Band alignment of transition metal dichalcogenide heterostructures

Two-dimensional heterostructures and superlattices consisting of transition metal dichalcogenides offer huge potential in the next generation of optoelectronic devices. For such transition metal dichalcogenides, a predictive theory of their properties, based upon in-depth understanding of the interlayer interactions, is desirable due to the huge potential number of combinations. These weakly interacting heterobilayers provide an excellent case study to understand how interlayer interactions interfere with the Anderson/Schottky picture of band alignment at interfaces. We demonstrate here, using a combination of first principles and tight-binding methods, how the band alignment can be predicted in terms of a modified form of Anderson's rule. We explain how two physically based corrections to Anderson's rule, $\mathrm{\Delta }{E}_{\mathrm{\Gamma }}$ and $\mathrm{\Delta }{E}_{\text{IF}}$, are necessary for accurate band alignment prediction. We identify the interlayer interactions that affect band alignment prediction and show how these take the form of long range interactions between the $d_{z^2}$ and/or $p_z$ orbitals and induced fields between layers. Finally, we apply this theorem to Moiré and strained structures to predict these structures band alignment and provide a comprehensive guide to accurately predict the resultant band gap of the various transition metal dichalcogenide heterostructures.

Figure 1

The overlay band structures of all nine studied heterobilayers. From (a) to (i) they are (a) ${\mathrm{CrS}}_2$/${\mathrm{MoS}}_2$, (b) $\mathrm{W}{\mathrm{Se}}_2$/${\mathrm{MoS}}_2$, (c) ${\mathrm{MoS}}_2$/${\mathrm{WS}}_2$, (d) $\mathrm{W}{\mathrm{Se}}_2$/${\mathrm{WS}}_2$, (e) ${\mathrm{ZrS}}_2$/${\mathrm{HfS}}_2$, (f) $\mathrm{Cr}{\mathrm{Se}}_2$/$\mathrm{W}{\mathrm{Se}}_2$, (g) ${\mathrm{CrS}}_2$/${\mathrm{WS}}_2$, (h) ${\mathrm{MoS}}_2$/${\mathrm{ZrS}}_2$, (i) ${\mathrm{WS}}_2$/${\mathrm{ZrS}}_2$. Each panel consists of the band structure of the heterostructure and the individual constituents and are aligned by reference to the vacuum level.

Figure 2

(a) The electronic dispersion of the heterostructure $\mathrm{W}{\mathrm{Se}}_2/{\mathrm{MoS}}_2$ (black dashed) overlaying the electronic dispersions of the two constituents $\mathrm{W}{\mathrm{Se}}_2$ (red) and ${\mathrm{MoS}}_2$ (blue). (b) Dispersion of the heterostructure ${\mathrm{MoS}}_2/{\mathrm{CrS}}_2$ (black dashed) overlaying the electronic dispersions of the ${\mathrm{MoS}}_2$ (blue) and ${\mathrm{CrS}}_2$ (yellow). (c) Band edge diagram of heterostructures and their constituents showing VBM and CBM. All energies are plotted relative to the vacuum energy and calculated with the HSE06 functional.

Figure 3

The relationship between the valence band offset $({\chi }_{h,B}-{\chi }_{h,A})$ and the resultant shift in the band energies ($\mathrm{\Delta }{E}_{\text{IF}}$). The inset gives the equation and parameters for the empirical fit (blue dashed line).

Figure 4

(Top) The relation between $\mathrm{\Delta }{E}_{\mathrm{\Gamma }}$ and $\mathrm{\Delta }d$. (Bottom) The tabulated fitting parameters that reproduce the simulation data, corresponding to those in Eq. (2) for ${\eta }_0$ (eV) and ${\eta }_1$ (eV/Å). Both ${\mathrm{ZrS}}_2$ and ${\mathrm{HfS}}_2$ are metallic for $d\le 6$ Å and therefore have no ${\eta }_i^{\prime }$ values.

Figure 5

Conduction and valence band edges calculated within PBE for the 2D TMDCs. The colored bar edges represent the valence bands (shades of blue/cyan) and the conduction bands (shades of red/yellow) for each monolayer; all energies are taken with respect to the vacuum level. Due to the band gap underestimate by PBE, the electron affinities are overestimated.

Figure 6

(a),(b) An overlay band structure showing the constituent bands (blue and orange) overlayed by the heterobilayer bands (black dashed) with (a) showing the Moiré structure and (b) showing the one-to-one cell matching for ${\mathrm{MoS}}_2$/${\mathrm{HfS}}_2$. (c),(d) Constituent bands (blue and gold) with heterobilayer bands (black dashed) overlayed. (c) Showing the Moiré structure and (d) showing a one-to-one cell matching for ${\mathrm{CrS}}_2$/${\mathrm{MoS}}_2$.