Electronic structures and optical properties of realistic transition metal dichalcogenide heterostructures from first principles

We calculate from first principles the electronic structure and optical properties of a number of transition metal dichalcogenide (TMD) bilayer heterostructures consisting of MoS${}_2$ layers sandwiched with WS${}_2$, MoSe${}_2$, MoTe${}_2$, BN, or graphene sheets. Contrary to previous works, the systems are constructed in such a way that the unstrained lattice constants of the constituent incommensurate monolayers are retained. We find strong interaction between the $\Gamma$-point states in all TMD/TMD heterostructures, which can lead to an indirect gap. On the other hand, states near the $K$ point remain as in the monolayers. When TMDs are paired with BN or graphene layers, the interaction around the $\Gamma$-point is negligible, and the electronic structure resembles that of two independent monolayers. Calculations of optical properties of the MoS${}_2$/WS${}_2$ system show that, even when the valence- and conduction-band edges are located in different layers, the mixing of optical transitions is minimal, and the optical characteristics of the monolayers are largely retained in these heterostructures. The intensity of interlayer transitions is found to be negligibly small, a discouraging result for engineering the optical gap of TMDs by heterostructuring.

Figure 1

(a) Top view of MoS${}_2$/MoSe${}_2$ bilayer heterostructure as modeled in our supercell approach. The construction of the supercell basis vectors ($n_i{a}_i+m_i{b}_i$) is also illustrated. (b) Side view of the MoS${}_2$/WS${}_2$ heterostructure showing the adopted stacking similar to the 2H polytype of MoS${}_2$. The definition for layer distance $d$ is indicated. (c) Schematic demonstration of the overlap of the primitive-cell Brillouin zones from the MoS${}_2$/MoSe${}_2$ system.

Figure 2

(a) Band structure of MoS${}_2$/WS${}_2$ heterostructure. Projection to MoS${}_2$ layer is denoted by red (circles) and to WS${}_2$ by blue (crosses). The spin orientations of the wave functions at the $K/K^{\prime }$ point are also denoted. (b) The energies for the band-edge states as a function of the interlayer distance $d$ [cf. Fig. 1b]. The vertical dotted lines denote distances calculated with various functionals (the two VV10-type functionals give nearly identical results) and experimental value evaluated from the average of bulk MoS${}_2$ and WS${}_2$.

Figure 3

Partial charge density isosurfaces (blue) at $0.37$ (transparent) and $1.2$ (solid) $e$/nm${}^3$ for selected wave functions of monolayer MoS${}_2$.

Figure 4

The optical absorption spectrum of MoS${}_2$/WS${}_2$ heterostructure (middle) together with the spectra from monolayer MoS${}_2$ (top) and WS${}_2$ (bottom). The vertical lines denote the actual calculated transition energies and intensities. Absorption spectra are evaluated through application of 0.02-eV Lorentzian broadening. Overlaid with the explicitly calculated heterostructure spectrum, the sum of the monolayer spectra is also plotted (dashed line). Interlayer transitions are present but have negligible intensities.

Figure 5

Band structures of (a) MoS${}_2$/MoSe${}_2$ and (b) MoS${}_2$/MoTe${}_2$ heterostructures. Circles (red) denote projections to MoS${}_2$ layer (opacity) and to the corresponding reciprocal space directions (size of the marker). Crosses (blue) denote similarly projections to (a) MoSe${}_2$ and (b) MoTe${}_2$ layers.

Figure 6

Band structures of (a) MoS${}_2$/BN and (b) MoS${}_2$/graphene heterostructures. Notation as in Fig. 5. For each layer, the band structure is plotted up to its first Brillouin zone boundary. Due to different lattice constants, this boundary is located at different wave vectors.

Figure 7

Diagrams illustrating the interaction of band-edge states around $\Gamma$ and $K$ points as the heterostructure is constructed from the respective monolayers. The energies are given with respect to the vacuum level.