Robust altermagnetism and compensated ferrimagnetism in MnPX$_3$-based (X = S or Se) heterostructures

The recent research interests in the non-relativistic spin splitting of electronic band structures have led to the exploration of altermagnets and other compensated magnets. Here, we show that various types of non-relativistic spin splitting can be robustly induced by constructing Van der Waals heterostructures consisting of materials with intra-plane anti-ferromagnetic orders and suitable substrates. Using MnPX$_3$ (X = 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 MnPX$_3$/MPX$_3$ (M = Cd, Mg, or Zn) heterostructures. For the AB-stacking heterostructures that are semiconducting, ferrimagnetic-type spin splitting emerges, and the fully compensated magnetization is protected by the Luttinger theorem. By combining with a Van der Waals ferroelectric substrate like CuInP$_2$S$_6$, MnPX$_3$-based heterostructures can show tunable spin splitting and spin-related properties that depend on the electronic band structures and ferroelectric polarization, which can be non-volatilely reversed by applying an out-of-plane electric field. Our study provides a route to induce tunable non-relativistic spin splitting in experimentally synthesizable two-dimensional magnets.