Ab initio investigations of magnetic properties of ultrathin transition ...

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Appendix Heisenberg model on Bravais lattice: The classical Heisenberg Hamiltonian of two localized spins, Mi and Mj, on lattice sites i andj, can be written as H = � (A-1) i,j −JijMi · Mj with the assumption that the magnetic atoms have spin, with the same magnitude, on all lattice sites M 2 i = M 2 , for all i. (A-2) By using Fourier transforms, it becomes very convenient to express any quantity on a periodic lattice with boundary conditions. Therefore, spins which are localized on N lattice sites can be described by their reciprocal lattice vectors (q) and real space coordinate (Ri) of lattice site i: Mi = � Mqe iqRi (A-3) then, the inverse Fourier transform is given by Mq = 1 N q � Mie −iqRi (A-4) Since Mi is real, then the Fourier components of the spins fulfill the equation i Mq = M−q ∗ (A-5) If we replace the real spins Mi in equation (A-1) by their Fourier components (eq.A-3), the Heisenberg Hamiltonian becomes 119
120 where, the relation � i H = � i,j = � i,j −Jij −Jij = −N � q � q,q ′ � q,q ′ Mq · Mq ′eiqRi e iq ′ Rj Mq · Mq ′ei(q+q′ )Ri e iq ′ (Rj−Ri) Mq · M−q � � δ J0δe −iqRδ � (A-6) e i(q+q′ )Ri = Nδq,−q ′ holds for a sum over all lattice sites, Rδ = Rj − Ri and the exchange constants are defined to be real quantities with J(q) = � J0δe −iqRδ ∗ = J(−q) =J(q) then, equation (A-6) will become δ H = −N � J(q)Mq · M−q q (A-7) (A-8) Minimizing the energy (eq. A-6) under the condition that all lattice sites have the same spin magnitude (eq. A-2) will lead to determination of the magnetic ground state for N independent equations. This is equivalent to a system of N equations with Fourier spin components � Mq · M−q = M 2 (A-9) and � q q Mq · Mq ′ −q = 0, with q ′ �= 0 (A-10) Which means that all Mq vanish except for MQ and M−Q, where ±Q maximize J(Q) and the lowest energy is then given by E = −NM 2 J(Q) (A-11) The spin structure which corresponds to MQ and MQ can be covered by introducing the real and imaginary parts, RQ and IQ: MQ = RQ + iIQ, M−Q = RQ − iIQ Then, using eq. (A-9) and (A-10), we obtain MQ · M−Q = R 2 Q + I 2 Q = M 2 (A-12) (A-13)
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Mitglied der Helmholtz-Gemeinschaft
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Forschungszentrum Jülich GmbH Inst
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Abstract In this thesis, we investi
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”A human being is part of the who
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II Contents 4.2.1 Relaxations and m
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2 INTRODUCTION (ao =3.80 ˚A) is in
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4 INTRODUCTION understanding of our
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6 INTRODUCTION
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8 1 Density functional theory (DFT)
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10 1 Density functional theory (DFT
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16 2 The FLAPW method 1 0 0 1 Figur
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18 2 The FLAPW method nonlinear pro
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20 2 The FLAPW method metal systems
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22 2 The FLAPW method Then one can
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24 2 The FLAPW method Evac is the v
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26 2 The FLAPW method of the operat
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28 2 The FLAPW method a matrix expr
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30 3 Magnetism of low dimensional s
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52 4 Collinear magnetism of 3d-mono
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Page 81 and 82: 68 4 Collinear magnetism of 3d-mono
Page 87 and 88: 74 5 Fe monolayers on hexagonal non
Page 113 and 114: 100 5 Fe monolayers on hexagonal no
Page 115 and 116: 102 5 Fe monolayers on hexagonal no
Page 117 and 118: 104 6 Co MCA from monolayers to ato
Page 129 and 130: 116 rather subtle. To shed more lig
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Page 135 and 136: 122 Figure 6.8: The real (left) and
Page 137 and 138: 124 E(q) = −M 2 ( 2J2 +2J6 +cos(
Page 139 and 140: 126 For a 2D hexagonal lattice, the
Page 141 and 142: 128 Bibliography [12] P. Ferriani,
Page 143 and 144: 130 Bibliography [39] A. Dallmeyer,
Page 145 and 146: 132 Bibliography [68] J. Perdew and
Page 147 and 148: 134 Bibliography [101] L. Nordströ
Page 149 and 150: 136 Bibliography [130] O. Le Bacq,
Page 151 and 152: 138 Bibliography [160] P. M. Marcus
Page 153 and 154: 140 Bibliography [190] R. Germar, W
Page 156: Acknowledgement I don’t know how
Page 159: Schriften des Forschungszentrums J
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