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

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5.2 Results of Fe monolayer on different hexagonal substrates from non-collinear cal. 83 E = −M 2 (+2J1 − 2J2 − 2J3 − 4J4)+2B1M 4 − 12K1M 4 if we use QMΓ/2 = 1 2 in eq. (A-21) for the single-QMΓ/2 . If we use the double-Q biquadratic and 4-spin terms derived in Ref. [38] Ebiquardr,2Q = −2B1M 4 and E4−spin,2Q = −4K1M 4 (5.10) (5.11) and then plug them in the uudd–model Hamiltonian up to the fourth nearest neighbor, we get for the uudd(3ΓK/4)-state and E = −M 2 (−2J1 +2J2 − 2J3 − 4J4) − 2B1M 4 − 4K1M 4 E = −M 2 (+2J1 − 2J2 − 2J3 − 4J4) − 2B1M 4 − 4K1M 4 (5.12) (5.13) for the uudd(MΓ/2)-state. A a result (5.12)-(5.9)=(5.13)-(5.10), which means that we will have the same energy difference between the single- and double-Q-state in both Γ-K-M and Γ-M directions, i. e Euudd − E 3ΓK/4 = Euudd − E MΓ/2 =4{2K1 − B1} (5.14) for the difference between the uudd � 3ΓK/4 � (uudd � MΓ/2 � ) and the single-Q3ΓK/4 (QMΓ/2 ) along Γ-K-M (M-Γ) direction. By that, we are able to calculate B1 and K1 from equations (5.14) and (5.8). Using equations (A-20) and (A-21), different cuts through the zero temperature magnetic phase diagrams can be calculated analytically. The magnetic phase diagram, shown in figure 5.6, are calculated only for the first three exchange interaction terms J1, J2 and J3. I. e., the exchange interactions taken into account are up to the third nearest neighbor. From these phase diagrams, it can be seen that not only collinear but mainly non-collinear magnetic structures exist on 2D hexagonal lattices. If we calculate the phase diagram in arbitrary units of J1 (5.6(a)), the Néel ground state is expected for any positive value of J2 if J1 is negative. A spin spiral structure is predicted for large positive J1 if J2 is smaller than −J1. On the other hand, if the third nearest neighbor interaction term J3 is included, the possibility of predicting more complex structures increases (c. f. 5.6(b) and (c)). For example, the possibility to have non-collinear magnetic structures is very large if both J2 and J3 values are negative and larger than ≈ |J1| 2 . In practice, these phase diagrams can be used as a starting point to predict the magnetic order on triangular lattices by tuning the substrate or the overlayer. A very simple, and computationally less expensive approach is to employ the virtual crystal approximation (VCA) [148], in which one studies a crystal with the primitive periodicity, but composed
84 5 Fe monolayers on hexagonal nonmagnetic substrates 3-Q state: 3Q combination uudd uudd � 3Γ K 4 � Γ M 2 � : 2Q combination � : 2Q combination Figure 5.5: Double- and multi-Q states spin structures in real space (left) and reciprocal space (right).
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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
Page 45 and 46: 32 3 Magnetism of low dimensional s
Page 65 and 66: 52 4 Collinear magnetism of 3d-mono
Page 87 and 88: 74 5 Fe monolayers on hexagonal non
Page 95: 82 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 133 and 134: 120 where, the relation � i H =
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
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134 Bibliography [101] L. Nordströ
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136 Bibliography [130] O. Le Bacq,
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138 Bibliography [160] P. M. Marcus
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140 Bibliography [190] R. Germar, W
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Acknowledgement I don’t know how
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Schriften des Forschungszentrums J
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Ab initio investigations of magnetic properties of ultrathin transition ...

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