arXiv:1001.0510v1  [cond-mat.stat-mech]  4 Jan 2010
Interplay among helical order, surface eﬀects and range of i nteracting layers in
ultrathin ﬁlms.
F. Cinti (1, 2, 3), A. Rettori (2, 3), and A. Cuccoli (2)
(1) Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1
(2)CNISM and Department of Physics, University of Florence, 50019 Sesto Fiorentino (FI), Italy. and
(3)CNR-INFM S3 National Research Center, I-41100 Modena, Italy
(Dated: June 8, 2022)
The properties of helical thin ﬁlms have been thoroughly inv estigated by classical Monte Carlo
simulations. The employed model assumes classical planar s pins in a body-centered tetragonal
lattice, where the helical arrangement along the ﬁlm growth direction has been modeled by nearest
neighbor and next-nearest neighbor competing interaction s, the minimal requirement to get helical
order. We obtain that, while the in-plane transition temper atures remain essentially unchanged with
respect to the bulk ones, the helical/fan arrangement is sta bilized at more and more low temperature
when the ﬁlm thickness, n, decreases; in the ordered phase, increasing the temperatu re, a softening
of the helix pitch wave-vector is also observed. Moreover, w e show also that the simulation data
around both transition temperatures lead us to exclude the p resence of a ﬁrst order transition for all
analyzed sizes. Finally, by comparing the results of the pre sent work with those obtained for other
models previously adopted in literature, we can get a deeper insight about the entwined role played
by the number (range) of interlayer interactions and surfac e eﬀects in non-collinear thin ﬁlms.
PACS numbers: 64.60.an,64.60.De,75.10.Hk,75.40.Cx,75.70.Ak.
I. INTRODUCTION
The study of low dimensional frustrated magnetic
systems1 still raises great interest, both in consequence
of theoretical aspects, related to their peculiar criti-
cal properties2, and in view of possible technological
applications3. Indeed, beside conventional ferromagnetic
or antiferromagnetic phase transitions, in many new ma-
terials other nontrivial and unconventional forms of or-
dering have been observed4,5. A quantity of particular
interest in this context is the spin chirality, an order pa-
rameter which turned out to be extremely relevant in,
e.g., magnetoelectric materials6, itinerant MnSi 7, binary
compounds as FeGe 8, glass transition of spins 9, and XY
helimagnets, as Holmium, Terbium or Dysprosium 10. In
the latter case, a new universality class was predicted be-
cause aZ2 × SO(2) symmetry is spontaneously broken
in the ordered phase 2: In fact, when dealing with such
systems, in addition to the SO(2) symmetry of the spin
degrees of freedom ⃗Si, one has to consider also the Z2
symmetry of the spin chirality κij ∝
[
⃗Si × ⃗Sj
] z
.
For these rare-earth elements, the development of new
and sophisticated experimental methods 11 has allowed to
obtain ultra-thin ﬁlms where the non-collinear modula-
tion is comparable with the ﬁlm thickness. Under such
conditions the lack of translational invariance due to the
presence of surfaces results decisive in order to observe
a drastic change of the magnetic structures12. Recent
experimental data on ultra-thin Holmium ﬁlms 13 have
been lately interpreted and discussed 14,15 on the basis
of detailed classical Monte Carlo (MC) simulations of a
spin Hamiltonian, which is believed to give a realistic
modeling of bulk Holmium. Such Hamiltonian, proposed
by Bohr et al.16, allows for competitive middle-range in-
teractions by including six diﬀerent exchange constants
along thec crystallographic axis, and gives a helix pitch
wave-vector Qz such that Qzc′ ≃ 30◦, where c′ = c/2 is
the distance between nearest neighboring spin layers par-
allel to theab crystallographic planes, henceforth denoted
also as x − y planes, while z will be taken parallel to c.
For n > 16, n being the number of spin layers in the ﬁlm,
a correct bulk limit is reached, while for lower n the ﬁlm