Tunable multiple-state polarized antiferromagnets

Momentum-dependent nonrelativistic spin splitting in altermagnets is determined by crystal symmetry and thus it is difficult to tune spin splitting for a specific material. The discovery and design of antiferromagnets with tunable ferromagnetic-like nonrelativistic spin splitting is of great significance for both basic science and practical applications. In this work, we propose the multiple-state polarized antiferromagnets (PAFMs), which couple multiple polarization states and antiferromagnetic order and thus realize tunable ferromagnetic-like spin splitting. For realizing multiple polarization states, we propose a symmetry design concept by considering two factors: broken spatial inversion symmetry for ferroelectricity, and low interface symmetry for multiple potential wells. Based on these two considerations, we choose the displacement-type ferroelectric monolayer with the out-of-plane polarization and different atomic arrangements on the upper and lower surfaces. As an example, in the displacement-type antiferroelectric ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ bilayers, two opposite sliding ferroelectric states induce opposite spin splitting. Furthermore, in displacement-type ferroelectric bilayers, the displacement-type ferroelectricity and sliding ferroelectricity can coexist. Due to the spontaneous broken spatial inversion symmetry and low interface symmetry, multiple polarization states with a suitable transition barrier in between are obtained. Considering the coupling between the displacement-type ferroelectricity and sliding ferroelectricity, there exist six polarization states corresponding to different spin splitting, which can be controlled by an external electric field. Therefore, the nonvolatile electric-field controllable ferromagnetic-like spin splitting over the whole Brillouin zone can be achieved in PAFMs, providing strong theoretical support for future multistate information storage and multifunctional spintronic devices based on multiple polarization states and tunable spin polarization.