Explore the Spin Structure and Field-Induced Bilinear Magnetoelectric Effect in Ca₂MgFe₂(PO₄)₄

Linear magnetoelectric materials, in which an external magnetic field induces electric polarization, have garnered significant attention due to their potential applications in spintronics, sensors, and memory devices. The Ca₂MgFe₂(PO₄)₄ material crystallizes in a centrosymmetric orthorhombic structure with the Pbca space group. The structure consists of Fe1O₆ octahedra and (Fe(2,3)/Mg(1,2))O₅ polyhedra connected via corner-sharing PO₄ tetrahedra. This compound undergoes a long-range antiferromagnetic ordering at T_N = 14 K, as confirmed by magnetization and specific heat data. Furthermore, a field-driven metamagnetic transition is observed at a critical field H_c ≈ 3 T. Powder neutron diffraction analysis below T_N indicates that the three crystallographically distinct Fe atoms form a noncollinear antiferromagnetic structure with a magnetic propagation vector k = (0, 0, 0). Dielectric measurements exhibit an anomaly at T_N, while magnetodielectric data show clear features corresponding to the metamagnetic transition at H = 3 T. Interestingly, a field-induced, switchable electric polarization with two distinct linear magnetoelectric responses is observed across H_c, indicating a bilinear magnetoelectric effect. This behavior is consistent with the predicted Pb'ca magnetic space group. These results demonstrate that Ca₂MgFe₂(PO₄)₄ is a type-II multiferroic material exhibiting finite magnetoelectric coupling.