Magnetic coupling in CoCr$_2$O$_4$ and MnCr$_2$O$_4$: an LSDA+$U$ study

Competing magnetic interactions can lead to very interesting magnetic structures and pronounced effects due to spin-lattice coupling. Chromium spinels with additional magnetic $A$-site cations are prime examples for such systems, and have recently regained attention due to the discovery of a small electric polarization in CoCr$_2$O$_4$ [1]. The multiferroic character of CoCr$_2$O$_4$ is supposedly a result of its inversion symmetry-breaking magnetic structure, which has been classified as a ``ferrimagnetic spiral'' [2]. To achieve a better understanding of the complicated magnetic structure in CoCr$_2$O$_4$ and similar systems, an {\it ab initio} determination of the magnetic coupling constants is very desirable. Here, we present results of LSDA+$U$ calculations of the magnetic coupling constants in both CoCr$_2$O$_4$ and MnCr$_2$O$_4$ [3]. We carefully assess the predictive power of such calculations, and then give quantitative estimates for the strengths of the most prominent magnetic interactions. Our results highlight the possible importance of $AA$ interactions in spinel systems with magnetic ions on both $A$ and $B$ sites, and provide an important link between previous theory and experimental observations. \par \noindent [1] Y. Yamasaki {\it et al.}, Phys. Rev. Lett. 96, 207204 (2006). [2] D. H. Lyons {\it et al.}, Phys. Rev. 126, 540 (1962). [3] C. Ederer and M. Komelj, Phys. Rev. B 76, 064409 (2007).