Dielectric properties of graphene/${\mathrm{MoS}}_2$ heterostructures from ab initio calculations and electron energy-loss experiments

High-energy electronic excitations of graphene and ${\mathrm{MoS}}_2$ heterostructures are investigated by momentum-resolved electron energy-loss spectroscopy in the range of 1 to 35 eV. The interplay of excitations on different sheets is understood in terms of long-range Coulomb interactions and is simulated using a combination of ab initio and dielectric model calculations. In particular, the layered electron-gas model is extended to thick layers by including the spatial dependence of the dielectric response in the direction perpendicular to the sheets. We apply this model to the case of graphene/${\mathrm{MoS}}_2$/graphene heterostructures and discuss the possibility of extracting the dielectric properties of an encapsulated monolayer from measurements of the entire stack.

Figure 1

Momentum-resolved energy-loss experiments on (a) $1H-{\mathrm{MoS}}_2$ monolayers and (b) G/${\mathrm{MoS}}_2$/G heterostructures for momentum transfers $\overline{q}=0$, 0.3, and $0.6{\AA }^{-1}$ along the $\mathrm{\Gamma }M$ direction of ${\mathrm{MoS}}_2$. The measured energy-loss function (ELF; gray filled curve) is compared to corresponding ab initio RPA calculations of $1H-{\mathrm{MoS}}_2$ (red solid line) and graphene (black dashed line). Blue dotted curves indicate the background originating from unscattered electrons.

Figure 2

Sketch of different model systems for a three-layer heterostructure (sandwich). (a) Fully microscopic description of the dielectric response including lattice mismatch within the building-block approach (BBA). (b) In-plane homogeneous (IPH) model with a $z$-dependent response for each layer. (c) Stack of strictly two-dimensional layers in the layered electron-gas (LEG) model.

Figure 3

Validation of different model calculations for EEL spectra of bulk ${\mathrm{MoS}}_2$ at in-plane momentum transfer $\overline{q}=0.3{\AA }^{-1}$ along the $\mathrm{\Gamma }M$ direction: Comparison between BBA, IPH, and LEG models (green solid, blue dash-dotted, and red dashed lines, respectively). As a reference, a full TDDFT calculation (RPA; black solid line) and the measured EEL spectrum (gray filled curve) are shown. Additionally, the RPA spectrum for a single ${\mathrm{MoS}}_2$ monolayer is shown (red dotted line). The inset shows the calculated position of the $\pi +\sigma$ peak in bulk ${\mathrm{MoS}}_2$ with an artificially increased layer distance $d$. The result of the rigorous RPA calculation for bulk ${\mathrm{MoS}}_2$ (black cross) can be reproduced by model calculations only if the $z$ dependence of the monolayers is considered (IPH; blue dash-dotted line).

Figure 4

Variation of the interlayer distance $d$ in the G/${\mathrm{MoS}}_2$/G sandwich structure. Experimental EEL spectra (gray filled curve) for a momentum transfer of $\overline{q}=0.3{\AA }^{-1}$ are compared to IPH model calculations for $d=3.35\AA , d=4.0\AA$, and $d\to \infty$.

Figure 5

Comparison of experimental EEL spectra (gray filled curve) of the G/${\mathrm{MoS}}_2$/G sandwich structure with corresponding IPH and LEG model calculations for a momentum transfer of $\overline{q}=0.3{\AA }^{-1}$ and an interlayer distance $d=4.95\AA$ (see text).