Nature of magnetic phase transitions and spin-driven ferroelectricity in BaHoFeO4

BaHoFeO₄ is an emergent multiferroic material that exhibits unique magnetic-field (H)-induced ferroelectric behavior. However, the origin of this effect has remained elusive, due to the role of complex magnetic interactions in driving the system's behavior. To shed light on this, we investigated the magnetic structures of the material using neutron powder diffraction, as well as their magnetic-field-induced evolutions through a combination of complementary techniques, including magnetic measurements, M\"ossbauer spectroscopy, and terahertz time-domain spectroscopy. The orthorhombic crystal structure of the compound, characterized by the structural interconnection between two-dimensional Fe chains and Ho chains, remains stable upon cooling to 3 K. Below TN1≈50 K, Fe³⁺ spins form a collinear incommensurate spin-density-wave antiferromagnetic order with a magnetic propagation vector k₁=(0,0,0.329), and the magnetic moments align along the b axis. Upon further cooling, the enhancement of the 3d-4f exchange coupling induces long-range order in the Ho sublattice below 25 K, transforming the magnetic structure into a noncollinear commensurate antiferromagnetic order with k₂=(0.5,0,0.5), with all Fe and Ho moments lying in the $ac$ plane. As the temperature decreases further, strengthening of the 4f-4f interactions leads to the emergence of a new commensurate antiferromagnetic order with a magnetic propagation vector k₃=(0,0,0.5) at 3 K, coexisting with the antiferromagnetic structure associated with k₂. Similar to the incommensurate spin-density-wave structure, the Ho and Fe spins in the k₃ phase are oriented along the b axis, with their magnetic moments modulated along the c axis. The spin configuration of the Ho rings is highly sensitive to temperature and magnetic field, in contrast to the more robust spin configuration of the Fe rings. We attribute the H-induced magnetic and ferroelectric transitions in BaHoFeO₄ to the H-induced modifications in the Ho sublattice. Two metamagnetic phase transitions occur at H₁≈1 T and H₂≈2 T, associated with a spin-flop transition of the Ho magnetic sublattice. The H-induced canted magnetic configuration in the Ho rings above H₁ drives the H-induced ferroelectricity in BaHoFeO₄. This is subsequently suppressed by the growth of a higher-field paraelectric antiferromagnetic phase, characterized by a fully ferromagnetic arrangement of the Ho chains above H₂. Our study provides important insights into the origins of the metamagnetic phase transitions and the associated magnetoelectric effects.