# Valley-Selective Response of Nanostructures Coupled to 2D Transition-Metal Dichalcogenides
## 2 Optical Properties of 1L-TMDCs
Examples include the observation of strong plasmon-exciton coupling [63; 70; 71; 72], pronounced Fano resonances [58], and plasmon-induced resonance energy transfer [73], which are very attractive for various quantum optics and nanophotonic applications. These effects benefit from the localization of light within plasmonic and dielectric resonators, enabled by their small mode volumes and the strong dipole moment of excitons in 1L-TMDCs. Direct bandgap transitions in 1L-TMDCs occur at the energy-degenerate K (K') points at the edges of the 2D hexagonal Brillouin zone (schematically shown in Figure 1a). Due to inversion symmetry breaking and strong spin-orbital coupling in 1L-TMDCs, the electronic states of the two valleys have different chirality, which leads to valley-selective circular dichroism [15; 74; 75; 76; 77]. This effect is key for valleytronics applications, which focus on the manipulation of valley pseudo-spins to encode signals and information [78; 77; 12; 23]. The operation principles of valleytronics are mainly based on the valley Hall effect [75], which refers to the opposite Hall currents for carriers located in opposite valleys (Figure 1f). Excitation of 1L-TMDCs by \(\sigma^{+}\) and \(\sigma^{-}\) light causes a nonequilibrium population of the +K or \(-\)K valley. These carriers have opposite Berry curvature, which acts like an effective magnetic field in the momentum space. As a result, these two sorts of valley particles (+K or \(-\)K) propagate in different directions. If the particles are electrically charged (electrons, holes, or trions), this effect gives rise to the Hall voltage. If the particles are neutral, the Hall effect can be detected by measuring the circularly polarized components of the emitted photoluminescence (PL) intensity. In this case, the effect can be characterized by the degree of valley polarization:  
\[\rho(\omega)=\frac{I_{+}(\omega)-I_{-}(\omega)}{I_{+}(\omega)+I_{-}(\omega)}, \tag{1}\]  
where \(I_{+}(\omega)\) and \(I_{-}(\omega)\) refer to the PL intensity with \(\sigma^{+}\) and \(\sigma^{-}\) polarization, respectively. The Hall effect can also be detected in reflectance experiments via so-called Kerr rotation spectroscopy, when a system illuminated by linearly polarized light causes a rotation of polarization in a scattered signal [79; 80; 81]. By spatially resolving the Kerr signal, the valley polarization can be mapped out in a sample, and accumulation of the +K and \(-\)K valley polarizations can be observed. A general scheme for generating and detecting valley polarization was first proposed and realized in another 2D material--graphene--with broken inversion symmetry [82; 83]. Then, a free-carrier valley Hall effect in monolayer MoS\({}_{2}\) was demonstrated using electric readout methods [75] in the form of transverse motions of valley-polarized carriers in the presence of an in-plane electric field. More recently, an exciton valley Hall effect was reported for a monolayer MoS\({}_{2}\) where the exciton motion was driven by a temperature gradient [77]. Moreover, other factors such as uncontrolled strain can also lead to the separation of valley-polarized excitons [12]. More recently, the valley-polarized exciton dynamics in a 2D semiconductor van-der-Waals heterostructure was also studied and was proven to be a versatile platform for valleytronics [84]. For a full description and understanding of 1L-TMDC behavior, it is important to know the exciton and coherence lifetimes. The exciton lifetime describes the average time for which an exciton exists, and it usually defines the excitonic spectral linewidth. This time also limits the propagation distance of excitons. The coherence time defines the time during which an exciton remembers the state of the excitation field (for example, polarization) [85]. If the system is excited by \(\sigma^{+}\) or \(\sigma^{-}\) light, the coherence lifetime says how long the +K or \(-\)K valley is inequivalently polarized (nonzero valley polarization).