Noncollinear magnetism and single-ion anisotropy in multiferroic perovskites

The link between the crystal distortions of the perovskite structure and the magnetic exchange interaction (J), the single-ion anisotropy (SIA), and the Dzyaloshinsky-Moriya (DM) interaction are investigated by means of density-functional calculations in AFeO$_3$ systems. We explore the effect of the crystal distortions (Antiferrodistortive-AFD and Ferroelectric) as well as the effect of the A-cation chemistry on the 3 magnetic properties, J, SIA and DM. Our analysis shows a never explored but possible switching of the weak ferromagnetism in the $R3c$ phase of BiFeO$_3$ through the competition of the SIA shapes induced by the AFD and the ferroelectric distortions. We also found that-in spite of the d$^5$ electronic configuration of Fe$^{3+}$, the SIA is very large in some structures and is surprisingly strongly sensitive to the chemistry of the A-site cation of the ABO$_3$ perovskite. To clarify the origin of this unexpected effect, we analyze the crystal field splitting by means of Wannier functions.