Chiral Excitonics in Monolayer Semiconductors on Patterned Dielectrics

Monolayer transition metal dichalcogenides feature tightly bound bright excitons at the degenerate valleys, where electron-hole Coulomb exchange interaction strongly couples the valley pseudospin to the momentum of the exciton. Placed on a periodically structured dielectric substrate, the spatial modulation of the Coulomb interaction leads to the formation of exciton Bloch states with real-space valley pseudospin texture displayed in a mesoscopic supercell. We find this spatial valley texture in the exciton Bloch function is pattern locked to the propagation direction, enabling nano-optical excitation of directional exciton flow through the valley selection rule. The left-right directionality of the injected exciton current is controlled by the circular polarization of excitation, while the angular directionality is controlled by the excitation location, exhibiting a vortex pattern in a supercell. The phenomenon is reminiscent of the chiral light-matter interaction in nanophotonics structures, with the role of the guided electromagnetic wave now replaced by the valley-orbit coupled exciton Bloch wave in a uniform monolayer, which points to new excitonic devices with nonreciprocal functionalities.

Figure 1

(a) Schematic of a monolayer TMD on a patterned substrate where the dielectric $ϵ$ has a triangular superlattice profile as a function of location $\mathbf{R}$. The bottom plots show local dispersions of exciton at the high and low $ϵ$ regions. Electron-hole Coulomb exchange splits exciton dispersion into $L$ (orange) and $T$ (gray) branches featuring opposite textures of valley pseudospin (green arrows). The $L$ branch has a massless dispersion with velocity $\propto {ϵ}^{-1}$, while the $T$ branch has the usual exciton mass independent of $ϵ$. For $L$ branch the modulation $ϵ(\mathbf{R})$ leads to the formation of minibands. (b) Typical example of $L$-branch dispersion shown in the superlattice minizone [the same as Fig. 3 where parameters used in the calculation are listed].

Figure 2

(a) Example of $L$ exciton Bloch function [at wave vector ${\mathbf{k}}_0$ denoted by the gray dot in Fig. 1]. Left to right: probability density of the $\mathbf{K}$, $-\mathbf{K}$ valley components, and their sum, shown in a supercell [cf. dashed diamond in Fig. 1]. (b) This Bloch function has six dominant Fourier components at ${\mathbf{k}}_0+{\mathbf{g}}_i$ of the unfolded $L$ branch, which have nearly degenerate kinetic energy, now hybridized by Fourier component $V_{\mathbf{g}}$ of the electron-hole Coulomb exchange. ${\mathbf{g}}_i$ are reciprocal superlattice vectors. (c) Spatially resolved valley polarization $P_{\mathbf{k}}(\mathbf{R})$ in the supercell, shown for three exciton Bloch functions on the contour of energy $E_t$ [cf. Fig. 1]. The orientation of the texture is locked to the superlattice crystal momentum. (d) Through circular polarized nano-optical excitation at ${\mathbf{R}}_0$, out of the three Bloch states shown in (c), the one at ${\mathbf{k}}_0$ is selectively excited because of its valley pseudospin texture, leading to a directional exciton flow. Green dots in plots of (a),(c),(d) mark the location of nano-optical excitation.

Figure 3

(a) Exciton dispersion in dielectric superlattice characterized by ${ϵ}^{-1}(\mathbf{R})={ϵ}_c^{-1}[\alpha (0)+{\sum }_{i=1}^6\alpha ({\mathbf{g}}_i)\exp (i{\mathbf{g}}_i·\mathbf{R})]$, color coded by the projection of the Bloch functions to the $T$ (gray) and $L$-branch (orange) modes. $\alpha (0)=1$, $\alpha ({\mathbf{g}}_i)=-0.04$, corresponding to a dielectric modulation between $0.9{ϵ}_c$ and $1.3{ϵ}_c$. The superlattice period $l=100\mathrm{nm}$, and electron-hole exchange strength $J=2\mathrm{eV}$ [cf. Eq. (3)]. (b)–(f) Injection of excitons by a $\sigma +$ polarized nano-optical excitation with a field profile $\mathcal{G}(\mathbf{R})=\exp [-2(\mathbf{R}-{\mathbf{R}}_0{)}^2/w^2]$, with $w=10\mathrm{nm}$ centered at ${\mathbf{R}}_0$. (b) ${\eta }_{L/T}$ denotes the overall injection rate of $L/T$ excitons as a function of excitation energy $E_t$. (c) Population distribution of the injected $L$ excitons in momentum space at the excitation energy $E_t=9\mathrm{meV}$. The anisotropic distribution in momentum space corresponds to an exciton current, primarily contributed by the inner contour (cf. red arrows). (d) Angular distribution of the injected exciton flow. (e) The current injection rate ${\mathbf{j}}_{+}$ as a function of $E_t$. (f) ${\mathbf{j}}_{+}$ as a function of excitation location ${\mathbf{R}}_0$ in a supercell. Green dot marks the excitation location for the plots of (b)–(e). (g) Angular distributions of the injected exciton flow under various linearly polarized excitations at ${\mathbf{R}}_0$.

Figure 4

(a) The patterned dielectric can also lead to a weak spatial modulation of exciton energy $\mathrm{\Delta }V(\mathbf{R})$ independent of valley pseudospin. (b) Exciton minibands taking into account $\mathrm{\Delta }V(\mathbf{R})$ with a modulation amplitude of 9 meV. Other parameters are the same as in Fig. 3. Above 6 meV, there are minibands predominantly from the $L$ branch (orange). (c) Population distribution of the injected $L$ excitons in momentum space at $E_t=8\mathrm{meV}$ [cf. (a)]. (d) Angular distribution of the injected exciton flow. (e) The current injection rate ${\mathbf{j}}_{+}$ as a function of excitation energy $E_t$.