Effects of surface exchange anisotropy in Heisenberg ferromagnetic insulators

We consider an fcc semi-infinite ferromagnetic insulator displaying an anisotropic exchange interaction between spins on the (111) surface plane of the form J_∥(SᵢˣSⱼˣ+SᵢʸSJʸ+ηSᵢᶻSⱼᶻ). We assume all other interactions isotropic. A self-consistent random-phase-approximation calculation is performed, with a Green's-function method valid for any spin S, up to the bulk transition temperature Tcᵇ, by the assumption that the magnetization of the third layer equals the bulk value. For η sufficiently large, the surface magnetization is nonzero for T>Tcᵇ, up to a transition temperature Tcˢ(η) whenever η≥ηc>1, where Tcˢ(ηc)=Tcᵇ. For T>Tcᵇ the system is equivalent to a film of three layers, where the magnetization of the third one is identically zero as a boundary condition. A discontinuity of the derivative in the curve of the magnetization of the first two layers versus temperature is found at Tcᵇ. The results show clearly a crossover from Heisenberg to Ising behavior at the surface.