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Strain-Induced Magnetic Anisotropy in Epitaxial Thin Films of the Spinel CoCr$_2$O$_4$

We show that the magnetic anisotropy in spinel-structure CoCr$_2$O$_4$ thin films exhibits a strain dependence in which compressive strain induces an out-of-plane magnetic easy axis and tensile strain an in-plane easy axis, exactly opposite to the behavior reported for the related compound CoFe$_2$O$_4$. We use density functional theory calculations within the LSDA+U approximation to reproduce and explain the observed behavior. Using second-order perturbation theory, we analyse the anisotropy tensor of the Co$^{2+}$ ions in both octahedral and tetrahedral coordination, allowing us to extend our results to spinels with general arrangements of Co$^{2+}$ ions.