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

98 5 Fe monolayers on hexagonal nonmagnetic substrates
Figure 5.15: Phase diagrams of the classical Heisenberg model for a 2D hexagonal lattice
for states on the high symmetry lines. (a) J1-J2 plane for J3 =0. J2-J3 plane for (b)
J1 > 0 and (c) J1 < 0. (Filled symbols denote values obtained from fits for the Fe ML on
4d-TM substrates).
From these interesting results, we can modify the Heisenberg model Hamiltonian (eq. 3.22),
by adding an additive term, which includes the magnetic moments variation multiplied by
a Stoner parameter, IFe (+Isub):
with
for the single-Q state and
�
i
IiM 2 i
H2-spin = − �
�
i
i,j
i>j
Jij Mi · Mj + 1
2
�
i
IiM 2 i
(5.16)
IiM 2 i = IFeM Fe 1Q + IsubM sub 1Q (5.17)
= IFe[(MFe1Q + δMFe1Q 2Q
−Fe )
1
2 +(M2 Fe1Q + δ2M 2Q
Fe1 −Fe2Q 2
+Isub[(Msub1Q + δMsub1Q 2Q
−sub )
1
2 +(Msub1Q + δM 2Q
sub1 −sub2Q 2
) 2 ]
) 2 ] (5.18)