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

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34 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 3.2.2 Weak Localization in Quantum Wires If the host lattice of the electrons provides SO interaction, quantum corrections to the conductivity have to be calculated in the basis of eigenstates of the Hamiltonian with SO interaction H = 1 2m e (p+eA) 2 +V(x)− 1 2 γ gσ(B+B SO (p)), (3.32) where m e is the effective electron mass (see AppendixA for examples in semiconductors). A is the vector potential due to the external magnetic field B. B T SO = (B SOx ,B SOy ) is the momentum dependent SO field. σ is a vector, with components σ i , i = x,y,z, the Pauli matrices, γ g is the gyromagnetic ratio with γ g = gµ B with the effective g factor of the material, and µ B = e/2m e is the Bohr magneton constant. In Sec.2.3.3 we presented the dominant SO interactions in semiconductors: For example, the breaking of inversion symmetry in III-V semiconductors causes a SO coupling, which for quantum wells grown in the [001] direction is given by [Dre55] − 1 2 γ gB SO,D = α 1 (−ê x p x +ê y p y )+γ D (ê x p x p 2 y −ê yp y p 2 x ). (3.33) Here, α 1 = γ D 〈p 2 z〉 is the linear Dresselhaus parameter, which measures the strength of the term linear in momenta p x ,p y in the plane of the 2DES. When 〈p 2 z 〉 ∼ 1/a2 z ≥ k2 F (a z is the thickness of the 2DES and k F is the Fermi wavenumber), that term exceeds the cubic Dresselhaus terms which have coupling strength γ D . Asymmetric confinement of the 2DES yields the Rashba term which does not depend on the growth direction − 1 2 γ gB SO,R = α 2 (ê x p y −ê y p x ), (3.34) with α 2 the Rashba parameter.[BR84, Ras60] We consider the standard white-noise model for the impurity potential, V(x), which vanishes on average 〈V(x)〉 = 0, is uncorrelated, 〈V(x)V(x ′ )〉 = δ(x−x ′ )/2πντ, and weak, E F τ ≫ 1. Going to momentum (Q) and frequency (ω) representation, and proceeding as presented Sec.3.2.1 but now taking into account the spin degree of freedom for a two electron interference, we yield the quantum correction to the static conductivity as [HLN80] ∆σ = −2 e2 D e ∑ 2π Vol Q ∑ α,β=± C αββα,ω=0 (Q), (3.35) where α,β = ± are the spin indices, and the Cooperon propagator Ĉ is for E Fτ ≫ 1 (E F , Fermi energy), given by Eq.(3.30). The summation over the spins is described in more
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 35 detail in AppendixC.1. Thus, the problem reduces to the calculation in presence of SOC of the correlation function ∑ q E,p + q E ′ ,p ′ −q = 1 2πντ ∑ GE,σ R (p+q)G E A ′ ,σ ′(p′ −q), (3.36) q which simplifies for weak disorder ǫ F τ ≫ 1 to ∫ dΩ 1 ≈ 2π 1−iτˆΣ , (3.37) where ˆΣ = ǫ p ′ +q,σ ′ −ǫ p−q,σ. (3.38) For diffusive wires, for which the elastic mean-free path l e is smaller than the wire width W, the integral is over all angles of velocity v on the Fermi surface. Using ǫ p = (p+eA)2 − 1 2m e 2 γ gσ(B+B SO (p)), v = p−q+eA , m e S = 1 2 (σ +σ′ ), Q = p+p ′ , we obtain to lowest order in Q, ˆΣ = −v(Q+2eA+2m e âS)+(Q+2eA)âσ ′ + 1 2 γ g(σ ′ −σ)B. (3.39) Here, the SO couplings are combined in the matrix ⎛ Thus, the Cooperon becomes Ĉ(Q) −1 = 1 τ â = ( ∫ dΩ 1− 2π ⎝ −α 1 +γ D ky 2 −α 2 α 2 α 1 −γ D kx 2 1 1+iτ(v(Q+2eA+2m e âS)+H σ ′ +H Z ) ⎞ ⎠. (3.40) ) , (3.41) where H σ ′ = −(Q+2eA)âσ ′ and the Zeeman coupling to the external magnetic field yields H Z = − 1 2 γ g(σ ′ −σ)B. (3.42)
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 35 and 36: Chapter 3 WL/WAL Crossover and Spin
Page 37 and 38: Chapter 3: WL/WAL Crossover and Spi
Page 43: 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
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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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