Topological Exciton Bands in Moiré Heterojunctions

Moiré patterns are common in van der Waals heterostructures and can be used to apply periodic potentials to elementary excitations. We show that the optical absorption spectrum of transition metal dichalcogenide bilayers is profoundly altered by long period moiré patterns that introduce twist-angle dependent satellite excitonic peaks. Topological exciton bands with nonzero Chern numbers that support chiral excitonic edge states can be engineered by combining three ingredients: (i) the valley Berry phase induced by electron-hole exchange interactions, (ii) the moiré potential, and (iii) the valley Zeeman field.

Figure 1

(a) Illustration of $AA$ stacking with a small twist angle $\theta$ and an in-plane displacement $\mathit{d}$. (b) Variation of ${\mathrm{MoS}}_2$ band gap as a function of $\mathit{d}$ in $AA$ stacked ${\mathrm{MoS}}_2/{\mathrm{WS}}_2$ bilayer with zero twist angle. (c) Variation of ${\mathrm{MoS}}_2$ band gap as a function of position in $AA$ stacked twisted ${\mathrm{MoS}}_2/{\mathrm{WS}}_2$. (d)–(f) Corresponding plots for $AB$ stacking. (g) First-shell reciprocal lattice vectors ${\mathit{G}}_j$ of a monolayer TMD triangular lattice and the corresponding Brillouin zone (red hexagon). (h) Moiré reciprocal lattice vectors and corresponding Brillouin zone

Figure 2

(a) Exciton moiré bands along paths connecting high symmetry points in the MBZ. The twist angle $\theta$ used for this calculation is 1°. The sizes of the dots at the $\gamma$ point correspond to the weight of state $|K_{\pm }^{(0)}⟩$. The color of the dots encodes the main character of the states, with blue, red, green, and yellow corresponding to character (1), (2), (3), and (4). See text. (b) Optical conductivity at several twist angles $\theta$. Curves for different $\theta$ are shifted vertically for clarity and the dashed curves track the peak evolution. The two minipeaks in (4) are amplified 10 times. The broadening factor $\eta$ is taken to be 0.5 meV. The exciton potential parameters of $AA$ stacked ${\mathrm{MoS}}_2/{\mathrm{WS}}_2$ were used in these calculations.

Figure 3

(a) Topological exciton bands with quantized Chern numbers $\mathcal{C}$ for twist angle $\theta =1°$. The gray bar identifies the gap between the first and second exciton moiré band. (b) Stripe geometry quasi-1D bands for the same twist angle, an extended edge along the $x$ direction, and finite width in the $y$ direction. The red and green lines show the dispersions of chiral exciton states that are localized on opposite edges of the stripe. In (a) and (b), $h_z$ is 1.5 meV, and ($V$, $\psi$) take the parameter values of $AB$ stacked ${\mathrm{MoS}}_2/{\mathrm{WS}}_2$.

Figure 4

Spatially resolved optical conductivity on the same stripe geometry as studied in Fig. 3. The optical conductivity is averaged over $x$. (a) Response from bulk states. The energy splitting between the two peaks is due to the Zeeman energy. An energy broadening factor of 1 meV is used. (b) Response from both bulk and edge states. There is an enhancement of the optical response around the edge due to edge states. The arrows indicate the energy level of bulk states (gray arrow) and edge state (red arrow) in the absence of energy broadening, and correspond to the three arrows shown in Fig. 3.