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

TL;DR

This work addresses why MoS$_2$ transitions from a direct-bandgap semiconductor in the monolayer to an indirect-bandgap material in multilayer form. By combining first-principles calculations with Wannier-based tight-binding modeling, it constructs an interlayer Hamiltonian from bulk parameters and analyzes orbital contributions across the Brillouin zone. The key finding is that S $p_z$ orbitals dominate interlayer coupling and band splitting, particularly at $\Gamma$ and $Q$, while in-plane S $p_x$/$p_y$ orbitals contribute through hybridization, affecting the detailed evolution of the valence and conduction band edges. The results provide a microscopic mechanism for the layer-dependent band-gap evolution and offer guidance for orbital- and interlayer-controlled band engineering in MoS$_2$-based devices.

Abstract

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. To clarify the microscopic mechanism behind this transition, we performed first-principles calculations combined with Wannier-based modeling, focusing on the roles of atomic orbitals and interlayer interactions. While orbitals oriented perpendicular to the plane -- such as Mo-$d_{z^2}$ and S-$p_z$ -- have been considered the primary contributors, our analysis reveals that in-plane $p_x$ and $p_y$ orbitals of S atoms also play a significant role. 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.

Figures (6)

• Figure 1: Crystal structure of 2H-MoS$_2$: (a) unit cell and (b) side view of layer structure. This figure are drawn by VESTAVESTA.
• Figure 2: Density of states of bulk MoS$_2$
• Figure 3: Comparison of band structure by DFT(red solid lines) and TB model(green broken lines) : (a) bulk and (b) monolayer MoS$_2$. The arrows indicate the band-gap transitions. (c) Brillouin Zone of MoS$_2$
• Figure 4: Band structure for monolayer (green), bilayer (red), and 6-layers (gray) MoS$_2$.
• Figure 5: (a)(b) The Mo $d$ orbital character of the MoS$_2$ band structure for (a)monolayer and (b)6-layer. The orbital weights are represented as follows:$d_{z^2}$ (red), $d_{xz}$ and $d_{yz}$ (green), and $d_{xy}$ and $d_{x^2-y^2}$ (blue). (c)(d) The S $p$ orbital character of the MoS$_2$ band structure for (c)monolayer and (d)6-layer. The orbital weights are represented as follows:$p_{z}$ (red), $p_{x}$ (green), and $p_{y}$ (blue).