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

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96 Chapter 5: Spin Hall Effect Using the recursion relations of the polynomials, T 0 (x) = 1, T −1 (x) = T 1 (x) = x, T m+1 (x) = 2xT m (x)−T m−1 (x), (5.34) and correspondingly for the polynomials of the second kind U 0 (x) = 1, U −1 (x) = 0, U m+1 (x) = 2xU m (x)−U m−1 (x), (5.35) one can calculate the expansion coefficients µ n iteratively. Replacing the variable x by the Hamiltonian one can calculate various spectral quantities. The simplest example is the calculation of the spectral density, with the coefficients given by ρ(E) = 1 D µ n = ∫ 1 −1 D−1 ∑ k=0 δ(E −E k ), (5.36) dxρ(x)T n [x] (5.37) = 1 D Tr[T n( ˜H)], (5.38) where ˜H is the rescaled Hamiltonian with all D eigenvalues inside the interval [−1,1]. The efficiency of the procedure is not yet evident. This changes if one realizes the following aspects: • Self averaging properties allow for replacing the trace over the operator by a relatively small number R ≪ D of random vectors |r〉 = D−1 ∑ i=0 ζ ri |i〉, (5.39) where the amplitudes ζ ri = e iφ are random phases on site i. This makes the effort for the calculation of M coefficients µ n linear in D. • The most time consuming operation in this procedure is the matrix-vector multiplication (AppendixF). Due to the fact that the number of neighbors which a site has in the presented systems, the full multiplication can be replaced by sparse-matrix
Chapter 5: Spin Hall Effect 97 operations using a decomposition of the operators into a matrix which contains the connectivity information of the sites and a matrix where the hopping matrices according to the kind of hopping (e.g. with next nearest neighbor) are stored, as explained in more detail in AppendixF. Adjusting the number of Moments in the KPM Using exact diagonalization to calculate e.g. SHC as presented in Sec.5.2 and Sec.5.3.1, we were forced to adjust the cutoff η in the Kubo formula, Eq.(5.5) to get the 2D limit. Otherwise we would be left with results which are highly oscillatory dependent on the Fermi energy. In theKPM the Hamiltonian is approximated by a finitepolynomial, therefor the cutoff is implicitly set by choosing the number of moments M. Such a cutoff, which is inevitable in numerical calculations, leads i.g. to Gibbs oscillations, especially when the expandedfunction includesdiscontinuities orsingularities. Thereforetheexpandedfunction is convoluted with a particular kernel (in our case the Jackson kernel) which damps this oscillations. Exact Calculations on a finite lattice result in δ peaks in the DOS which are broadened due to the polynomial cutoff. The broadening can be approximated by Gauss curves of widths σ. To do finite-size analysis it is crucial to keep the same number of states within the kernel, i.e. within the distance of σ.[SSB + 10] Choosing a number of moments M in the KPM which is too large will lead to oscillations which are due to finite size of the system as can be seen in Fig.(5.6). On the other hand, a too small number will smear out features of the system which are independent of the size. This consideration leads to the condition σ ! > ∆, with the averaged level spacing ∆. It is helpful to analyze the relation between M and the broadening σ. This can be done by a convolution of a δ-distribution with the mentioned kernel, which leads to[WWAF06] M = πt σ , (5.40) and at the boundaries M = ( ) πt 2/3 . σ
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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 55 and 56: 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
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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: Chapter 5: Spin Hall Effect 95 0.4
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 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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