Exciton transport in a moiré potential: From hopping to dispersive regime

The propagation of excitons in transition metal dichalcogenide (TMD) monolayers has been intensively studied revealing interesting many-particle effects, such as halo formation and nonclassical diffusion. Initial studies have investigated how exciton transport changes in twisted TMD bilayers, including Coulomb repulsion and Hubbard-like exciton hopping. In this work, we investigate the twist-angle dependent transition of the hopping regime to the dispersive regime of effectively free excitons. Based on a microscopic approach for excitons in the presence of a moiré potential, we show that the hopping regime occurs up to an angle of approximately $2^{\circ }$ and is well described by a moiré intercell tunneling model. At large angles, however, this nearest-neighbor approximation fails due to increasingly delocalized exciton states. Here, the quantum-mechanical dispersion of free particles with an effective mass determines the propagation of excitons. Overall, our work provides microscopic insights into the character of exciton propagation in twisted van der Waals heterostructures.

Figure 1

Schematic illustration of exciton transport in a moiré potential created by a twisted heterostructure. For different twist angles $\vartheta$ we encounter different transport regimes. (a) At small angles, we observe a hopping-driven transport while (b) at large twist angles we reach a dispersive regime.

Figure 2

Exciton propagation in a twisted ${\text{MoSe}}_2/{\text{WSe}}_2$ heterostructure. (a),(b) The initial exciton density distribution at $2^{\circ }$ and $5^{\circ }$, respectively. The contour lines represent the moiré potential. (c),(d) Exciton distribution after 2.4 ps. For smaller twist angles excitons are strongly localized leading to a suppressed propagation compared to larger angles, where the Wannier orbitals have a larger overlap.

Figure 3

(a) Hopping term over the twist angle $\vartheta$ for nearest-neighbor (NN) and next-neighbor (2NN) interaction. We depict the hopping term for the ground and first excited exciton states, i.e., $\nu =0,1$. Due to the more pronounced overlap between the Wannier wave function [insets in (a)] the nearest-neighbor interaction is the dominant term for the hopping interaction. (b),(c) Moiré exciton subbands for interlayer excitons in the twisted ${\text{MoSe}}_2/{\text{WSe}}_2$ heterostructure at $\vartheta =3^{\circ }$ and $\vartheta =1^{\circ }$, respectively. (d) Time evolution of the dispersion length for the exact solution (blue) in comparison with the tunneling model (red) for two different twist angles. For the lower angle of $\vartheta =1.0^{\circ }$, the length is the same, while already for $\vartheta =2^{\circ }$ we find clear differences.

Figure 4

Dispersion parameter $\alpha$ [Eq. (8)] as a function of the twist angle $\vartheta$ showing a direct comparison of the exact numerical solution for the propagation of excitons in a moiré potential (red line) with the moiré tunneling model and the free-exciton solutions. The exact solution converges to the free solution (dashed line) for $\vartheta >4^{\circ }$ and describes the dispersive regime that is characterized by a delocalized exciton distribution (inset at $7^{\circ }$). The exact solution goes over to the tunneling solution for $\vartheta >1.8^{\circ }$ that is characterized by a localized exciton distribution (inset at $1^{\circ }$).

Figure 5

Effective exciton mass as a function of the twist angle $\vartheta$. For larger angles, we find that the effective mass corresponds to the free exciton mass. The insets illustrate that the parabolic approximation (dashed line) fails for small twist angles (here $1^{\circ }$), while it is a perfect assumption at larger angles (here $7^{\circ }$).