Coupled exciton internal and center-of-mass motions in two-dimensional semiconductors by a periodic electrostatic potential

We theoretically investigate the coupling between the exciton internal and center-of-mass motions in monolayer transition-metal dichalcogenides subjected to a periodic electrostatic potential. The coupling leads to the emergence of multiple absorption peaks in the exciton spectrum, which are the hybridizations of $1s, 2s$, and $2p_{\pm }$ Rydberg states with different center-of-mass momentums. The energies and wave functions of hybrid states can be strongly modulated by varying the profile of the periodic electrostatic potential, which well reproduces the recent experimental observations. Combined with the electron-hole exchange interaction, nondegenerate valley-coherent bright excitons can be realized by applying an in-plane electric field, with the valley coherence determined by the field direction.

Figure 1

(a) A schematic illustration of the van der Waals heterostructure formed by monolayer TMDs and an adjacent TBG moiré pattern vertically separated by $d$. The periodic charge density in the moiré pattern generates an electrostatic potential in the monolayer TMDs, where the electron and hole feel potentials $U\left({\mathbf{r}}_{\text{e}}\right)$ and $-U\left({\mathbf{r}}_{\mathrm{h}}\right)$, respectively. (b) The 2D map of the charge distribution, with the density maxima located at $\mathbf{AA}$ sites in TBG. The inset shows the superlattice mini-Brillouin zone (black hexagon) and reciprocal lattice vectors $\pm {\mathbf{b}}_n$ (red arrows) and $\pm {\mathbf{b}}_n^{\prime }$ (blue arrows). (c) Dispersions of free exciton Rydberg states $1s, 2s$, and $2p_{\pm }$. $\mathbf{Q}$ is the exciton CoM momentum. Double arrows denote couplings induced by the periodic electrostatic potential. (d) A schematic of the oscillator strength redistribution. The $\mathbf{Q}=0$ bright $1s$ and $2s$ excitons are coupled directly (indirectly) to dark $2p_{\pm }$ ($1s/2s$) excitons with finite CoM momentums, and thus redistribute their oscillator strengths into multiple hybrid states.

Figure 2

(a) The coupling strength $|t_{2s}^{2p}|$ as a function of $\nu$ under $\epsilon =5$ and $\lambda =12$, 23.5, and 35 nm, which corresponds to $1.{18}^{\circ }, 0.6^{\circ }$, and $0.4^{\circ }$ TBG, respectively. (b) The calculated optical absorption spectra as functions of $\nu$, for $\lambda =23.5\mathrm{nm}$. (c) The Rydberg state and CoM momentum compositions for the bright states $❶$ to $❻$ in (b). Different colors denote different magnitudes of the CoM momentum.

Figure 3

(a) The real-space CoM density distributions $\int d\mathbf{r}|{\mathrm{\Phi }}_{n_X,\mathbf{Q}=0}{(\mathbf{R},\mathbf{r})|}^2$ for the bright exciton states $❶$ to $❻$ in Fig. 2. The density distribution of $❹$* is similar to that of $❹$. (b) The real-space wave function for the electron-hole relative motion of state $❹$ when the CoM position $\mathbf{R}$ is fixed at $\mathbf{AB}$. (c) The relative wave function of state $❹$ when $\mathbf{R}$ is fixed at $\mathbf{AA}$.

Figure 4

(a) The calculated optical spectrum under a relatively short wavelength $\lambda =12\mathrm{nm}$. (b) The moiré wavelength dependence of the maximum energy shift $\mathrm{\Delta }{E}_{\text{m}}$ and the critical density ${\nu }_{\text{m}}$. The top horizontal axis indicates the twist angle of TBG for the corresponding $\lambda$ value. (c) The calculated optical spectrum under a large wavelength $\lambda =35\mathrm{nm}$. The dashed cyan curves correspond to the result after turning off the coupling $|\mathbf{Q},1s\rangle \leftrightarrow |{\mathbf{Q}}^{\prime },2p_{\pm }\rangle$, i.e., artificially setting $t_{1s}^{2p}=0$. (d) The Rydberg state and CoM momentum compositions for the four bright states $❶$ to $❹$ in (c).

Figure 5

(a) The diagram for the intervalley electron-hole exchange interaction. (b) The coupling between ${\widehat{C}}_3$- and inversion-symmetric basis states in $\pm \mathbf{K}$ valleys induced by ${\widehat{U}}_X, {\widehat{H}}_{\text{ex}}$, and $\mathbf{E}$. ${\widehat{U}}_X$ is the periodic electrostatic potential from the adjacent TBG, ${\widehat{H}}_{\text{ex}}$ is the electron-hole exchange interaction, and $\mathbf{E}$ is an externally applied in-plane electric field. $|0,ns\rangle$ in the $\pm \mathbf{K}$ valley can emit a ${\sigma }_{\pm }$ circularly polarized photon. (c) The simplified three-level model. Red (blue) denotes the state in the $+\mathbf{K}$ ($-\mathbf{K}$) valley, whereas purple denotes a superposition of two valleys. (d) The dependence of emission linear polarizations for the two split valley-coherent excitons with the direction of $\mathbf{E}=(E_x,E_y)$.

Figure 6

(a)–(e) Theoretically calculated absorption spectra of excitons under various parameter selections, whose values are summarized in Table 2. (f) The experimental reflectance contrast spectrum of a TMDs/TBG sample with $\lambda =23.5\mathrm{nm}$. Details about the experiment can be found in Ref. [40].