Static magnetization direction under perpendicular surface anisotropy

Abstract

The problem of depth dependence of the magnetization tilt angle \ensuremath{\theta}(z) is solved analytically for an exchange-coupled ferromagnet with dipole energy, applied field, and surface anisotropy ${\mathit{K}}_{\mathit{s}}$. For ${\mathit{K}}_{\mathit{s}}$\ensuremath{\le}(\ensuremath{\pi}${\mathit{M}}^{2}$A${)}^{1/2}$=${\mathit{K}}_{\mathit{c}}$, the surface magnetization direction remains in the plane, ${\mathrm{\ensuremath{\theta}}}_{\mathit{s}}$=\ensuremath{\theta}(0)=\ensuremath{\pi}/2. For ${\mathit{K}}_{\mathit{s}}$\ensuremath{\ge}${\mathit{K}}_{\mathit{c}}$, the magnetization vector moves out of the surface, initially with d${\mathrm{\ensuremath{\theta}}}_{\mathit{s}}$/${\mathit{dK}}_{\mathit{s}}$=-\ensuremath{\infty}, then slowly approaches ${\mathrm{\ensuremath{\theta}}}_{\mathit{s}}$=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.