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

82 5 Fe monolayers on hexagonal nonmagnetic substrates
For the single-Q we can obtain values for the nearest-neighbor biquadratic interaction,
B1, and four-spin interaction, K1. The first order contributions of the biquadratic terms,
B1, to the FM, RW-AFM and 3Q states are[38]
Ebiquardr,FM = −6B1M 4
Ebiquardr,RW−AFM = −6B1M 4
Ebiquardr,Néel = − 3
Ebiquardr,3Q =
4
B1M
2
− 2 4
B1M
3
while the contributions if the 4-spin interaction exchange constants, K1, are
E4−spin,1Q = −12K1M 4
(5.6)
E4−spin,3Q = − 4 4
K1M (5.7)
3
The degeneracy of single- and multi-Q states within the Heisenberg model is lifted by
the higher-order interactions, by calculating the energy differences between suitable single-
Q and multi-Q states. One set of degenerate states are spin spirals at the M-points of the
2D-BZ and the 3Q-state constructed from the three independent M-points [31], then the
difference between the single-QM and the 3Q is a constant as shown in eq. (5.8).
E3Q − ERW−AFM = (16/3){2K1 + B1} (5.8)
for the difference between the 3Q and the single-QM , which are degenerate in the Heisenberg
model.
To calculate B1 and K1 we need a second equation, which can be found as following:
A superposition of two ±Q-points at the high symmetry line, which is degenerate with
each of them, called the double-Q-state. There are two possible choices, the first one is to
take the Q3ΓK/4-point, which represents a rotating spirals with ± π along Γ-K direction. A
2
superposition of ±Q3ΓK/4 will lead to an double-RW-AFM structure in real space, called
uudd(3ΓK/4)-state as shown in figure 5.5. A second choice is to take the QMΓ/2-point, which represents also rotating spirals with ± π but this time along M-Γ direction. A
2
superposition of ±QMΓ/2 will lead to an double-RW-AFM structure in real space along
[112], called uudd(MΓ/2)-state (see fig. 5.5).
in eq. (A-20) we will have
This means if we plug Q 3ΓK/4 = 3
4
for the single-Q 3ΓK/4 and
× 2
3
E = −M 2 (−2J1 +2J2 − 2J3 − 4J4)+2B1M 4 − 12K1M 4
(5.9)