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20 Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems precession term, which yields the term 〈 s(x,t)× ∫ ∆t 0 dt ′ B SO (k(t ′ ))× ∫ ∆t 0 〉 dt ′′ B SO (k(t ′′ )) , (2.29) where 〈...〉 denotes the average over all angles due to the scattering from impurities. Since the electrons move ballistically at times smaller than the elastic scattering time, the momenta are correlated only on time scales smaller than τ, yielding 〈k i (t ′ )k j (t ′′ )〉 = (1/2)k 2 δ ij τδ(t ′ −t ′′ ). (2.30) Noting that (A×B×C) m = ǫ ijk ǫ klm A i B j C l and ∑ ǫ ijk ǫ klm = δ il δ jm −δ im δ jl we find that Eq. (2.29) simplifies to − ∑ i (1/τ sij)S j , where the matrix elements of the spin relaxation terms are given by [DP71c], 1 (ˆτ s ) ij = τ ( 〈B SO (k) 2 〉δ ij −〈B SO (k) i B SO (k) j 〉 ) , (2.31) where 〈...〉 denotes the average over the direction of the momentum k. In Chapter3 we will focus on this kind of spin relaxation and show that these nondiagonal terms can diminish the spin relaxation and even result in vanishing spin relaxation. In the context of weak localization, which is presented in the next Chapter, we will show that the relaxation tensor Eq.(2.31) can be also derived from Cooperon equation (see AppendixC.3). 2.4.2 DP Spin Relaxation with Electron-Electron and Electron-Phonon Scattering It has been noted, that the momentum scattering which limits the D’yakonov- Perel’ mechanism of spin relaxation is not restricted to impurity scattering, but can also be due to electron-phonon or electron-electron interactions[GI02, GI04, PF06, DR04]. Thus the scattering time τ is the total scattering time as defined by, [GI02, GI04], 1/τ = 1/τ 0 + 1/τ ee + 1/τ ep , where 1/τ 0 is the elastic scattering rate due to scattering from impurities ∫ with potential V, given by 1/τ 0 = 2πνn i (dθ/2π)(1 − cosθ) | V(k,k ′ ) | 2 , where ν is the DOS per spin at the Fermi energy, n i is the concentration of impurities with potential V, and kk ′ = kk ′ cos(θ). Concerning the temperature dependence of the spin relaxation, for degenerate electrons in semiconductors (E k = E F , with the Fermi energy E F ) it is given by the temperature dependence of τ(T). However, for a non-degenerate statistics one finds 1/τ s ∼ T 3 τ m (T), where τ m = 〈τ(E k )E k 〉/〈E k 〉.
Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 21 2.4.3 Elliott-Yafet Spin Relaxation Because of the spin-orbit interaction the conduction electron wave functions are not eigenstates of the electron spin, but have an admixture of both spin up and spin down wave functions. Thus, a nonmagnetic impurity potential V can change the electron spin, by changing their momentum due to the spin-orbit coupling. This results in another source of spin relaxation which is stronger, the more often the electrons are scattered, and is thus proportional to the momentum scattering rate 1/τ[Ell54, Yaf63]. For degenerate III-V semiconductors one finds[Cha75, PT84] 1 τ s ∼ ∆ 2 SO (E G +∆ SO ) 2 E 2 k E 2 G 1 τ(k) , (2.32) where E G is the gap between the valence and the conduction band of the semiconductor, E k the energy of the conduction electron, and ∆ SO is the spin-orbit splitting of the valence band. Thus, the Elliott-Yafet spin relaxation (EYS) can be distinguished, being proportional to 1/τ, and thereby to the resistivity, in contrast to the DP spin scattering rate, Eq. (2.31), which is proportional to the conductivity. Since the EYS decays in proportion to the inverse of the band gap, it is negligible in large band gap semiconductors like Si and GaAs. The scattering rate 1/τ is again the sum of the impurity scattering rate [Ell54], the electron-phonon scattering rate [Yaf63, GF97], and electron-electron interaction [Bog80], so that all these scattering processes result in EY spin relaxation. In degenerate semiconductors and in metals, the electron-electron scattering rate is given by the Fermi liquid inelastic electron scattering rate 1/τ ee ∼ T 2 /E F . The electron-phonon scattering time 1/τ ep ∼ T 5 decays fasterwith temperature. Thus, at low temperatures theElliott-Yafet spinrelaxation, Eq.(2.32), is dominated by elastic impurity scattering τ 0 . In non-degenerate semiconductors, where the Fermi energy is below the conduction band edge, one finds 1/τ s ∼ T 2 /τ(T). 2.4.4 Spin Relaxation due to Spin-Orbit Interaction with Impurities The spin-orbit interaction, as defined in Eq.(2.6), arises whenever there is a gradient in an electrostatic potential. Thus, the impurity potential gives rise to the spin-orbit interaction V SO = 1 2m 2 c2∇V ×k s. (2.33)
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 29: 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 77 and 78: Chapter 4 Direction Dependence of S
Page 79 and 80: Chapter 4: Direction Dependence of
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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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