## 2 Few-Layer Transition Metal Dichalcogenides  
### Fundamental Materials Science  
TMDs are a group of 40+ layered materials, with the chemical structure MX\({}_{2}\). The layers consist of a single plane of hexagonally-arranged transition metal (M) atoms (e.g., molybdenum, Mo, tungsten, W) covalently bonded between two hexagonal planes of chalcogen (X) atoms (e.g., sulphur, S, selenium, Se); the layers themselves are weakly bound together by van der Waals forces (Figure 2a) [22]. Semiconducting TMDs (e.g., MoS\({}_{2}\), MoSe\({}_{2}\), WS\({}_{2}\)) are currently experiencing renewed interest for photonic and optoelectronic applications, expanding upon earlier studies conducted in the 1960s [22; 23], with the use of modern fabrication and characterization techniques allowing new insight.  
As is characteristic for 2D materials, the properties of few-layer TMDs depend on the number of layers in the material [11]. For instance, bulk MoS\({}_{2}\) has an indirect \(\sim\)1.29 eV (961 mm) bandgap, which increases due to stronger quantum confinement as the number of layers is reduced, eventually becoming a direct 1.80 eV (689 nm) bandgap semiconductor in isolated monolayer form [24]. Similar behaviour is observed for other semiconducting TMDs; for example, MoSe\({}_{2}\) exhibits a cross-over from an indirect 1.1 eV (1128 nm) gap to a direct 1.55 eV (800 nm) gap [25], and WS\({}_{2}\) shows an indirect 1.4 eV (886 nm) to direct 2.1 eV (590 nm) transition [26]. An additional result of the reduced dimensionality is the emergence of strong excitonic effects.  
In a semiconductor, photoexcitation places an electron into the conduction band, leaving a hole in the valence band. If the electron-hole interaction is weak (e.g., due to long-distance separation), they can each be considered as free carriers. However, their proximity can result in an attractive Coulombic interaction, creating a bound-state quasiparticle known as an exciton [27]. The photon energy requiredto create a bound exciton is less than the threshold for creating free carriers. Therefore, excitons can be understood as occupying energy levels that lie below the conduction band (Figure 2b, where each excitonic transition is labelled with a letter, according to standard nomenclature [22]), although this visualization is not physically correct, as it combines a one-electron and two-particle picture [27], the hybrid scheme is widely adopted in the literature, as it provides an intuitive interpretation of the level structure. This leads to two definitions of the bandgap: the electronic bandgap (or transport gap) \(E_{g}\), which defines the energy required to inject an electron-hole pair into the material, and the optical bandgap \(E_{o}\), which is the lowest absorbed photon energy. The energy difference is known as the exciton binding energy: \(E_{b}=E_{g}-E_{o}\)[28].  
In a typical inorganic semiconductor, the exciton binding energy is small (\(<\)10 meV) [28]. Therefore, generated excitons can rapidly ionize via collisions with optical phonons (lattice vibrations); hence, their influence is negligible at room temperature [27, 28]. However, for few-layer TMDs, increased quantum confinement and reduced dielectric screening results in large excitonic binding energies (\(>\)100 meV) and strong excitonic effects [11]. In photonics, it is common to use the term bandgap to describe the optical gap. Additionally, in few-layer TMDs, spin-orbit coupling leads to a splitting of the valence band, as shown in Figure 2b, which offers potential for spintronic devices [11].