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

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90 Chapter 5: Spin Hall Effect as an effective magnetic field, which however does not break time reversal symmetry, this is very similar to the case where a Peierls substitution is applied,[Pei33] i.e. where a Peierls phase is added to the electron whenever it hops in the direction of finite vector field. The sign changes according to the sign of α 2 2 − ˜α2 1 and exhibits the value e/(8π) for small filling with the condition that both bands, E ± , are filled, independent of the strength of SOC. This can bee seen by expanding the spectrum around the Γ point which yields[She04] σ SH = ∫ e 2π 16m e π 2 = e 8π with k x = kcos(ϕ) and k x = ksin(ϕ). 0 dϕ (α2 2 − ˜α2 1 )cos2 (ϕ)(k + −k − ) (α 2 2 + ˜α2 1 −2α , (5.25) 2˜α 1 sin(2ϕ)) 3 2 α 2 2 − ˜α2 1 |α 2 2 − (5.26) ˜α2 1 |, Looking at the results from the calculation on a clean lattice, Fig.(5.2) (b), it can be seen that the value e/(8π) decreases with increasing SOC strength in the case of pure Rashba SOC. This can be understood by noticing that the value of the SHC[SCN + 04] σ SH = e 16m e πα 2 (k F+ −k F− ) (5.27) is diminished when we add corrections to the parabolic assumption: On the lattice we have ( ) 2 (k F+ −k F− ) = arccos 1+ ( α 2 ) 2 −1 (5.28) 2t = 2m e α 2 − 2 3 (m eα 2 ) 3 +O(m e α 2 ) 4 (5.29) and therefore the diminishment is given by σ SH = e 8π − e 24π (m eα 2 ) 2 . (5.30) 5.3 Numerical Analysis of SHE 5.3.1 Exact Diagonalization For linear Rashba coupling, the value σ SH = e 2 /(8π), as presented in the previous section, has been obtained both by analytical calculations in the continuum model, and by numerical calculations of the tight binding model[Sch06]. However, in the presence of nonmagnetic impurities, the DC spin Hall conductance is diminished to exactly zero,
Chapter 5: Spin Hall Effect 91 as soon as the system size L exceeds the elastic mean free path l e . In the case of δ- function potential and pure linear Rashba SOC in the thermodynamic limit this result is even independentof theimpurity strength, as discovered by Schwab andRaimondi[RS05] by usingselfconsistentBornapproximation: Thevertex correction intheladderapproximation cancel exactly the term coming from the one-loop part. Following Rashba, Ref.[Ras04], this can be understood as follows: Corresponding to Eq.(5.10) and Eq.(5.22) we have, at low filling, positive contribution to σ SH due to interbranch transitions between the occupied (E + ) and unoccupied (E − ) states. Adding a perturbation by applying an infinitesimal small magnetic field, intrabranch transitions at the Fermi energy give rise to a negative contribution. Surprisingly, this additional contribution cancels the first and we are left with zero SHC. This counterintuitive finding led to several controversies because the first numerical calculations in this field showed finite SHC when extrapolated to infinite large samples[NSJ + 05]. However, new numerical results, e.g. Ref.[NSJ + 06] as an erratum to Ref.[NSJ + 05], showed agreement with the analytical predictions. In contrast, taking into account also spin-orbit terms which are cubic in momentum, as they are present in any system with broken inversion symmetry (cubic Dresselhaus terms), or in quantum wells with strongly asymmetric confinement (cubic Rashba terms), the spin Hall conductance has a quite universal value, of σ SH = Ne 2 /(8π), where N is the number of times the spinorbit field B SO (k) winds around a circle, as the momentum is moved once around the Fermi surface[Sch06]. Resonant Impurities The effect of resonant impurities and of magnetic impurities on the SHC has not been studied yet in that detail[LX06, WLZ07a]. Especially, it is unclear how large its magnitude is, when the Fermi energy is in the vicinity of the resonant levels close to the metal-insulator transition, where it has been observed that the spin relaxation rate is minimal[DKK + 02], making it a potentially attractive regime for spintronic applications. In the following we analyze the reduction of the SHC in a finite system with periodic boundary conditions in presence of non-magnetic impurities of binary type, i.e. on-site potential V i = p i V where p i = 1 for impurity sites and p i = 0 otherwise. The lattice is assumed to be contaminated with 10% of impurities. To have a better understanding of how the SHC changes with the strength of impurities, V, we first calculate the DOS. The
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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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Page 49 and 50: Chapter 3: WL/WAL Crossover and Spi
Page 77 and 78: Chapter 4 Direction Dependence of S
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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: Chapter 5: Spin Hall Effect 89 1.0
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
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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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