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

More documents

Recommendations

Info

30 Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems (a) (b) Figure 3.3: Exemplification of the second term in Eq.(3.4): Interference of electrons traveling in the opposite direction along the same path causes an enhanced back-scattering, the WL effect. (a) Closed electron paths enclose a magnetic flux from an external magnetic field, indicated as the red arrow, breaking time reversal symmetry, breaking constructive interference. (b) The entanglement of spin and charge by SO interaction causes the spin to precess inbetween two scatterers around an axis which changes with the momentum vector of the itinerant electron. This effective field can cause WAL. With the definition G R/A E (p′ ,p) = 〈 ∣ ∣∣∣ p ′ 1 E −H 0 ∓iη the conductivity can be rewritten to the following form σ = with the propagator of density 〉 ∣ p , (3.8) e 2 ∑ πm 2 p x p ′ x eVol ×〈GR (p,p ′ )G A (p ′ ,p)〉 imp , (3.9) p,p ′ Γ(p,p ′ ) = 〈G R (p,p ′ )G A (p ′ ,p)〉 imp , (3.10) where impurity averaging products of Green’s functions of the type 〈G R G R 〉 and 〈G A G A 〉 yield small corrections of order 1/E F τ and will be neglected (AppendixB.1). The first approximation one can apply is to assume 〈G R (p,p ′ )G A (p ′ ,p)〉 imp ≈ 〈G R (p)〉 imp 〈G A (p)〉 imp , (3.11)
Chapter 3: WL/WAL Crossover and Spin Relaxation in Confined Systems 31 where we used the definition G R/A (p) ≡ G R/A (p,p ′ )δ p,p ′. The average over impurities can be depicted diagrammatically, ( 〈G〉 imp = + + ) ( + + + + + + ) where the fermion line +··· , (3.12) denotes the unperturbed Green’s function. For uncorrelated disorder potential, 〈V(x)V(x ′ )〉 = δ(x−x ′ )/2πντ, as we will use in the following, we perform the disorder average in first-order Born approximation and get G R E(p) = = 1 E −H 0 (p)+i 1 , (3.13) 2τ whereG A E (p) is its complex conjugate, respectively. H 0 is the Hamiltonian without disorder potential V. The impurity vertex (the cross) is given by 1/2πντ. Until now we have the same information in the scattering time τ as we would gain from the Drude formula. Assuming low temperature, we can simplify Eq.(3.9) to e 2 ∑ σ = πm 2 e Vol p 2 x ×〈G R (p)〉 imp 〈G A (p)〉 imp (3.14) p e 2 ∑ = πm 2 e Vol p 2 x ×G R (p)G A (p) (3.15) = p e 2 ∑ p 2 x πm 2 e Vol p (E F −E p ) 2 + ( ) 1 2 (3.16) 2τ which can be simplified in the metallic regime, E F ≫ 1/τ, where the dominant contribution is given by energies close to E F , to e 2 ∫ ∞ ( ) E2me ≈ πm 2 dE(Volρ(E)) eVol d ≈ 2 e2 m e ρ(E F )E F 1 d 0 ∫ ∞ ∞ dE 1 (E F −E) 2 + ( 1 2τ 1 (E F −E) 2 + ( 1 2τ ) 2 (3.17) ) 2 (3.18) = e2 nτ m e , (3.19)
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 39: 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 91 and 92:
Chapter 4: Direction Dependence of
Page 93 and 94:
Chapter 5 Spin Hall Effect 5.1 Intr
Page 95 and 96:
Chapter 5: Spin Hall Effect 85 curr
Page 97 and 98:
Chapter 5: Spin Hall Effect 87 with
Page 99 and 100:
Chapter 5: Spin Hall Effect 89 1.0
Page 101 and 102:
Chapter 5: Spin Hall Effect 91 as s
Page 103 and 104:
Chapter 5: Spin Hall Effect 93 V⩵
Page 105 and 106:
Chapter 5: Spin Hall Effect 95 0.4
Page 107 and 108:
Chapter 5: Spin Hall Effect 97 oper
Page 109 and 110:
Chapter 5: Spin Hall Effect 99 the
Page 111 and 112:
Chapter 5: Spin Hall Effect 101 str
Page 113 and 114:
Chapter 5: Spin Hall Effect 103 Com
Page 115 and 116:
Chapter 5: Spin Hall Effect 105 Fin
Page 117 and 118:
Chapter 6: Critical Discussion and
Page 119 and 120:
Chapter 6: Critical Discussion and
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
Page 145 and 146:
Appendix B: Linear Response 135 Pro
Page 147 and 148:
Appendix C Cooperon and Spin Relaxa
Page 149 and 150:
Appendix C: Cooperon and Spin Relax
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?