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

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92 Chapter 5: Spin Hall Effect result is presented in Fig.5.3. For small V one would expect the Van Hove singularity at half filling. Due to finite Rashba SOC it is split to finite energies E = ±(2t− √ 4+α 2 2 t), indicated in Fig.5.1 for the energy below half filling. If the impurity strength is increased, a preformed impurity band is created. Using exact diagonalization 1 and applying the Kubo formalism, Eq.5.22, we calculate the SHC. Exemplarily we show the result for σ SH (E) at V = −2.8t in Fig.5.4. The SHC is strongly reduced but shows an additional maximum at energy where the preformed impurity band is located, as can be seen in comparison with the DOS, Fig.5.4(b). Toanalyze thereduction ofSHCfor agiven fillingn, wevary theimpuritystrength up to V = −5t and keep the concentration constant at 10%. Similar to the results in the case of block distribution of impurity strength, we see a monotone suppression at all fillings, even in the preformed impurity band, Fig.5.5. 5.3.2 Kernel Polynomial Method The numerical calculations presented in the previous section, which were based on exact diagonalization using LAPACK[LAP] routines, are limited to small system sizes. This leads to finite size effects like oscillations in the SHC, e.g. Fig.5.4(b) above half filling. For further calculations concerning the role of the impurity band it is necessary to do a finite size scaling analysis and consider system-sizes beyond L = 64 which makes an exact treatment on current hardware impossible: for a D-dimensional matrix such a calculation requires memory of the order of D 2 , and the LAPACK routine scales as D 3 . Another problem is the adjustment of the cutoff η, see Eq.(5.22), which has to be taken with care as analyzed e.g. by Nomura et al., Ref.[NSSM05]. Toovercome thelimitation onsmall systems, therearedifferent numerical order-Dmethods. Oneprocedureisthetimeevolution projection methoddevelopedbyTanakaandItoh[TI98], which was already used to calculate SHC[MMF08, MM07]. However, the algorithm requires both the choice of a sufficient number of time steps and an adjustment of cutoff η. A more effective method, which uses Chebyshev expansion based on Kernel Polynomial Method, will be presented in the following. 1 using LAPACK[LAP] and OpenMP[OMP] in C++
Chapter 5: Spin Hall Effect 93 V⩵0.2 V⩵2.8 0.20 0.15 Ρ 0.15 0.10 0.05 Ρ 0.10 0.05 0.00 10 5 0 5 10 E 0.00 10 5 0 5 10 E V⩵ 3 V⩵ 3.6 0.15 0.15 Ρ 0.10 Ρ 0.10 0.05 0.05 0.00 10 5 0 5 10 E 0.00 10 5 0 5 10 E V⩵4 V⩵5 0.15 0.15 0.10 0.10 Ρ Ρ 0.05 0.05 0.00 10 5 0 5 10 E 0.00 10 5 0 5 10 E Figure 5.3: DOS, as a function of Fermi energy in units of t, in presence of impurities of binary type with a concentration of 10% calculated using exact diagonalization. The system size is L 2 = 32 2 , and the SOC is Rashba type with α 2 = 1.2t with cutoff η = 0.02t.
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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 51 and 52: 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
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: Chapter 5: Spin Hall Effect 91 as s
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 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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