Interlayer excitons in bilayer ${\mathrm{MoS}}_2$ with strong oscillator strength up to room temperature

Coulomb bound electron-hole pairs, excitons, govern the optical properties of semiconducting transition-metal dichalcogenides like ${\mathrm{MoS}}_2$ and ${\mathrm{WSe}}_2$. We study optical transitions at the $K$ point for $2H$ homobilayer ${\mathrm{MoS}}_2$ in density functional theory including excitonic effects and compare them with reflectivity measurements in high-quality samples encapsulated in hexagonal BN. In both calculated and measured spectra we find a strong interlayer exciton transition in energy between A and B intralayer excitons, observable for $T=4–300$ K, whereas no such transition is observed for the monolayer in the same structure in this energy range. The interlayer excitons consist of an electron localized in one layer and a hole state delocalized over the bilayer, which results in the unusual combination of high oscillator strength and a static dipole moment. We also find signatures of interlayer excitons involving the second-highest valence band (B) and compare absorption calculations for different bilayer stackings. For homotrilayer ${\mathrm{MoS}}_2$ we also observe interlayer excitons and an energy splitting between different intralayer A excitons originating from the middle and outer layers, respectively.

Figure 1

Schematics of intralayer and interlayer excitons in ${\mathrm{MoS}}_2$ homobilayers. In (a) intralayer excitons consist of an electron (red) and a hole (blue) in the same layer, while in (b) an electron localized on one layer interacts with a hybridized hole state to form an interlayer exciton. Optical selection rules, represented by wavy lines, for intralayer and interlayer transitions in $k$ space are also given for $K$ points, respecting spin conservation. For clarity only one interlayer exciton (spin up) is shown; there is also a spin-down state with the same energy.

Figure 2

Calculated absorption spectra for a single mono-, bi-, and trilayer. Interlayer exciton transitions are marked by green circles. Orange arrows for the TL case indicate intralayer transitions involving only the middle layer (${\mathrm{L}}_2$) of the three layers, while the maroon ones stress the ${\mathrm{L}}_1$ and ${\mathrm{L}}_3$ intralayer transitions. See the text and Appendix pp1 for computational details.

Figure 3

Calculated absorption spectra for a bilayer, using three different stackings. The green lines mark interlayer transitions for ${\mathrm{AA}}^{\prime }$ stackings.

Figure 4

Monolayer (ML), bilayer (BL), and trilayer (TL) ${\mathrm{MoS}}_2$ encapsulated in hBN. (a) Optical microscope image of the hBN/${\mathrm{MoS}}_2$/hBN heterostructure. (b) AFM measurements confirm atomic steps. (c) Differential reflectivity of the three different thicknesses (ML, BL, and TL) at a sample temperature of $T=4$ K; spectra are offset for clarity.

Figure 5

Optical spectroscopy results: Temperature-dependent differential reflectivity spectra for (a) monolayer, (b) bilayer, and (c) trilayer ${\mathrm{MoS}}_2$ encapsulated in hBN.

Figure 6

Quasiparticle band structures at the $G_3{W}_0$ level in (a) ${\mathrm{AA}}^{\prime }$ BL, (b) AB BL, and (c) ${\mathrm{AA}}^{\prime }\mathrm{A}$ TL stackings.