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

90 5 Fe monolayers on hexagonal nonmagnetic substrates
Table 5.3: LDA and GGA Fe and Rh(111) induced moments at the interface and the subsurface
rhodium layers, Rh(I) and Rh(I-1), for the double- and multi-Q states in units of μB using.
GGA LDA
Rh(I) Rh(I-1) Rh(I) Rh(I-1)
uudd � 3ΓK/4 � : 0.11 0.04 0.11 0.04
uudd � MΓ/2 � : 0.32,0.09 0.04 0.27,0.08 0.03
3Q: 0.11 0.02 0.10 0.02
LDA predict again similar double- and multi-Q rhodium induced moments for the 3Q
and uudd � 3ΓK/4 � . If we take GGA results of Rh induced moments, we see that the
double-Q uudd � 3ΓK/4 � , has the same polarization between Rh(I) and Rh(I-1) induced
moments as the Q 3ΓK/4 -point (≈0.05 μB), where the 3Q has larger induced moments
(≈0.08 μB) than the Q M/2 (≈0.02 μB) state in the substrate Rh(I) and Rh(I-1) layers.
The largest polarization for Rh(I) and Rh(I-1) occurs for the uudd � MΓ/2 � state, where
it has also larger polarization (≈0.32 μB) than the Q M/2 -point (≈0.16 μB), even larger
than the polarization at Γ-point (≈0.25 μB). From these results, we can now explain why
the equality between the degenerate double-Q states in equation (5.14) is broken and we
cannot use this equations to calculate the higher order terms B1 and K1, to use them to
fit the energy dispersion curve along M-,Γ, using equation (A-21). This means that, if we
use equations (5.8 and 5.14) to calculate B1, we will have B1 values which, if we use in
equation (A-21) to fit the energy dispersion curve, will produce inconsistent J values which
cannot be used to predict the true ground state.
.
The reason of this inconsistency is the following: When we used uudd � 3ΓK/4 � results to
calculate B1 and then use it in equation (A-20), we got consistent J, B1 and K1 values due
to the homogeneous induced moments in the Rh(I), which agrees with what was mentioned
in ref.[31], and that the model Hamiltonian for higher order terms is only applicable for
constant magnetic moments. This means that we cannot use this model for for highly
polarized substrates like Rh(111) substrate to calculate higher order interactions constants,
because the substrate moments are not equally induced in all directions, which leads to
a failure in calculating the correct exchange interaction parameters and then to wrong
prediction of the system ground state from the magnetic phase diagrams. Instead, our
results suggest that a term proportional to M 2 or M 4 is missing in the functional, similar
to what was proposed in Ref.[149].
To see if we can explain Fe magnetic ground state on Rh(111) using the LDOS, we
compare the LDOS of RW-ARM, FM and the uudd � MΓ/2 � configurations, shown in figure
5.10. If we look at Fe LDOS, we notice that the RW-AFM and the uudd � MΓ/2 �
majority spins are very similar with the FM LDOS majority shifted toward Fermi level
(EF ). Most important is the fact that smallest LDOS is obtained for uudd � MΓ/2 � minority
spin states, while FM and the RW-AFM minority spin states remain of similar height at
Fermi energy, with sharper FM minority band. The Rh(I) LDOS have opposite trend: The