Finite size effects on the symmetry of metastable configurations in the classical one-dimensional planar spin model with competing exchange interactions

The classical one-dimensional (1D) planar spin model with competing nearest neighbor (nn) and next nearest neighbor (nnn) exchange interactions (Jnn>0 and Jnnn<0, respectively) was introduced decades ago [1] to account for the observation of a modulated phase (a spiral or helicoid) in a class of magnetic crystals and alloys, including rare-earth elements and manganese compounds. In the thermodynamic limit, the modulated phase was proved to exist provided that G=Jnn/(4|Jnnn|)<1 and the relative angle between neighboring spins is given by +arccos(G) or -arccos(G). Opposite signs correspond to equivalent helicoids with opposite sense of rotation (or chirality). In the present work, we investigate the effect of finite size on the equilibrium states of such a model. We are driven by the interest for artificially created nanoscale magnetic structures: for example, an ultrathin film of Ho [2], made of N parallel ferromagnetic planes, where the vector magnetization of each atomic layer is confined to the film plane and is exchange coupled to the magnetization of neighboring layers, with opposite signs of the exchange constant depending on the layers' position (positive between nn layers and negative between nnn ones). Finding the magnetization profile across the film thickness, while accounting for the discrete location of atomic layers, is a difficult task even in a mean field approximation, where the problem is reduced to a 1D one, since it requires the necessity to solve a system of (N-1) equations, for the (N-1) relative orientation angles, obtained after the minimization of the thermodynamic potential. Except for very small values of N, finding the exact solution is quite demanding; thus, to obtain an estimate of the equilibrium configurations, most authors resorted either to time-consuming iterative procedures [2] or to a continuous approximation which allowed to obtain analytical results [3]. In this work we make use of a theoretical method [4], recently developed to find the noncollinear canted magnetic states of ultrathin ferromagnetic films with competing surface and bulk anisotropies [5], to calculate the magnetization profile in the case of our model with competing nn and nnn exchange interactions. The essence of the method is to reduce the difficult problem of finding minima of the thermodynamic potential in the (N-1)-dimensional space of the (N-1) relative orientation angles, to the much simpler problem of finding the (N-1) roots of a function in the one-dimensional space of the first relative orientation angle. Subsequently, the roots are analyzed in order to determine which of them correspond to stable, metastable or unstable states. In this way, we were able to determine, in a very quick and quite accurate way, the equilibrium states of the model up to N=15. In addition to the ground state, which is symmetric with respect to the center of the chain (or, equivalently, to the center of the film), we found metastable states of two kinds: either antisymmetric or without a definite symmetry ("ugly" states). In the ground state, the modulated configuration is non uniform along the finite size of the chain, but the chirality of the helicoid does not change. In contrast, the metastable states are characterized either by a change of chirality in the middle of the chain (antisymmetric state) or a change of chirality located away from the middle of the chain ("ugly" state). The above interpretation was confirmed performing a further analysis of the various modulated configurations in the framework of a discrete nonlinear mapping approach developed years ago [6]. The most interesting result, coming from our exact calculations, is that the antisymmetric states are metastable for even values of N and unstable for odd values of N, while the "ugly" states are always metastable. This fact, being a consequence of discretization and finite size, can by no means be evidenced using a continuum model [3]. Clearly, as N grows, any difference between even and odd number of N is found to decrease, and for N tending to infinity it is expected to vanish.[1] T. A. Kaplan, Phys. Rev. 116, 888 (1959); A. Yoshimori, J. Phys. Soc. Jpn. 14, 807 (1959); J. Villain, J. Phys. Chem. Solids 11, 303 (1959).[2] E. Weschke et al., Phys. Rev. Lett. 93, 157204 (2004).[3] P. I. Melnichuk, A. N. Bogdanov, U. K. Roessler, and K.-H. Mueller, J. Magn. Magn. Mater. 248, 142 (2002).[4] A. P. Popov, A. V. Anisimov, O Eriksson, and N. V. Skorodumova, Phys. Rev. B 81, 054440 (2010).[5] A. P. Popov, J. Magn. Magn. Mater. 324, 2736 (2012).[6] L. Trallori, P. Politi, A. Rettori, M. G. Pini, and J. Villain, Phys. Rev. Lett. 72, 920 (1994).