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

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where, the relation �
i
H = �
i,j
= �
i,j
−Jij
−Jij
= −N �
q
�
q,q ′
�
q,q ′
Mq · Mq ′eiqRi e iq ′ Rj
Mq · Mq ′ei(q+q′ )Ri e iq ′ (Rj−Ri)
Mq · M−q
� �
δ
J0δe −iqRδ
�
(A-6)
e i(q+q′ )Ri = Nδq,−q ′ holds for a sum over all lattice sites, Rδ = Rj − Ri
and the exchange constants are defined to be real quantities with
J(q) = �
J0δe −iqRδ ∗
= J(−q) =J(q)
then, equation (A-6) will become
δ
H = −N �
J(q)Mq · M−q
q
(A-7)
(A-8)
Minimizing the energy (eq. A-6) under the condition that all lattice sites have the same
spin magnitude (eq. A-2) will lead to determination of the magnetic ground state for N
independent equations. This is equivalent to a system of N equations with Fourier spin
components
�
Mq · M−q = M 2
(A-9)
and
�
q
q
Mq · Mq ′ −q = 0, with q ′ �= 0 (A-10)
Which means that all Mq vanish except for MQ and M−Q, where ±Q maximize J(Q) and
the lowest energy is then given by
E = −NM 2 J(Q) (A-11)
The spin structure which corresponds to MQ and MQ can be covered by introducing
the real and imaginary parts, RQ and IQ:
MQ = RQ + iIQ, M−Q = RQ − iIQ
Then, using eq. (A-9) and (A-10), we obtain
MQ · M−Q = R 2 Q + I 2 Q = M 2
(A-12)
(A-13)