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

88 5 Fe monolayers on hexagonal nonmagnetic substrates
below the M, the RW-AFM phase, value using GGA (LDA). Then we calculated B1 and
K1 values using equations (5.8 and 5.14) for the Q 3ΓK/4 and got B1 = −0.6 and K1 =0.0
meV/Fe atom, we then added these values to the model Hamiltonian constructed along
Γ-K-M, and used it to fit our calculated energy dispersion curve to calculate the exchange
interaction parameters, which are shown in table 5.2.
Table 5.2: GGA and LDA results of Heisenberg exchange constants for the hcp Fe ML on Rh(111)
substrate obtained by fitting the total-energy dispersion along Γ-K-M and the higher order terms
B1 and K1
(meV) J1 J2 J3 J4 B1 K1
GGA 3.8 −0.6 −1.6 0.3 −0.6 0.0
LDA 3.9 −0.7 −1.1 0.4 −1.3 0.5
We see that GGA and LDA values of J1, J2 and J3 are very similar, but different from what
we got when we exclude the higher order interaction terms, using Eq. (5.15). These results
show how important is to include the higher order terms in the model Hamiltonian to get
accurate values of the exchange interaction parameters. If we use the values of J3/|J1| and
J2/|J1| from the shown results in table 5.2, we find that Fe has non-collinear spin spiral
along Γ-K-M on Rh(111) substrate, but still very close to to FM phase transition, referring
to the calculated phase diagrams in figure 5.6. This is consistent with the minimum −5
meV/Fe atom total energy difference below the Γ point, i.e. the FM solution. From the
calculated higher order interactions parameters, we notice that the biquadratic interaction
term B1 is in the same order as the next nearest neighbor term J2, where as the 4-spin
interaction term K1 is zero. In this case we can interpret our results keeping in mind that
if the Fe atoms nearest neighbor interactions are relatively weak compared to next nearest
neighbor interactions, the higher order terms are very important and cannot be ignored
due to their highly possible contribution of stabilizing non-collinear ground state.
Because of the small ground state total energy difference, the above interpretation is
not a final answer since we are not sure if this small energy difference yields the global
minimum or not, especially since the difference k-points sampling between the spin spiral,
collinear (for double-Q) and the real space (for 3Q) calculations will produce about 5
meV/Fe atom numerical error as it was checked for free-standing Fe monolayer. Because
of that, we calculated the total energy of the double-Q state along the line M-Γ, which
is constructed by a superposition of two Q points at Q MΓ/2 , which represents a collinear
magnetic structure in real space called uudd � MΓ/2 � state (see Fig. 5.5). The GGA (LDA)
results show, surprisingly, that the uudd � M Γ/4 � gained about −32 (−27) meV/Fe atom
below the Q M/2 point, and became a global minimum by −30 (−21) meV/Fe atom relative
to FM solution. These results were not expected because uudd � MΓ/2 � and uudd � 3ΓK/4 �
are degenerate according to Heisenberg picture explained in section (3.2), and supposed to
provide the same B1 and K1 values.
Trying to understand the unexpected obtained results for uudd � MΓ/2 � , we plotted the
Fe, the interface layer Rh(I), and the subsurface rhodium layer Rh(I-1) GGA and LDA