Modulated interlayer exciton properties in a two-dimensional moiré crystal

Twisted van der Waals heterostructures and the corresponding superlattices, moiré superlattices, are remarkable material platforms, in which electron interactions and excited-state properties can be engineered. Particularly, the band offsets between adjacent layers can separate excited electrons and holes, forming interlayer excitons that exhibit unique optical properties. In this work, we employ the first-principles GW-Bethe-Salpeter equation method to calculate quasiparticle band gaps, interlayer excitons, and their modulated excited-state properties in twisted $\mathrm{MoS}{\mathrm{e}}_2/\mathrm{WS}{\mathrm{e}}_2$ bilayers that are of broad interest currently. In addition to achieving good agreement with the measured interlayer exciton energies, we predict a more than 100-meV lateral quantum confinement on quasiparticle energies and interlayer exciton energies, guiding the effort on searching for localized quantum emitters and simulating the Hubbard model in two-dimensional twisted structures. Moreover, we find that the optical dipole oscillator strength and radiative lifetime of interlayer excitons are modulated by a few orders of magnitude across moiré supercells, highlighting the potential of using moiré crystals to engineer exciton properties for optoelectronic applications.

Figure 1

The schematic plots of two main types of twisting $\mathrm{MoS}{\mathrm{e}}_2/\mathrm{WS}{\mathrm{e}}_2$ bilayer heterostructures. (a) is the H-type twisting structure rotated from the AA′ stacking style, and (b) is the R-type twisting structure rotated from the AA stacking style. Six local stacking styles are identified and amplified with top and side views.

Figure 2

(a) Quasiparticle band structure of the $H_h^h$ twisted $\mathrm{MoS}{\mathrm{e}}_2/\mathrm{WS}{\mathrm{e}}_2$ bilayer. The energy of the top of valence band is set to be zero. The projected components of electronic states to those of $\mathrm{MoS}{\mathrm{e}}_2$ and $\mathrm{WS}{\mathrm{e}}_2$ are represented by different colors, respectively. (b1)–(b3 and (c1)–(c3) Optical absorption spectra [with (red) and without e-h (blue) interactions] of six identified local stacking styles in a $\mathrm{MoS}{\mathrm{e}}_2/\mathrm{WS}{\mathrm{e}}_2$ heterostructure. The energy of the interlayer exciton ($I{X}_0$) is marked by the black arrow. SOC is not included. A 13-meV smearing to spectral widths is applied.

Figure 3

(a) and (b) are optical oscillator strength of interlayer excitons at the six local sites with labeled stacking styles displayed on the logarithmic scale, respectively. (c) and (d) are the interpolated dipole oscillator strength of the H-type and R-type twisted $\mathrm{MoS}{\mathrm{e}}_2/\mathrm{WS}{\mathrm{e}}_2$ bilayers, respectively. The valley-dependent selection rules of interlayer excitons are marked in (a) and (b) according to Ref. [6].

Figure 4

Interlayer exciton wave functions of the $H_h^h$ and $H_h^M$ stacking styles. (a1) and (b1) are top views of the exciton wave functions. (a2) is the side view of the bright interlayer exciton. The electron (the cross) is fixed in the upper ($\mathrm{MoS}{\mathrm{e}}_2$) layer, and the hole wave function is plotted. The side panel is the integrated hole wave function. (a3) is the same exciton of (a2), but we fix the hole (the triangle) in the lower ($\mathrm{WS}{\mathrm{e}}_2$) layer and plot the electron wave function. The side panel is the integrated electron wave function. (b2) and (b3) are the similar plots for the dark interlayer exciton at the $H_h^M$ stacking style.

Figure 5

(a) Quasiparticle band structure of the $H_h^h$ twisted bilayer $\mathrm{MoS}{\mathrm{e}}_2/\mathrm{WS}{\mathrm{e}}_2$ heterostructure with SOC included. The energy of the top of valence band is set to be zero. The projected components to the $\mathrm{MoS}{\mathrm{e}}_2$ and $\mathrm{WS}{\mathrm{e}}_2$ are represented by different colors. (b) Schematic band alignment of the CBM and VBM for unstrained monolayer ($\mathrm{MoS}{\mathrm{e}}_2$ and $\mathrm{WS}{\mathrm{e}}_2$) and strained ones to match the lattice constant of heterostructures. The absolute band energies are DFT results.

Figure 6

(a) and (b) are energies of interlayer excitons at the six local sites H-type and R-type twisted $\mathrm{MoS}{\mathrm{e}}_2/\mathrm{WS}{\mathrm{e}}_2$ bilayers, respectively. (c) and (d) are the interpolated interlayer exciton energy of the H-type and R-type twisted bilayers, respectively. The lattice mismatch and SOC are included.