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

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136 Appendix B: Linear Response Furthermore we assume low temperature which is why the derivative of the Fermi function fixes the energy of the Green’s functions to the Fermi energy, − ∂f T=0 ∂E = δ(E −E F). (B.34) We end up with σ impx,x (0) = 1 πV e 2 m 2 e ∑ k x k ′ 〈 x G R E (k,k ′ )G A E (k′ ,k) 〉 . (B.35) imp k,k ′
Appendix C Cooperon and Spin Relaxation C.1 Sum Formula for the Cooperon Writing the Cooperon (Eq.3.43) Ĉ(Q) = 1 D e (Q+2eA+2eA S ) 2 +H γD . (C.1) in singlet, |S = 0;m = 0〉 = (|↑↓〉 − |↓↑〉)/ √ 2 ≡ |⇄〉 and triplet |S = 1;m = 0〉 = (|↑↓〉 + |↓↑〉)/ √ 2 ≡ |⇉〉,|S = 1;m = 1〉 ≡ |⇈〉,|S = 1;m = −1〉 ≡ |〉 representation, without magnetic field the singlet sector is decoupled from the triplet one. To sum over C αββα in the case of a finite magnetic field and having calculated the eigenvectors |i〉 and eigenvalue λ i for the Cooperon Hamiltonian, we can use the following simplification: ∑ C αββα = ∑ ∑ 〈αβ|mS〉〈 m ′ S ′ |βα 〉〈 mS|C|m ′ S ′〉 . αβ αβ mS m ′ S ′ Only several of the prefactors 〈αβ|mS〉〈m ′ S ′ |βα〉 are non-zero: (C.2) For αβ =⇈ 〈⇈ |⇈〉〈⇈ |⇈〉 = 1, (C.3) For αβ = 〈 |〉〈 |〉 = 1, (C.4) 137
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Itinerant Spin Dynamics in Structur
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Itinerant Spin Dynamics in Structur
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Contents v 3.2.2 Weak Localization
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Citations to Previously Published W
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Dedicated to Linda, my parents and
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Chapter 1 Introduction Structure of
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Chapter 1: Introduction 3 the coupl
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Chapter 1: Introduction 5 Throughou
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Chapter 2: Spin Dynamics: Overview
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Chapter 3 WL/WAL Crossover and Spin
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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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Page 117 and 118: Chapter 6: Critical Discussion and
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Page 121 and 122: List of Figures 111 3.3 Exemplifica
Page 123 and 124: List of Figures 113 4.2 The spin de
Page 125 and 126: List of Tables 3.1 Singlet and trip
Page 127 and 128: Bibliography 117 [BB04] [BBF + 88]
Page 129 and 130: Bibliography 119 [DP71b] [DP71c] [D
Page 131 and 132: Bibliography 121 [HSM + 06] [HSM +
Page 133 and 134: Bibliography 123 [MACR06] T. Mickli
Page 135 and 136: Bibliography 125 [PMT88] [PP95] [PT
Page 137 and 138: Bibliography 127 [Tor56] H. C. Torr
Page 139 and 140: Appendix A SOC Strength in the Expe
Page 141 and 142: Appendix A: SOC Strength in the Exp
Page 143 and 144: Appendix B: Linear Response 133 in
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Page 149 and 150: Appendix C: Cooperon and Spin Relax
Page 157 and 158: Appendix D: Hamiltonian in [110] gr
Page 159 and 160: Appendix E: Summation over the Ferm
Page 161: Appendix F: KPM 151 integer running
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