Spherical model for anisotropic ferromagnetic films

The corrections to the Curie temperature Tc of a ferromagnetic film consisting of N layers are calculated for N ≫ 1 for the model of D-component classical spin vectors in the limit D → ∞, which is exactly soluble and close to the spherical model. The present approach accounts, however, for the magnetic anisotropy playing the crucial role in the crossover from 3 to 2 dimensions in magnetic films. In the spatially inhomogeneous case with free boundary conditions the D = ∞ model is nonequivalent to the standard spherical one and always leads to the diminishing of Tc(N) relative to the bulk. The application of the spherical model [1] to spatially-inhomogeneous magnetic systems such as ferromagnetic films with free boundary conditions by Barber and Fisher [2] has revealed the unphysical behaviour of the solution being the consequence of the global spin constraint. The dependence T c (N) for a d-dimensional hypercubic lattice infinite in d ′ = d − 1 dimensions and having N layers in the dth dimension has been found to be for d ≥ 4 a non-monotonous function with a maximum, i.e., T c (N) for N ≫ 1 was larger than in the bulk. Other singular features of the spherical model were found by Abraham and Robert [3] by considering the problem of phase separation (i.e., the domain wall formation). Besides the numerous publications using the spherical model for inhomogeneous systems in its original form (see, e.g., [4, 5]), there is a work by Costache, Mazilu und Mihalache [6] in which the global spin constraint was replaced for a ferromagnetic film by separate constraints in each layer. Although this model is less convenient for analytical calculations, it was shown that for d ≥ 4 the value of T c (N) monotonically increases to its bulk value T c (∞), as it should from the physical grounds. Earlier Knops [7] had proved that in a general inhomogeneous situation the spherical model with a spin constraint on each lattice site is equivalent to the D-component classical vector model by Stanley [8] in the limit D → ∞. The latter is not only more physically appealing than the original spherical model, but it also allows one to take into account the spin anisotropy [9] and to produce the 1/D expansions [10, 11, 12]. A convenient tool to handle the D-vector model is the classical spin diagram technique [12, 13]. …