Multiferroics are known to be classified into two types. However, type I lacks sufficient magnetoelectric coupling and type II lacks sufficient electric polarization, making both practically difficult. In this Letter, we explore the possibility of type-III multiferroics, where the origins of ferroelectricity and magnetism are highly intertwined but not causally related, with a combination of strong magnetoelectric coupling and large polarization. Our first-principles calculations predict that monolayer ${\mathrm{TiCdO}}_4$ is such a type-III ferroelectric-ferromagnetic multiferroic with both electronic and magnetic orders originating from competing electron populations on oxygen atoms. It shows an electric polarization of $50\mathrm{\mu }\mathrm{C}/{\mathrm{cm}}^2$, while the maximum linear and quadratic magnetoelectric response are as high as $35000\mathrm{ps}/\mathrm{m}$ and $1.59\times {10}^{-14}\mathrm{s}/\mathrm{A}$, respectively. Our Letter opens up new perspectives for the discovery and design of much-anticipated multiferroics that can be used for cross-modulation.

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• Magnetoelectric effect
• Multiferroics
• Density functional theory
• First-principles calculations