Spin-Lattice Coupling and Third Neighbor Magnetic Interactions in EuTiO$_3$

An ongoing challenge in materials physics is to identify materials that display a strong coupling between the electrical polarization and magnetism. EuTiO$_3$ is one such material that has been of much recent interest. This novel material is antiferromagnetic and paraelectric in bulk but becomes simultaneously ferromagnetic and ferroelectric under biaxial strain due to a rather large spin-lattice (phonon) coupling. In this talk we will present the results of our first-principles study on the effect of ferroelectric distortions and octahedral rotations on the magnetic exchange interactions in EuTiO$_3$. We elucidate the evolution of the octahedral rotation pattern with strain and show how they influence the properties of the multiferroic phase. Going beyond the proposed cation-mediated exchange for EuTiO$_3$, which has been linked to the large spin-lattice coupling in this material, we uncover the importance of third-neighbor magnetic interactions and illustrate how it is responsible for the ``giant'' cross-field magnetoelectric effect recently demonstrated.