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Magnetic structure and magnetoelectric coupling in bulk and thin film FeVO$_4$

We have investigated the magnetoelectric and magnetodielectric response in FeVO$_4$, which exhibits a change in magnetic structure coincident with ferroelectric ordering at $T_{N2}$$\approx$15 K. Using symmetry considerations, we construct a model for the possible magnetoelectric coupling in this system, and present a discussion of the allowed spin structures in FeVO$_4$. Based on this model, in which the spontaneous polarization is caused by a trilinear spin-phonon interaction, we experimentally explore the magnetoelectric coupling in FeVO$_4$ thin films through measurements of the electric field induced shift of the multiferroic phase transition temperature, which exhibits an increase of 0.25 K in an applied field of 4 MV/m. The strong spin-charge coupling in \fvo\, is also reflected in the significant magnetodielectric shift, which is present in the paramagnetic phase due to a quartic spin-phonon interaction and shows a marked enhancement with the onset of magnetic order which we attribute to the trilinear spin-phonon interaction. We observe a clear magnetic field induced dielectric anomaly at lower temperatures, distinct from the sharp peak associated with the multiferroic transition, which we tentatively assign to a spin reorientation cross-over. We also present a magnetoelectric phase diagram for FeVO$_4$.