Are there type-III multiferroics?

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 work, 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 TiCdO$_{4}$ is such a type-III ferroelectric-ferromagnetic multiferroics with both electronic and magnetic orders originating from competing electron populations on oxygen atoms. It shows an electric polarization of 50 $\mu$C/m$^{2}$ while the maximum linear and quadratic magnetoelectric response are as high as 35000 ps/m and 1.59 $\times$ 10$^{-14}$ s/A, respectively. Our study opens up new perspectives for the discovery and design of much-anticipated multiferroics that can be used for cross-modulation.