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

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