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

B(q) =2B1(1 + 2 cos(2πq)) (A-25)
for the M-Γ line. Using equations (A-24) or (A-25), the biquadratic interactions for the
high symmetry points are summarized below:
Ebiq,FM = −6B1M 4
Ebiq,RW-AFM = −6B1M 4
Ebiq,Néel = − 3 4
B1M
2
Ebiq,2Q = −2B1M 4
Ebiq,3Q = − 2
Then the single-Q total energy becomes
4
B1M
3
125
(A-26)
E(q) =−NM 2 (J(q) − M 2 B(q)) (A-27)
noting that M 2 B(q) is always coupled with J3 and they can’t be decoupled.
The 4–spin interaction:
The 4–spin interaction Hamiltonian, on four lattice sites, has the form
H4−spin = − �
Kijkl[(Mi ·Mj)(Mk ·Ml)+(Mj ·Mk)(Ml ·Mi)−(Mi ·Mk)(Mj ·Ml)] (A-28)
ijkl
Figure 6.9: Schematic representation of the first order 4–spin interaction, for 2Q 3
ΓK (left)
4
and 2Q MΓ (right), on ijkl lattice sites. For better view, only four diamonds (out of twelve)
2
are shown, and the related lattice sites, to the total twelve diamonds, are only labeled.