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

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Appendix F KPM The recursion relations of the polynomials of first and second kind, Eq.(5.34) and Eq.(5.35), allow for a simple iteration procedure: The core of KPM is the iterative construction of the states |α n 〉 = T n ( ˜H)|α〉, described by the following steps[WWAF06], |α 0 〉 = |α〉, (F.1) |α 1 〉 = ˜H|α 0 〉, (F.2) |α n+1 〉 = 2 ˜H |α n 〉−|α n−1 〉, (F.3) where |α〉 is the starting vector. For instance, to calculate the local DOS at site i, our starting vector would be the site-occupation vector |i〉. Then the coefficients for the T n (E) polynomial are given by µ n = 〈i|i n 〉 and ρ i (E) is reconstructed by using Eq.(5.32). Consequently, the most time consuming part is the matrix-vector multiplication H|α n 〉. Because H is sparse in our case, a very efficient multiplication is done by decomposing the operator • in a matrix A which contains the information about connected sites, which comprises their site number and the type of hopping between them • and a matrix SOH which contains all hopping matrices which are two-dimensional due to SOC. The type of hopping is coded in a number h ij , e.g. h ij = 0 can be defined as “hopping from i to j ⇐⇒ hopping in the positive x-direction”. To give an example: The site i is connected with site j = A[i][2 ∗k] 1 and the type of hopping is h ij = A[i][2∗k +1], with k being an 1 we use C-type of writing matrices 150
Appendix F: KPM 151 integer running from 0 up to the number of neighbors. The hopping matrix t(i,j), which includes the SOC, is then given by ⎛ t(i,j) = ⎝ SOH[h ij][0][0] SOH[h ij ][0][1] SOH[h ij ][1][0] SOH[h ij ][1][1] This leads finally to the following algorithm for the operation H.v = v out : ⎞ ⎠. (F.4) Listing F.1: Optimized matrix-vector multiplication for ( m=0; m
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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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Chapter 5: Spin Hall Effect 85 curr
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Chapter 5: Spin Hall Effect 87 with
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Chapter 5: Spin Hall Effect 89 1.0
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Chapter 5: Spin Hall Effect 91 as s
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Chapter 5: Spin Hall Effect 93 V⩵
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Chapter 5: Spin Hall Effect 95 0.4
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Chapter 5: Spin Hall Effect 97 oper
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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]
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Page 133 and 134: Bibliography 123 [MACR06] T. Mickli
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Page 137 and 138: Bibliography 127 [Tor56] H. C. Torr
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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 157 and 158: Appendix D: Hamiltonian in [110] gr
Page 159: Appendix E: Summation over the Ferm
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