Noncollinear magnetism and single-ion anisotropy in multiferroic perovskites

The link between the crystal distortions of the perovskite structure and the magnetic exchange interaction, the single-ion anisotropy (SIA), and the Dzyaloshinsky-Moriya (DM) interaction are investigated by means of density functional calculations. Using BiFeO₃ and LaFeO₃ as model systems, we quantify the relationship between the oxygen octahedra rotations, the ferroelectricity, and the weak ferromagnetism (wFM). We recover the fact that the wFM is due to the DM interaction induced by the oxygen octahedra rotations. We find a simple relationship between the wFM, the oxygen rotation amplitude, and the ratio between the DM vector and the exchange parameter such that the wFM increases with the oxygen octahedra rotation when the SIA does not compete with the DM forces induced on the spins. Unexpectedly, we also find that in spite of the d⁵ electronic configuration of Fe³⁺, the SIA is very large in some structures and is surprisingly strongly sensitive to the chemistry of the A-site cation of the ABO₃ perovskite. In the ground $R3c$ state phase we show that the SIA shape induced by the ferroelectricity and the oxygen octahedra rotations are in competition such that it is possible to turn the wFM ``on'' and ``off'' through the relative size of the two types of distortion.