An electron-counting rule to determine the interlayer magnetic coupling of the van der Waals materials

In layered magnetic materials, the magnetic coupling between neighboring van der Waals layers is challenging to understand and anticipate, although the exchange interaction inside a layer can be well rationalized for example by the superexchange mechanism. In this work, we elucidate the interlayer exchange mechanism and propose an electron-counting rule to determine the interlayer magnetic order between van der Waals layers, based on counting the $d$-orbital occupation ($d^n$, where $n$ is the number of $d$-electrons at the magnetic cation). With this rule, we classify magnetic monolayers into two groups, type-I ($n<5$) and type-II ($n\geq5$), and derive three types of interlayer magnetic coupling for both insulators and metals. The coupling between two type-II layers prefers the antiferromagnetic (AFM) order, while type-I and type-II interfaces favor the ferromagnetic (FM) way. However, for two type-I layers, they display competition between FM and AFM orders and even lead to the stacking dependent magnetism. Additionally, metallic layers can also be incorporated into this rule with a minor correction from the free carrier hopping. Therefore, this rule provides simple guidance to understand the interlayer exchange and further design van der Waals junctions with desired magnetic orders.