Two-dimensional ferromagnetic-ferroelectric multiferroics in violation of the d0 rule

Contribution of a $d$ electron to ferroelectricity of type-II multiferroics causes strong magnetoelectric coupling and distinguishes them from the conventional type-I multiferroics. However, their therein polarization is too small because the ferroelectricity is merely a derivative from the magnetic order. Here we report a class of multiferroic materials, monolayer $\mathrm{VO}{X}_{2}$ ($X=\mathrm{Cl}$, Br, and I), which combine the advantages of type-I and type-II multiferroics. Both ferroelectricity and magnetism arise from the same V cation, where the filled $d$ orbital is perpendicular to an a priori ferroelectric polarization and thus poses no hindrance to ferroelectricity, indicating a violation of the usual ${d}^{0}$ rule. This makes the combination of large polarizations and strong magnetoelectric coupling possible. Our findings not only add new ferromagnetic-ferroelectric multiferroics, but also point to a unique mechanism to engineer multiferroics.