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

A linear combination of two or three single-qs will also correspond to a real space spin
structure satisfying equation (A-5) on all lattice sites with spin Mqi = Aˆx + Bˆy + Cˆz.
Since the product of Qi · Rj is a multiple of π, then cos(Qi · Rj) =±1 and the new spin
on the lattice sites j can be written
Mj =
3,N�
i,j=1,1
123
Mqi {cos(qi · Rj), sin(qi · Rj), 0} (A-18)
with M2 qi = M 2 , (A-9) and (A-10) q = 2π
a (0, 1/(2√3)) and q = 2π
a (0, −1/(2√3)) ( ΓM),
or 2
q = 2π
a (0, 1/(2√3)) and q = 2π
a (0, −1/(2√3)) ( 3ΓK),
will correspond to four distinct real
4
space lattice sites.
Energetics of the high symmetry states:
The energetics of the magnetic states on the two-dimensional hexagonal lattice can be
described within the Heisenberg model, using the Fourier transform of the exchange constants
J(q) according to equation (A-19). This can be done by expanding the vector q into
the primitive vectors of the reciprocal lattice, q = q1b1 + q2b2. Then, up to the seventh
nearest neighbor, J(q) can be written as
J(q) = −M 2 [2J1[cos(2πq1) + cos(2πq2) + cos(2π(q1 − q2))]
+ 2J2[cos(2π(q1 + q2)) + cos(2π(2q1 − q2) + cos(2π(−q1 +2q2))]
+ 2J3[cos(4πq1) + cos(4πq2) + cos(2π(2q1 − 2q2))]
+ 2J4[cos(2π(3q1 − q2)) + cos(2π(2q1 + q2)) + cos(2π(3q1 − 2q2))
+ cos(2π(q1 +2q2)) + cos(2π(−q1 +3q2)) + cos(2π(2q1 − 3q2))]
+ 2J5[cos(6πq1) + cos(6πq2) + cos(2π(3q1 − 3q2))]
+ 2J6[cos(2π(2q1 − 4q2)) + cos(2π(4q1 − 2q2) + cos(2π(2q1 +2q2))]
+ 2J7[cos(2π(−q1 +4q2)) + cos(2π(q1 +3q2)) + cos(2π(3q1 + q2))
+ cos(2π(4q1 − q2)) + cos(2π(4q1 − 3q2)) + cos(2π(3q1 − 4q2))] (A-19)
By that one can write the energies of all relevant magnetic states within the Heisenberg
model. Along the high symmetry line Γ-K-M, the q = q(b1 + 1
2 b2) holds (q1 = q,q2 = 1
2 q).
The Γ-point is at q = 0, the K-point is at q = 2
3
and the M-point is at q = 1. By that one
can express the energy along the high symmetry line Γ-K-M, using eq. (A-19), up to the
seventh nearest neighbor exchange constants