Spin-splitting antiferromagnets with spin-polarized band structures in momentum space have garnered intensive research attention due to their zero net magnetic moments, ultrafast spin dynamics as conventional antiferromagnets, and spin-polarized transport properties akin to ferromagnets, making them promising candidates for antiferromagnetic spintronics. However, unlike spin-torque switching of ferromagnets by electric current, efficient electric control of spin-splitting antiferromagnetic order remains challenging. In this work, we identify prototypes of multiferroic spin-splitting antiferromagnets, including $\mathrm{BiFe}{\mathrm{O}}_3, {\mathrm{Fe}}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$, and compensated ferrimagnet $\mathrm{GaFe}{\mathrm{O}}_3$ with ferroelectric polarization as well as spin-polarized electronic structures. We establish design principles for the spin-splitting multiferroic antiferromagnets and compensated ferrimagnet, elucidating the band symmetry features in the Brillouin zone. We demonstrate that the spin polarization in spin-splitting magnets, despite zero net magnetic moment, can be switched by ferroelectric polarization, providing an efficient means of controlling the antiferromagnetic order. Our work may inspire future development of novel multiferroic functional magnets with zero magnetic moments and pave the way for their applications in magnetoelectric spintronic devices.

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• Density of states
• Electronic structure
• Ferroelectricity
• First-principles calculations
• Magnetism
• Spintronics