# Valley-Selective Response of Nanostructures Coupled to 2D Transition-Metal Dichalcogenides
## 1 Introduction  
As the growth of electronics and computing infrastructure nears the limits of the so-called Moore's law [1], new directions in information processing and harvesting become of crucial importance in today's technology. This issue inspired new fields of research, such as spintronics and valleytronics, in which spin or pseudospin is used as an alternative information carrier. Currently, valleytronics is of great interest due to the discovery of atomically thin single-layered (1L) transition-metal dichalcogenides (TMDCs) [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. Monolayer TMDCs are formed (Figure 1a) by a hexagonal network of transition metal atoms (M: Mo, W) hosted between two hexagonal lattices of chalcogenide atoms (X: S, Se). Electronically, 1L-TMDCs behave as two-dimensional semiconductors, with bandgaps lying in the visible and near-infrared (IR) range. In the monolayer limit, the bandgaps of these materials are direct, enabling enhanced interactions of dipole transitions with light. An essential property of 1L-TMDCs is the broken inversion symmetry of their hexagonal 2D crystals, which, in combination with time-reversal symmetry, leads to opposite spins at the +K and \(-\)K valleys, effectively locking the spin and valley degrees of freedom (pseudospin or Berry curvature) [15; 16; 17]. Optical manipulation of the valley degree of freedom can be realized via exciton resonances based on the valley contrasting optical selection rules (for instance with \(\sigma^{-}\) and \(\sigma^{+}\) light excitation). The unique opportunity ofaddressing the valley index makes it possible to explore this binary quantum degree of freedom as an alternative information carrier [12; 18; 19; 20], which may complement both classical and quantum computing schemes based on charge and spin. This possibility of using the valley degree of freedom for information processing and storage (similar to spin in spintronics) leads to conceptually new applications for valleytronics [12]. However, short valley depolarization times and relatively short exciton lifetimes at room temperature prevent using valley pseudospin in on-chip integrated valley devices. Recently, it was demonstrated how coupling these materials to optical nanoantennas and metasurfaces can overcome this obstacle. It was demonstrated that resonant optical nanostructures and nanoantennas can effectively enhance and direct the emission from opposite valleys in 1L-TMDCs into different directions [21; 22; 23; 24]. The proposed nanostructures for valley-selective directional emission can be divided into two groups: nanostructures for _spatial separation_ of valley degrees of freedom using surface waves, and for _separation in momentum-space_, based on which photons with opposite helicity are emitted in different directions. Spatial separation of valley degrees of freedom may be accompanied by _division of valley excitons in space_[24], or transformation of valley exciton pseudospin to photonic degrees of freedom (e.g., _transverse optical spin angular momentum_) via _optical spin-orbit coupling_[23] (or both these processes). This paper is devoted to reviewing state-of-the-art advances in 1L-TMDC valley degree-of-freedom separation (valley-selective directional emission and exciton sorting) mediated by nanostructures and nanoantennas. The paper is organized as follows: in Section 2, we briefly discuss  
Figure 1: **(a)** Monolayer (1L) transition-metal dichalcogenide crystal (TMDC) structure. Transition-metal atoms (M) appear in purple, and chalcogen atoms (X) are in yellow. The structure resembles that of graphene, but with broken inversion symmetry. A side view shows the three-dimensional (3D) structure. The hexagonal Brillouin zone is shown with the \(\Gamma\) point and the two inequivalent +K and \(-\)K points. **(b)** Energy structure of 1L-WS\({}_{2}\); blue and black curves demonstrate the valence and conduction bands. **(c)** Absorption spectrum (orange curve) and photoluminescence (blue curve) of 1L-WS\({}_{2}\) at room temperature.