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

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4.2 Results of 3d-Monolayers on Rh(001) Substrate 61 case the position of the majority d-band is determined by the band-filling and the whole exchange-split d-like DOS is energetically at a much higher position. This modifies the hybridization to the substrate, as can be seen in the relaxations: while Fe and Cr have very similar bulk lattice constants, the relaxation of FM Cr on Rh(001) is much bigger than FM Fe/Rh(001) relaxation (see Fig. 4.4). In the case of the AFM solutions the trend is exactly opposite, i.e. Fe relaxes much stronger than Cr. Additionally, the Rh(I) AFM LDOS majority and minority bands, in both Fe and Cr cases, are completely identical (since there is no induced moments), while its FM LDOS Rh(I) majority and minority bands are distorted. The AFM minority DOS at Fermi is smaller than the FM one in both Fe/Rh(001) and Cr/Rh(001) systems. This indicates that the stability (instability) of the Fe and Cr AFM (FM) solution is related to the bands values at the Fermi level. In the AFM case also the d-bandwidth is much smaller than in the FM case. In all these cases the magnetic moments are approximately the same (Fig. 4.5). 4.2.3 Magnetocrystalline anisotropy: Focusing on the calculation of the EMCA, there are two ways to do that. The first is using the total energy approach, were it should converge for each chosen spin quantization axis and then the energy difference is EMCA. Another way to calculate EMCA is using the force theorem[128] which should be, in principle, applicable up to linear changes in the charge and magnetization densities, as the magnetization is rotated from one direction to another[129]. This theorem states that EMCA is given by the difference in band energies (sum of valence eigenvalues) obtained with spin-orbit coupling for two different spin-quantization axes, using the same self consistent scalar-relativistic potential. E FT MCA = �occ i ɛ ⊥ �occ i − i ɛ � i (4.2) From this, EMCA is represented by forming the difference of the sum of the occupied eigenvalues of the Hamiltonian taken for the out-of-plane (⊥) and in-plane (�) spin directions respectively, provided that the same effective potential is used when the Kohn-Sham equation is solved. This method has the great advantage that only one nonrelativistic self-consistent calculation of the potential must be done. Numerical evidence was provided by O. Eriksson that the MCA of 3d elements are rather well reproduced using the force theorem, and that it gives results similar to total energy calculations[130]. In addition, difference between the total energy and force theorem calculations of the EMCA should be smaller than the numerical accuracy of the total energy calculations. By that, converged potentials are computed within DFT without spin-orbit coupling and then one extra iteration. Because of the lack of symmetry, a huge number of k points necessary to achieve convergence of the integrals in the Brillouin zone (BZ) involved in the calculation of the EMCA. Using self consistent (SC) calculations (Fig. 4.9), we tested the MCA convergence versus the number of k-points up to 1024 k�-points in the full 2DBZ. After that we calculated
62 4 Collinear magnetism of 3d-monolayers on Rh substrates MCA [meV/3d.atom] 2 1.5 1 0.5 0 −0.5 −1 −1.5 Fe/Rh(001) Co/Rh(001) Ni/Rh(001) −2 0 0.005 0.01 0.015 0.02 0.025 0.03 1/N k-points in the full BZ Figure 4.9: Solid symbols are the MCA values of Fe, Co and Ni monolayers on Rh(001) as a function of 1/k obtained from SC calculations. Open symbols represent the force theorem results with 2048 k�-points in the full BZ. MCA>0 (
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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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Page 29 and 30: 16 2 The FLAPW method 1 0 0 1 Figur
Page 31 and 32: 18 2 The FLAPW method nonlinear pro
Page 33 and 34: 20 2 The FLAPW method metal systems
Page 35 and 36: 22 2 The FLAPW method Then one can
Page 37 and 38: 24 2 The FLAPW method Evac is the v
Page 39 and 40: 26 2 The FLAPW method of the operat
Page 41 and 42: 28 2 The FLAPW method a matrix expr
Page 43 and 44: 30 3 Magnetism of low dimensional s
Page 65 and 66: 52 4 Collinear magnetism of 3d-mono
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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
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112 6 Co MCA from monolayers to ato
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114 6 Co MCA from monolayers to ato
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116 rather subtle. To shed more lig
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118
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120 where, the relation � i H =
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122 Figure 6.8: The real (left) and
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124 E(q) = −M 2 ( 2J2 +2J6 +cos(
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126 For a 2D hexagonal lattice, the
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128 Bibliography [12] P. Ferriani,
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130 Bibliography [39] A. Dallmeyer,
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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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