Origin of Magnetic Anisotropy in Cobalt-Substituted Magnetite

The large part of the ferromagnetic anisotropy of ${\mathrm{Co}}_x{\mathrm{Fe}}_{3-x}{\mathrm{O}}_4$ 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 $\alpha \left(\simeq 1\right)$ of ${\mathrm{Co}}^{2+}$ is constrained by the crystal electric field to lie parallel to the axis of trigonal symmetry. Spin-orbit energy $\lambda \mathrm{L}·\mathrm{S}$ couples the spin to this axis, accounting for the anisotropy energy. By fitting the theory to cubic anisotropy data one finds $\alpha \lambda =-132\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$. 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 ${\mu }_L$ of ${\mathrm{Co}}^{2+}$ is predicted to be large (${\mu }_L\simeq 0.5$ Bohr magneton) and anisotropic ($\Delta {\mu }_L\simeq 0.1$ Bohr magneton) at low temperatures.