarily diagonal in the presence of spin-orbit interaction as will be shown in Sec.2.4.1. In narrow quantum wells where the cubic Dresselhaus coupling is weak compared to the linear Dresselhaus and Rashba couplings, the spin-orbit field is given by ⎛ ⎞ −α 1 k x +α 2 k y B SO (k) = −2⎜ ⎝ α 1 k y −α 2 k x ⎟ ⎠ , (2.18) 0 whichchangesbothitsdirectionanditsamplitude| B SO (k) |= 2 √ (α 2 1 +α2 2 )k2 −4α 1 α 2 k x k y , as the direction of the momentum k is changed. Accordingly, the electron energy dispersion
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 21: 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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Chapter 3: WL/WAL Crossover and Spi
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Chapter 3: WL/WAL Crossover and Spi
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Chapter 4 Direction Dependence of S
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Chapter 4: Direction Dependence of
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Chapter 5 Spin Hall Effect 5.1 Intr
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Chapter 5: Spin Hall Effect 85 curr
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Chapter 5: Spin Hall Effect 87 with
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Chapter 5: Spin Hall Effect 89 1.0
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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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