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

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8 Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 2.2 Dynamics of a Localized Spin A localized spin ŝ, like a nuclear spin, or the spin of a magnetic impurity in a solid, precesses in an external magnetic field B due to the Zeeman interaction with Hamiltonian H Z = −γ g ŝB, where γ g is the corresponding gyromagnetic ratio of the nuclear spin or magnetic impurityspin, respectively, whichwewill set equal toone, unlessneededexplicitly. This spin dynamics is governed by the Bloch equation of a localized spin, ∂ t ŝ = γ g ŝ×B. (2.4) This equation is identical to the Heisenberg equation ∂ t ŝ = −i[ŝ,H Z ] for the quantum mechanical spin operator ŝ of an S = 1/2-spin, interacting with the external magnetic field B due to the Zeeman interaction with Hamiltonian H Z . The solution of the Bloch equation for a magnetic field pointing in the z-direction is ŝ z (t) = ŝ z (0), while the x- and y- components of the spin are precessing with frequency ω 0 = γ g B around the z-axis, ŝ x (t) = ŝ x (0)cosω 0 t+ŝ y (0)sinω 0 t, ŝ y (t) = −ŝ x (0)sinω 0 t+ŝ y (0)cosω 0 t. Since a localized spininteracts withits environment by exchange interaction and magnetic dipoleinteraction, the precession will dephase after a time τ 2 , and the z-component of the spin relaxes to its equilibrium value s z0 within a relaxation time τ 1 . This modifies the Bloch equations to the phenomenological equations, ∂ t ŝ x = γ g (ŝ y B z −ŝ z B y )− 1 τ 2 ŝ x ∂ t ŝ y = γ g (ŝ z B x −ŝ x B z )− 1 τ 2 ŝ y ∂ t ŝ z = γ g (ŝ x B y −ŝ y B x )− 1 τ 1 (ŝ z −s z0 ). (2.5) 2.3 Spin Dynamics of Itinerant Electrons 2.3.1 Ballistic Spin Dynamics Starting from Dirac equation we have seen that one obtains in addition to the Zeeman term a term which couples the spin s with the momentum p of the electrons, the spin-orbit coupling H SO = − µ B 2m e0 c 2ŝ p×E = −ŝB SO(p), (2.6)
Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 9 where we set the gyromagnetic ratio γ g = 1. E = −∇ϕ, is an electrical field, and B SO (p) = µ B /(2m e0 c 2 )p×E. Substitution into the Heisenberg equation yields the Bloch equation in the presence of spin-orbit interaction: ∂ t ŝ = ŝ×B SO (p), (2.7) so that the spin performs a precession around the momentum dependent spin-orbit field B SO (p). It is important to note, that the spin-orbit field does not break the invariance under time reversal ( ŝ → −ŝ,p → −p ), in contrast to an external magnetic field B. Therefore, averaging over all directions of momentum, there is no spin polarization of the conduction electrons. However, injecting a spin-polarized electron with given momentum p into a translationally invariant wire, its spin precesses in the spin-orbit field as the electron moves through the wire. The spin will be oriented again in the initial direction after it moved a length L SO , the spin precession length. The precise magnitude of L SO does not only depend on the strength of the spin-orbit interaction but may also depend on the direction of its movement in the crystal, as we will discuss below. 2.3.2 Spin Diffusion Equation Translational invariance isbroken bythepresenceofdisorderdueto impuritiesand lattice imperfections in the conductor. As the electrons scatter from the disorder potential elastically, their momentum changes in a stochastic way, resulting in diffusive motion. That results in a change of the the local electron density ρ(r,t) = ∑ α=± | ψ α(r,t) | 2 , where α = ± denotes the orientation of the electron spin, and ψ α (r,t) is the position and time dependent electron wave function amplitude. On length scales exceeding the elastic mean free path l e , that density is governed by the diffusion equation ∂ρ ∂t = D e∇ 2 ρ, (2.8) where the diffusion constant D e is related to the elastic scattering time τ by D e = v 2 F τ/d D, wherev F is the Fermi velocity, and d D the diffusion dimension 1 of the electron system. That diffusion constant is related to the mobility of the electrons, µ e = eτ/m e by the Einstein relation µ e ρ = e2νD e , where ν is the density of states (DOS) per spin at the Fermi energy E F and m e the effective electron mass. 1 d D can have a fractal value e.g. on quasi-periodic lattices
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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: Chapter 2: Spin Dynamics: Overview
Page 21 and 22: 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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