Interlayer Magnetic Coupling in FePS$_{3}$ and NiPS$_{3}$ Stacked Bilayers

Single layers of transition-metal thiophosphates (2D-TMPS$_{3}$) van der Waals magnets are an ideal platform for studying antiferromagnetic interactions in two dimensions. However, the magnetic coupling mechanism between two or more individual layers of these materials remains mostly unexplored. This study presents a density-functional based analysis and analytical models to describe the magnetic configurations of FePS$_{3}$ and NiPS$_{3}$ stacked bilayers. We explore the interplay between magnetic configurations and stacking shift, therefore identifying the mechanisms that result in either ferromagnetic or antiferromagnetic coupling between layers. Our findings indicate that the stacking with the lowest energy is metal-dependent, and the interlayer magnetic configuration (ferromagnetic or antiferromagnetic) varies based on the stacking type and the metal involved. Using an Ising-Hamiltonian model and a tight-binding model based on Wannier functions, we show that interlayer exchange interactions must be considered up to the third nearest neighbor and to elucidate the superexchange mechanism for the NiPS$_{3}$ system.