Interlayer multi-level orbital coupling in 2D materials: Beyond the two-level paradigm

The interlayer orbital interaction (IOI) of two-dimensional (2D) materials and their heterostructures triggers diverse property modifications, driving advancements in interlayer engineering. Previous investigations into IOI have primarily relied on a two-level interlayer interaction framework (one energy level per layer), which is insufficient for fully capturing band edge evolutions-even in prototypical 2D materials transitioning from monolayers to multilayers. The underlying reason lies in the multi-level nature of orbitals: taking the MoS2 monolayer as a paradigm, the pz orbitals of the two sulfur atoms (which dominate interlayer IOI) contribute to the wavefunctions of three energy levels (rather than one), enabled by pz-dz2-pz coupling along the S-Mo-S chemical bonds. Consequently, these three energy levels within a single layer can interact with a target energy level of interest (e.g., a band edge) in adjacent layers, provided they share the same orbital character. This gives rise to the "n-act-on-one" IOI mechanism, the core of multi-level interlayer interaction. Notably, this multi-level characteristic is inherent to general 2D materials. To address this, we extend the interlayer interaction model to a multi-level framework. This multi-level perspective offers deeper insights into the properties of 2D materials and helps property tuning from a perspective of combining intra- and interlayer orbital interactions.