Spatially modulated magnetic structures in thin films

The axial next–nearest neighbour Ising (ANNNI) model is studied for thin films of up to L = 10 layers, with a distinct phase diagram for each film thickness. The systematics of the ordered phases, as obtained from mean–field theory, Monte Carlo simulations, and low temperature expansions, is discussed. Results are compared to those for the ANNNI model in the limit L −→ ∞. PACS numbers: 68.35.Rh, 75.70.Mk, 64.70.Rh ¶ Laboratoire associé au CNRS UMR 7556 Modulated magnetic structures 2 In recent years magnetism in thin films of a few atomic layers has attracted much interest [1]. However, studies on the influence of the layer thickness, L, on spatially modulated magnetic structures seem to be very scarce [2]. In the following, we shall deal with this aspect by analysing phase diagrams of the axial next–nearest neighbour Ising (ANNNI) model [3, 4, 5] on a simple cubic lattice (setting the lattice constant equal to one) consisting of rather few, but large layers. The competing interactions are supposed to be ferromagnetic between neighbouring spins in each layer, J0 > 0, as well as between neighbouring spins in adjacent layers, J1 > 0, and to be antiferromagnetic, J2 < 0, between axial next–nearest neighbour spins, distinguishing one of the three cubic axes, say, the z–axis. The resulting magnetic structures may exhibit non–trivial spatial modulations along the z–axis. In the case of indefinitely many and indefinitely large layers, the model displays a phase diagram with a plenitude of commensurate phases, including those springing from the multiphase point at zero temperature and those emerging from structure combination branching processes at finite temperatures, as well as incommensurate structures, see the reviews [3, 4, 5]. Here we shall consider thin films of L layers, with free boundary conditions at the surface layers. Note that the ANNNI model on finite lattices, connecting the first and last layers by periodic boundary conditions, has been studied quite extensively before, attempting to predict properties for L −→ ∞. The scope of the present study is, however, to determine the phase diagram of the ANNNI model with free boundary conditions at fixed film thickness. Attention may be also drawn to recent work on the semi–infinite ANNNI model, analysing critical surface properties near the Lifshitz point [6], which partly motivated our investigation. To establish the phase diagrams for thin films, in particular for L = 3, 4, ..., 10, we used mean–field theory, Monte Carlo simulations and low temperature expansions. For simplicity, we set J0 = J1, both in the surface layers and in the bulk of the film. In mean–field approximation [7, 8, 9], we solved the set of coupled non–linear equations for the layer magnetizations, m(z = 1, ..., L), by standard iteration procedures starting from all possible combinations of fully ordered, m = 1 or m = −1, or completely disordered, m = 0, layers, i. e. from, in principle, 3 distinct configurations. To identify the stable phase, at given temperature, τ = kBT/J0, and ratio of competing interactions, κ = −J2/J1, we calculated the free energy for all resulting structures. This approach seems to be suitable to map correctly, in mean–field approximation, the entire phase diagram. The Monte Carlo simulations, using the single–spin flip algorithm, were performed for lattices with K ×K spins in each layer connected by periodic boundary conditions (for thicker films, an efficient layer flip algorithm may be useful [10]). To circumvent finite size effects due to the layer size, K has to be chosen sufficiently large. Actually, we varied K from 10 to 100. To identify the phases springing directly from the degenerate ground states at κ = 1/2, low temperature expansions to leading, first order [11, 12] were found to suffice in most cases. At zero temperature, both for L odd and even, the ferromagnetic state is stable Modulated magnetic structures 3 at κ < 1/2. For L even, at κ > 1/2, the ground state is the 〈2〉 structure, where two layers of ”up” (”down”) spins are followed by two layers of ”down” (”up”) spins, with totally (L/2) pairs of such layers (in general, 〈...kl...〉 refers to a phase at low temperatures where a ”k–band”, i. e. spins in k adjacent layers being oriented (predominantly) in one direction, say, ”+”, is followed by a l–band of spins being oriented (predominantly) in the opposite direction , say, ”–”[12]). For L odd, structures consisting of 2–bands and one 3–band have the lowest energy at 1/2 < κ < 1, while at κ > 1 the ground state contains 2–bands with a 1–band at one of the surfaces (of course, κ = 1 plays no special role for the ANNNI model with L −→ ∞). The transition line, at κ ≈ 1, between the corresponding phases is, at low temperatures, of first order. The degeneracy, DL, of the multiphase point, at (κ = 1/2, τ = 0), is related to the Fibonacci sequence by DL = DL−2 +DL−1, with D2 = D3= 2. We studied the vicinity of that point by using low temperature expansions [11, 12]. In the limit L −→ ∞ [11], indefinitely many periodic, commensurate phases spring directly from that point, namely the ferromagnetic phase and phases in each of which 3–bands are separated by a fixed number, j, of 2–bands, j = 0, 1, 2, ...∞. In thin films, obviously only a few phases may arise from the multiphase point, in particular, the ferromagnetic phase as well as additional phases formed mostly by 2–bands and 3–bands. For L even, the 〈2〉 phase will be among the additional phases, while for L odd, all of these phases will have at least one 3–band (when it is just one 3–band, then that band will be preferably in the center of the film). Note that structures consisting of 3–bands and a supplementary single 4–band or a single 5–band at one of the surfaces of the film occur when L = 3n+1 or L = 3n+2, n = 2, 3, ... Examples of those phases bordering, at low temperatures, the ferromagnetic phase are the 〈43〉, 〈53〉, and 〈433〉 phases. Again in contrast to the case L −→ ∞, phases with 2–bands and pairs of 3–bands may be stable as well; examples are the 〈2332〉 and 〈332〉 phases.– To understand the phase diagram in the vicinity of the multiphase point one should be aware of the fact that the stable phases correspond to the ground state structures with a low energy and a large number of lowest–energy excitations. The latter property favours 3–bands, where the spins in both edge layers can be excited easily when both the preceeding and the following band comprise at least two layers. Examples of entire phase diagrams are depicted in Figs. 1 and 2 (a more complete set of phase diagrams can be found elsewhere [13]). In general, for thin films, there are not many ordered phases. The structures close to the transition to the paramagnetic phase, Tc, may be obtained in mean–field approximation without difficulty from the linearized theory. One finds that the pattern of the layer magnetization m(z) is either symmetric or antisymmetric about the center of the film. Accordingly, some of the low temperature phases discussed above cannot extend up to Tc, e. g. the 〈43〉 phase (note that mean–field theory seems to reproduce the low temperature behaviour qualitatively correctly). Each of the symmetric or antisymmetric phases is usually stable in a rather wide region of the phase diagram, for the thin films we considered. These features, symmetrization and possible instability of low temperature phases close to Tc, are in Modulated magnetic structures 4