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

5.2 Results of Fe monolayer on different hexagonal substrates from non-collinear cal. 87
E Q −E ⎯Γ [meV/Fe]
40.0
30.0
20.0
10.0
0.0
−10.0
−20.0
−30.0
1 ML Fe/Rh(111)
hcp-Fe-Rh(I)
2Q
fcc-Fe-Rh(I) LDA
3Q 2Q
GGA
−40.0
⎯Γ ⎯Κ ⎯Μ ⎯Γ
Spin spiral vector Q (2π/a)
GGA
LDA
1 ML Fe/Rh(111)
2Q 3Q 2Q
hcp-Fe-Rh(I)
⎯Γ ⎯Κ ⎯Μ ⎯Γ
Spin spiral vector Q (2π/a)
Figure 5.7: Total energy of spin spirals for 1 ML Fe/Rh(111). Solid blue circles (green
squares) are the fcc (hcp) Fe-Rh(I) stacking GGA results (left). Solid indigo triangles are
the LDA results with hcp-Fe-Rh(I) stacking (right). The green (indigo) stars are the GGA
(LDA) double- and multi-Q total energies.
equations with three variables. By solving equation (5.15) we get values of J1, J2 and
J3 equal to +5.97(+4.18), −2.22 (−1.56), and −3.75 (−2.63) meV/Fe from GGA (LDA)
results. If we use these values and insert them in the phase diagram (Fig. 5.6), we find
out that Fe has indeed a non-collinear spin spiral ground state along Γ-K-M-Γ, using
both GGA and LDA exchange correlation potentials. This agrees well with what we see
in our spin spirals results. In addition to their effect on calculating J3 value accurately,
we know from eq. A-20, that including the higher order terms B1 and K1, in Heisenberg
model, requires us to go beyond the third nearest neighbor and calculate at least J4 for
qualitative results. In this case to calculate the fourth nearest neighbor we need to it to
the whole curve (Fig. 5.7) while to determine B1 and K1 equations (5.8 and 5.14) can be
used. Calculating the double- and multi-Q states will include the higher order interactions
in our calculations, and helps to get more accurate J values, and better prediction of the
true ground state.
The 3Q state can be constructed in real space by a superposition of three M points
on the hexagonal Brillouin zone (see Fig.5.5). The double-Q structure, along Γ-K-M, is a
collinear magnetic structure in real space which can be constructed by a superposition of
two spiral q-points at Q 3ΓK/4 , i.e. uudd � 3ΓK/4 � in real space (see Fig. 5.5). Because we
calculated the spin spirals using at ferromagnetically relaxed positions, we performed real
space (collinear) calculations to determine the total energies of the multi-Q (double-Q)
states with the same relaxed atomic positions. By that we got the value illustrated by
stars in figure 5.7 using LDA and GGA exchange correlation potentials. If we look at the
calculated double-Q value, we see that it has higher (lower) energy than the corresponded
q-point by about +3 (−30) meV/Fe atom for the Q 3ΓK/4 and Q MΓ/2 points, respectively,
which is surprising since they are expected to gain the same energy due to the including
of the higher order terms. The 3Q results show −2.5 (−8) meV/Fe atom energy gain