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

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B(q) =2B1(1 + 2 cos(2πq)) (A-25) for the M-Γ line. Using equations (A-24) or (A-25), the biquadratic interactions for the high symmetry points are summarized below: Ebiq,FM = −6B1M 4 Ebiq,RW-AFM = −6B1M 4 Ebiq,Néel = − 3 4 B1M 2 Ebiq,2Q = −2B1M 4 Ebiq,3Q = − 2 Then the single-Q total energy becomes 4 B1M 3 125 (A-26) E(q) =−NM 2 (J(q) − M 2 B(q)) (A-27) noting that M 2 B(q) is always coupled with J3 and they can’t be decoupled. The 4–spin interaction: The 4–spin interaction Hamiltonian, on four lattice sites, has the form H4−spin = − � Kijkl[(Mi ·Mj)(Mk ·Ml)+(Mj ·Mk)(Ml ·Mi)−(Mi ·Mk)(Mj ·Ml)] (A-28) ijkl Figure 6.9: Schematic representation <strong>of</strong> the first order 4–spin interaction, for 2Q 3 ΓK (left) 4 and 2Q MΓ (right), on ijkl lattice sites. For better view, only four diamonds (out <strong>of</strong> twelve) 2 are shown, and the related lattice sites, to the total twelve diamonds, are only labeled.
126 For a 2D hexagonal lattice, the 4–spin interaction represents the interaction between four spins on one diamond. Because there are twelve diamonds surrounding each lattice site; the first order 4–spin interaction will minimize the Hamiltonian by −12K1M 4 for any single-Q on the high symmetry line. Using eq. (A-28), one can find that the 4–spin interaction minimizes the Hamiltonian by −4K1M 4 for both double-Q states, 2Q 3 4 ΓK and 2Q MΓ (see Fig. 6.9). 2 By this, the 4–spin interaction lefts the degeneracy between the single- and double-Q by 8K 4 1.
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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 87 and 88: 74 5 Fe monolayers on hexagonal non
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Page 135 and 136: 122 Figure 6.8: The real (left) and
Page 137: 124 E(q) = −M 2 ( 2J2 +2J6 +cos(
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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Ab initio investigations of magnetic properties of ultrathin transition ...

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