Mean-field approximation model for multiferroicity and magnetoelectric coupling effects in polar magnet Co<sub>2</sub>Mo<sub>3</sub>O<sub>8</sub>

In recent years, polar magnets <i>M</i><sub>2</sub>Mo<sub>3</sub>O<sub>8</sub> (<i>M</i>: 3d transition metal) have emerged as a research focus in condensed matter physics and materials science due to their unique crystal structures, multiple continuous magnetoelectric-coupled state transitions, and potential applications. Notably, Co<sub>2</sub>Mo<sub>3</sub>O<sub>8</sub> exhibits a significant second-order nonlinear magnetoelectric coupling effect in its ground state, corresponding to a unique microscopic magnetoelectric coupling mechanism and practical value. In this work, based on a molecular field phenomenological model, two distinct antiferromagnetic sublattices for the Co<sub>2</sub>Mo<sub>3</sub>O<sub>8</sub> system constructed and the temperature-dependent variations of its spontaneous magnetic moment, spin-induced ferroelectric polarization, first-order linear magnetoelectric coupling coefficient, and second-order nonlinear magnetoelectric coupling coefficient are presented. Particularly, the parameters <i>t</i> = –1 and <i>o</i> = –0.96 indicate distinct exchange energies between the magnetic sublattices associated with tetrahedron (Co<sub>t</sub>) and octahedron (Co<sub>o</sub>). The Co<sup>2+</sup> ions in these two sublattices, which are characterized by different molecular field coefficients, synergistically mediate a spin-induced spontaneous polarization of <i>P</i><sub>S</sub> ~ 0.12 μC/cm<sup>2</sup> through the exchange striction mechanism and p-d hybridization mechanism in Co<sub>2</sub>Mo<sub>3</sub>O<sub>8</sub>. In addition, the significant second-order magnetoelectric coupling effect with a coefficient peaking at 7 × 10<sup>–18</sup> s/A near the <i>T</i><sub>N</sub> in Co<sub>2</sub>Mo<sub>3</sub>O<sub>8</sub>, with this coefficient being significantly larger than those of isostructural Fe<sub>2</sub>Mo<sub>3</sub>O<sub>8</sub> (1<i>.</i>81 × 10<sup>–28</sup> s/A) and Mn<sub>2</sub>Mo<sub>3</sub>O<sub>8</sub>, implies that this enhancement primarily arises from the weaker inter-sublattice antiferromagnetic exchange between its two sublattices, leading to a stabilizes metastable spin configuration. This also indicates that the Co<sub>2</sub>Mo<sub>3</sub>O<sub>8</sub> system possesses stronger irreversibility stability and exhibits a pronounced magnetoelectric diode effect, providing a solid theoretical and computational foundation for developing magnetoelectric diodes.