Theory of optical absorption by interlayer excitons in transition metal dichalcogenide heterobilayers

We present a theory of optical absorption by interlayer excitons in a heterobilayer formed from transition metal dichalcogenides. The theory accounts for the presence of small relative rotations that produce a momentum shift between electron and hole bands located in different layers, and a moiré pattern in real space. Because of the momentum shift, the optically active interlayer excitons are located at the moiré Brillouin zone's corners, instead of at its center, and would have elliptical optical selection rules if the individual layers were translationally invariant. We show that the exciton moiré potential energy restores circular optical selection rules by coupling excitons with different center of mass momenta. A variety of interlayer excitons with both senses of circular optical activity, and energies that are tunable by twist angle, are present at each valley. The lowest energy exciton states are generally localized near the exciton potential energy minima. We discuss the possibility of using the moiré pattern to achieve scalable two-dimensional arrays of nearly identical quantum dots.

Figure 1

(a) Illustration of AA stacking with a small twist angle $\theta$ and an in-plane displacement ${\mathit{r}}_0$. The blue and red hexagons mark the Wigner-Seitz cells of the two layers. (b) The corresponding illustration of AB stacking. AA and AB stacking are distinguished by a rotation of the top layer $\left(\mathrm{Mo}{X}_2\right)$ by $\pi$ around the metal (Mo) axis. (c) Unit cell of the bilayer when $\theta =0$. The vectors ${\mathit{r}}_{0,n}$ in this figure are high symmetry points as discussed in the main text. (d) Brillouin zone and the first shell reciprocal lattice vectors for the bilayer with $\theta =0$. (e) Brillouin zones associated with the top and bottom layers in a twisted bilayer.

Figure 2

(a) $\mathit{K}$-valley interlayer exciton optical matrix elements for heterobilayers with zero twist angle at special high symmetry displacements. (b) In-plane displacement dependence of the ab initio values of $|\mathit{D}\left({\mathit{r}}_0\right)·{\mathit{e}}_{+}^{*}|$ (top) and $|\mathit{D}\left({\mathit{r}}_0\right)·{\mathit{e}}_{-}^{*}|$ (bottom) for aligned AA stacking of ${\mathrm{WS}}_2/{\mathrm{MoS}}_2$ with zero twist angle. Values are given in units of the corresponding intralayer optical response parameter $\mathcal{D}$. $|\mathit{D}\left({\mathit{r}}_0\right)·{\mathit{e}}_{-}^{*}|$ at ${\mathit{r}}_{0,1}$ is small but nonzero. (c) Same plot as (b) but for AB stacking. (d) Leading harmonic approximation for $|\mathit{D}\left({\mathit{r}}_0\right)·{\mathit{e}}_{\pm }^{*}|$ explained in the text.

Figure 3

(a) The small red and blue dots represent Bravais lattices of the two layers with a relative rotation. The dashed lines mark the moiré unit cell. The big dots indicate regions where the local displacement between the two layers is ${\mathit{r}}_{0,n}$. Each ${\mathit{r}}_{0,1}$ region has three ${\mathit{r}}_{0,2}$ neighbors and each ${\mathit{r}}_{0,2}$ region has three ${\mathit{r}}_{0,1}$ neighbors. (b) Moiré Brillouin zone and the first-shell moiré reciprocal lattice vectors.

Figure 4

Results for a ${\mathrm{WS}}_2/{\mathrm{MoS}}_2$ heterobilayer with AA stacking. (a) Variation of the interlayer band gap as a function of ${\mathit{r}}_0$ when the twist angle is zero. (b) Variation of the band gap in real space when the twist angle is finite. (c) Exciton moiré band structure for $\theta =1^{\circ }$. (d) Optical spectrum of $\mathit{K}$-valley interlayer excitons for a series of twist angles. The red and blue curves respectively show $\text{Re}{\sigma }_{\mathit{K},+}/{\mathrm{\Lambda }}_{+}$ and $\text{Re}{\sigma }_{\mathit{K},-}/{\mathrm{\Lambda }}_{-}$. ${\mathrm{\Lambda }}_{+}/{\mathrm{\Lambda }}_{-}$ is about 18. The broadening parameter $\eta$ is taken to be 0.5 meV. Curves for different $\theta$ are shifted vertically. (e) The real-space probability function ${\left|\chi \left(\mathit{r}\right)\right|}^2$ of the interlayer excitons responsible for the first three absorption peaks at $\theta =1^{\circ }$.

Figure 5

Same plots as those in Fig. 4 but for the ${\mathrm{WS}}_2/{\mathrm{MoS}}_2$ heterobilayer in AB stacking. In (d), ${\mathrm{\Lambda }}_{+}/{\mathrm{\Lambda }}_{-}$ is about 6.

Figure 6

(a) Exciton potential with $\psi =2\pi /3$. $A$ and $B$ are potential minimum positions that form a honeycomb lattice. (b)Exciton band structure when $\psi =2\pi /3$. The red circles highlight the Dirac cones. (c) Optical spectrum for $\psi$ below, at, and above $2\pi /3$. The red and blue curves, respectively, show $\text{Re}{\sigma }_{\mathit{K},+}/{\mathrm{\Lambda }}_{+}$ and $\text{Re}{\sigma }_{\mathit{K},-}/{\mathrm{\Lambda }}_{-}$. Curves for different $\psi$ are shifted vertically. In (b) and (c), the twist angle is $1^{\circ }$. Other parameters take their values in AB stacked ${\mathrm{WS}}_2/{\mathrm{MoS}}_2$ bilayers.