Wannier based analysis of the direct-indirect bandgap transition by stacking MoS$_2$ layers

Molybdenum disulfide (MoS$_2$), a layered van der Waals material, has attracted considerable attention as a promising alternative to graphene for applications in field-effect transistors and nanophotonic devices because of its sizable band gap, high carrier mobility, large on/off ratio, and strong photoluminescence efficiency. A particularly intriguing property of MoS$_2$ is the transition of its band gap character with layer thickness: while the monolayer exhibits a direct gap, the band gap becomes indirect in multilayer and bulk forms.In this study, we clarify the microscopic mechanism underlying this transition. Focusing on the roles of atomic orbitals and interlayer interactions, we perform an analysis combining first-principles calculations with a Wannier-based model. Although interlayer $p_z$--$p_z$ coupling between neighboring sulfur atoms has been recognized as a key factor in this transition, we find that a complete quantitative description additionally requires interlayer $p_z$--$p_x$ and $p_z$--$p_y$ couplings between neighboring sulfur atoms. These findings highlight the importance of both out-of-plane and in-plane orbital contributions in governing the electronic structure of layered MoS$_2$, providing deeper insight into its band gap engineering for future device applications.