Static magnetization direction under perpendicular surface anisotropy

The problem of depth dependence of the magnetization tilt angle θ(z) is solved analytically for an exchange-coupled ferromagnet with dipole energy, applied field, and surface anisotropy ${\mathit{K}}_{\mathit{s}}$. For ${\mathit{K}}_{\mathit{s}}$≤(π${\mathit{M}}^2$A${)}^{1/2}$=${\mathit{K}}_{\mathit{c}}$, the surface magnetization direction remains in the plane, ${\mathrm{\theta }}_{\mathit{s}}$=θ(0)=π/2. For ${\mathit{K}}_{\mathit{s}}$≥${\mathit{K}}_{\mathit{c}}$, the magnetization vector moves out of the surface, initially with d${\mathrm{\theta }}_{\mathit{s}}$/${\mathit{dK}}_{\mathit{s}}$=-∞, then slowly approaches ${\mathrm{\theta }}_{\mathit{s}}$=0 as ${\mathit{K}}_{\mathit{s}}$ approaches infinity. For Fe, ${\mathit{K}}_{\mathit{c}}$ is of order 4 erg/${\mathrm{cm}}^2$. The results indicate that a perpendicular component of magnetization at the surface relaxes to its bulk in-plane orientation over a distance that is generally much shorter than the bulk domain-wall width. The surface remanence and initial susceptibility of the surface magnetization for in-plane applied field are calculated and provide useful expressions for interpreting experimental results. The experimentally measured surface magnetization increases linearly with probe sensitivity depth.