Interplay among helical order, surface effects and range of interacting 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: November 10, 2018) The properties of helical thin films have been thoroughly investigated by classical Monte Carlo simulations. The employed model assumes classical planar spins 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 interactions, the minimal requirement to get helical order. We obtain that, while the in-plane transition temperatures remain essentially unchanged with respect to the bulk ones, the helical/fan arrangement is stabilized at more and more low temperature when the film thickness, n , decreases; in the ordered phase, increasing the temperature, a softening of the helix pitch wave-vector is also observed. Moreover, we show also that the simulation data around both transition temperatures lead us to exclude the presence of a first order transition for all analyzed sizes. Finally, by comparing the results of the present 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 surface 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 systems 1 still raises great interest, both in consequence of theoretical aspects, related to their peculiar criti- cal properties $^{2}$, and in view of possible technological applications $^{3}$. Indeed, beside conventional ferromagnetic or antiferromagnetic phase transitions, in many new ma- terials other nontrivial and unconventional forms of or- dering have been observed $^{4,5}$. 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 materials $^{6}$, 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 a Z$_{2}$ × 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 ⃗ S$_{i}$ , one has to consider also the Z$_{2}$ symmetry of the spin chirality κ$_{ij}$ ∝ [ ⃗ S$_{i}$ × ⃗ S$_{j}$ ] $^{z}$. 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 structures $^{12}$. 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 the c crystallographic axis, and gives a helix pitch wave-vector Q$_{z}$ such that Q$_{z}$c ′ ≃ 30 $^{◦}$, where c ′ = c/ 2 is the distance between nearest neighboring spin layers par- allel to the ab 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 properties are clearly affected by the strong competition among the helical pitch and the surface effects, which in- volve the majority of the spin layers. In the thickness range n = 9 − 16, i.e. right for thickness values com- parable with the helical pitch, three different magnetic phases emerged, with the high-temperature, disordered, paramagnetic phase and the low-temperature, long-range ordered one separated by an intriguing, intermediate- temperature block phase, where outer ordered layers co- exist with some inner disordered ones, the phase tran- sition of the latter eventually displaying the signatures of a Kosterlitz-Thouless one. Finally, for n ≤ 7 the film collapses once and for all to a quasi-collinear order. The complex phase diagram unveiled by such MC sim- ulations awaken however a further intriguing question: to what extent the observed behavior may be considered a simple consequence of the competition between helical order and surface effects? I.e., is it just a matter of hav- ing such a competition or does the range of interactions also play a relevant role? Indeed, when the range of the interactions is large enough we have a greater number of planes which can be thought of as ”surface planes”, i.e. for which the number of interacting neighbors are sig- nificantly reduced with respect to the bulk layers; there- fore, we expect that the larger the interaction range, the stronger should be the surface effects. But, at the same time, the same modulation of the magnetic order can arXiv:1001.0510v1 [cond-mat.stat-mech] 4 Jan 2010