Robust altermagnetism and compensated ferrimagnetism in MnPX 3 -based ( X = S or Se) heterostructures

The discovery of altermagnets has led to the study of nonrelativistic spin splitting in materials with compensated magnetic moments. Despite the experimental validation of altermagnets in many bulk materials, two-dimensional altermagnets or compensated ferrimagnets are rare. Here, we show that nonrelativistic spin splittings can be robustly induced by constructing van der Waals heterostructures consisting of intraplane antiferromagnetic materials and suitable substrates. Using $\mathrm{MnP}{\mathrm{X}}_{3}$ ($\mathrm{X}=\mathrm{S}$ or Se) as an example, which has a N\'eel magnetic order, we demonstrate that altermagnetic spin splitting can arise in the AA-stacking $\mathrm{MnP}{\mathrm{X}}_{3}/\mathrm{MP}{\mathrm{X}}_{3}$ ($\mathrm{M}=\mathrm{Cd}$, Mg, or Zn) heterostructures. For the AB-stacking heterostructures, ferrimagnetic-type spin splitting emerges, and the fully compensated magnetization is guaranteed by the Luttinger theorem when the system has a band gap. By combining with a van der Waals ferroelectric substrate like $\mathrm{CuIn}{\mathrm{P}}_{2}{\mathrm{S}}_{6}, \mathrm{MnP}{\mathrm{X}}_{3}$-based heterostructures can show tunable spin splitting and spin-related properties that depend on the electronic band structures and ferroelectric polarization, which can be nonvolatilely reversed by applying an out-of-plane electric field. Our study provides a route to induce tunable nonrelativistic spin splitting in experimentally synthesizable two-dimensional magnets.