Origin of Magnetic Anisotropy in Cobalt-Substituted Magnetite

The large part of the ferromagnetic anisotropy of ${\mathrm{Co}}_{x}{\mathrm{Fe}}_{3\ensuremath{-}x}{\mathrm{O}}_{4}$ attributed to the presence of ${\mathrm{Co}}^{2+}$ is explained, for small $x$, by means of a one-ion model. The residual orbital angular momentum $\ensuremath{\alpha}(\ensuremath{\simeq}1)$ of ${\mathrm{Co}}^{2+}$ is constrained by the crystal electric field to lie parallel to the axis of trigonal symmetry. Spin-orbit energy $\ensuremath{\lambda}\mathrm{L}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{S}$ couples the spin to this axis, accounting for the anisotropy energy. By fitting the theory to cubic anisotropy data one finds $\ensuremath{\alpha}\ensuremath{\lambda}=\ensuremath{-}132$ ${\mathrm{cm}}^{\ensuremath{-}1}$. The assumption that cations are mobile at higher temperatures leads to a quantitative explanation of the annealing-induced anisotropy energy. The mean orbital magnetic moment ${\ensuremath{\mu}}_{L}$ of ${\mathrm{Co}}^{2+}$ is predicted to be large (${\ensuremath{\mu}}_{L}\ensuremath{\simeq}0.5$ Bohr magneton) and anisotropic ($\ensuremath{\Delta}{\ensuremath{\mu}}_{L}\ensuremath{\simeq}0.1$ Bohr magneton) at low temperatures.