Magnetization direction of a semi-infinite Heisenberg ferromagnet with perpendicular surface anisotropy

The phase transition of a semi-infinite Heisenberg ferromagnet having easy-axis (perpendicular) anisotropy of the form -${\mathit{D}}_{\mathit{s}}$(${\mathit{S}}^{\mathit{z}}$ ${)}^2$ (${\mathit{D}}_{\mathit{s}}$>0) in the surface and easy-plane anisotropy of the form -${\mathit{D}}_{\mathit{b}}$(${\mathit{S}}^{\mathit{z}}$ ${)}^2$ (${\mathit{D}}_{\mathit{b}}$<0) in the bulk is studied by the molecular-field approximation. For a fixed ${\mathit{D}}_{\mathit{b}}$, three ordered phases, i.e., canted spin, in-plane ordering, and perpendicular ordering phases, are obtained depending on the temperature and the strength of the surface anisotropy ${\mathit{D}}_{\mathit{s}}$.