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

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Chapter 2 Spin Dynamics: Overview and Analysis of 2D Systems 2.1 Short Reminder on the Origin of Spin Orbit Coupling Theinteraction whichmakes spintronicdevices sointeresting is arelativistic effect, theSOC.TheintrinsicdegreeoffreedomspinisadirectconsequenceoftheLorentzinvariant formulation of quantum mechanics. Expanding the relativistic Dirac equation in the ratio of the electron velocity and the speed of light up to the order (v/c) 2 (the derivation can be found in standard text books like Ref.[Sak67]) one gets ( p 2 2m e0 −eϕ− p4 8m 3 e0 c2 − e 4m 2 e0 c2σ ·(∇ϕ×p)− e ) 8m 2 e0 c2∆ϕ ψ = (E −m e0 c 2 )ψ, with the electrostatic potential ϕ, and the free electron mass m e0 . Our interest concerns the so called Thomas term −e/(4m 2 e0 c2 )σ·(∇ϕ×p). In atomic physics we assume that the electric field is a central field, E(r) = −(dϕ/dr)e r which leads to with ŝ = σ/2. − e e 4m 2 ·(∇ϕ×p) = − e0c2σ (− 1 4m 2 e0 c2 r e 1 = − 2m 2 e0 c2 r ) dϕ σ ·(r×p) (2.1) dr dϕ (2.2) drŝ·L ≡ λŝ·L, (2.3) Duetothelattice-periodic potential inacrystallinesolidthiseffect canhavestronginfluence 6
Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 7 Figure 2.1: Schematic representation of the band structure of GaAs around the Γ point (extracted from Ref.[Bur08]). on the energy band structure. Using the Kane Model it can be shown that the SOC due to lattice-periodic potential leads to a lifting of degeneracy with a gap ∆ SO = 3/(4λ) (with λ form Eq.(2.3)) between the Γ 8v states describing heavy and light holes and Γ 7v states, as sketched in Fig.2.1 where Γ 7v is represented as the split-off band.[DR93, Win04a] It is important to notice that in the following we do not focus on the SOC which is due to strong Coulomb potential of the atomic core regions, but on the appearance of SO effect in the conduction band due to additional external electric field and spatial symmetry breaking as explained in more detail in the next sections. Notice that the effects we present in this work are therefore on a different energy scale: The Pauli splitting ∆ SO can be large, 0.34 eV in GaAs, compared to splitting at the conduction band which is on the order of meV in GaAs. Before we review the spin dynamics of conduction electrons and holes in semiconductors and metals, let us first reconsider the spin dynamics of a localized spin, as governed by the Bloch equations.
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: Chapter 1: Introduction 5 Throughou
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
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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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