Two-dimensional rare-earth halide based single-phase triferroic

Two-dimensional multiferroic materials are highly sought after due to their huge potential for applications in nanoelectronic and spintronic devices. Here, we predict, based on first-principle calculations, a single-phase triferroic where three ferroic orders---ferromagnetism, ferroelectricity, and ferroelasticity---coexist simultaneously in the hole doped ${\mathrm{GdCl}}_{2}$ monolayer (a ferromagnetic semiconductor). This is achieved by substituting 1/3rd of the ${\mathrm{Gd}}^{2+}$ ions with ${\mathrm{Eu}}^{2+}$ in the hexagonal structure of the ${\mathrm{GdCl}}_{2}$ monolayer. The resulting metallic state undergoes a bond-centered charge ordering driving a distortion in the hexagonal structure, making it semiconducting again and ferroelastic. Further, the lattice distortion accompanied by a breaking of the lattice centrosymmetry renders a noncentrosymmetric charge distribution, which makes the monolayer ferroelectric, at the same time. The two ferroic orders, ferroelectricity and ferroelasticity, present in the Eu-substituted ${\mathrm{GdCl}}_{2}$ monolayer are found to be strongly coupled, making it a promising candidate for device applications. The Eu-substituted monolayer remains a ferromagnetic semiconductor with a large $4f$ magnetic moment just like the parent monolayer and possesses an even higher (out-of-plane) magnetic anisotropy energy than its pristine counterpart as desired for two-dimensional magnets to have high transition temperature.