Magnetoelectric multiferroics are systems that exhibit magnetic and ferroelectric order. In strongly coupled multiferroics, competing magnetic interactions can break spatial inversion symmetry and yield ferroelectricity through a magnetic phase transition. In this dissertation, I present original work on the multiferroic properties of Ni₃V₂O₈ and critical phenomena in TbMnO₃ and RbFe(MoO₄)₂.

Ni₃V₂O₈ is an insulating magnet where Ni-spins order in a longitudinal amplitude modulated pattern along the a axis in the high-temperature incommensurate (HTI) phase. Upon cooling to the low-temperature incommensurate phase, an additional spine site spin component along b results in a cycloidal structure that breaks spatial inversion symmetry, yielding ferroelectricity. Electric control of multiferroic domains is demonstrated quantitatively and qualitatively using polarized magnetic neutron diffraction. We show that magnetic and ferroelectric domains are strongly coupled in this system and that definite cycloid handedness is achieved by antisymmetric Dzyaloshinskii-Moriya interactions. Ni₃ V₂O₈ displays a memory effect where the system reverts to the previous polarization state upon exiting and re-entering the multiferroic phase through a first order phase transition. Our results suggest that small multiferroic domains in the paraelectric, commensurate phase retain the polarization history and reestablish it upon re-entering the multiferroic phase.

TbMnO₃ is a frustrated magnet similar to Ni₃V ₂O₈ which undergoes two magnetic phase transitions before becoming multiferroic. A temperature-dependent magnetic diffraction study near T_N reveals that this system orders through a continuous phase transition. Further studies in the HTI phase are needed to clarify whether a novel, weak transition exists at around 39 K. Our results support the single irreducible representation model as the system enters the HTI phase.

RbFe(MoO₄)₂ is a nearly 2D antiferromagnet that enters the multiferroic phase directly from the paramagnetic phase through a continuous phase transition. The obtained value of β = 0.297(1) places this system under the 3D chiral Heisenberg model for stacked triangular lattices with easy plane anisotropy, demonstrating that tridimensionality and chirality is required to explain its critical behavior. Electric fields up to 430 kV/m did not affect its critical properties.