2
x
z
y
J J
J
1 
0
2 
FIG. 1: (colors online) (a): body-centered tetragonal (BCT)
lattice with J0 in-plane coupling constant, and out-of-plane
J1, and J2 competing interactions.
be achieved with diﬀerent number of interacting layers:
notably, nearest and next-nearest layers competitive in-
teractions are enough to get a helical structure with a
whatever pitch wavevector. Such observation gives us a
possible way to solve the conundrum previously emerged,
as we have the possibility of varying the range of inter-
actions without modifying the helical pitch, thus decou-
pling the two relevant length scales along the ﬁlm growth
direction, and making accessible a range ofn of the or-
der of, or smaller than, the helical pitch, but still large
enough that a substantial number of layers can behave
as “bulk” layers. Therefore, while in the previous papers
we have studied the properties of ultrathin magnetic ﬁlms
of Ho assuming a model with six interlayer exchange in-
teractions, here we investigate by MC simulations the
properties of the same system by making use of the sim-
plest model Hamiltonian able to describe the onset of a
helical magnetic order in Holmium, i.e. we consider only
two inter-layer coupling constants, as previously done in
Ref. 11.
The paper is organized as follows: In Sec. II the model
Hamiltonian will be deﬁned, and the MC techniques, and
all the thermodynamic quantities relevant for this study,
will be introduced. In Sec. III the results obtained for
diﬀerent thicknesses will be presented, both in the matter
of the critical properties of the model and of the magnetic
ordered structures observed. Finally, in Sec. IV we shall
discuss such results, drawing also some conclusions.
II. MODEL HAMILTONIAN AND MONTE
CARLO OBSER V ABLES
The model Hamiltonian we use in our simulations is the
minimal one able to describe helimagnetic structures:
H = −

J0
∑
⟨ij⟩
⃗Si · ⃗Sj + J1
∑
⟨ik⟩
⃗Si · ⃗Sk + J2
∑
⟨il⟩
⃗Si · ⃗Sl

.
(1)
⃗Si are classical planar unit vectors representing the di-
rection of the total angular momentum of the magnetic
ions, whose magnitude
√
j(j + 1) ( j = 8 for Holmium
ions) is already encompassed within the deﬁnition of the
interaction constantsJ0, 1, 2. As sketched in Fig. 1, the
magnetic ions are located on the sites of a body-centered
tetragonal (BCT) lattice; the ﬁrst sum appearing in the
Hamiltonian describes the in-plane (xy) nearest neigh-
bor (NN) interaction, which is taken ferromagnetic (FM),
with exchange strengthJ0 > 0; the second sum rep-
resents the coupling, of exchange strength J1, between
spins belonging to nearest neighbor (NN) planes along
thez-direction (which we will assume to coincide with
the ﬁlm growth direction); ﬁnally, the third sum takes
into account the interaction, of exchange strengthJ2, be-
tween spins lying on next-nearest neighbor (NNN) planes
alongz. In order to have frustration, giving rise to non-
collinear order along z in the bulk, NN interaction J1
can be taken both ferro- or antiferromagnetic, but NNN
couplingJ2 has necessarily to be antiferromagnetic, and
the condition |J2| > |J1|/4 must be fulﬁlled. Such simpli-
ﬁed Hamiltonian was already employed to simulate he-
lical ordering in bulk systems by Diep1,17 and Loison 18.
In the bulk limit, the state of minimal energy of a sys-
tem described by Eq.(1) corresponds to a helical arrange-
ment of spins. The ground state energy per spin is equal
toeg(Qz) = [ −4J0 − 2J1 (4 cos (Qzc′) + δ cos (2Qzc′))]
where c′ is the distance between NN layers, δ = J2
J1
,
and Qzc′ = arccos
(
− 1
δ
)
is the angle between spins ly-
ing on adjacent planes along the z-direction. The ob-
served helical arrangement in bulk holmium corresponds
toQzc′ ≃ 30.5◦10: such value can be obtained from
the formula above with the set of coupling constants
J0=67.2 K, J1=20.9 K, and J2 = −24.2 K, that we have
employed in our simulations. The given values for the ex-
change constants are the same already used by Weschke