## 2 Electronic Band Structure  
The crystal structure of _MX\({}_{2}\)_ is schematically shown in Figure 1. A single layer is composed by an inner layer of metal \(M\) atoms ordered on a triangular lattice, which is sandwiched between two layers of chalcogen \(X\) atoms placed on the triangular lattice of alternating hollow sites. We use a notation such that \(a\) corresponds to the distance between nearest neighbor in-plane \(M\)-\(M\) and \(X\)-\(X\) atoms, \(b\) is the nearest neighbor \(M\)-\(X\) separation and \(u\) is the distance between the \(M\) and \(X\) planes. The _MX\({}_{2}\)_ crystal forms an almost perfect trigonal prism structure with \(b\simeq\sqrt{7/12}a\) and \(u\simeq a/2\). The lattice parameters of the bulk compounds corresponding to the more commonly studied TMDs are given in Table 1[36; 37; 38]. The in-plane Brillouin zone is a hexagon, and it is shown in Figure 2. It contains the high-symmetry points \(\Gamma=(0,0)\), K\(=4\pi/3a(1,0)\) and M\(=4\pi/3a(0,\sqrt{3}/2)\). The six Q points correspond to the approximate position of a local minimum of the conduction band. \begin{table}
\begin{tabular}{l c c c} \hline  & \(a\) & \(u\) & \(c^{\prime}\) \\ \hline MoS\({}_{2}\) & 3.160 & 1.586 & 6.140 \\ MoSe\({}_{2}\) & 3.288 & 1.664 & 6.451 \\ WS\({}_{2}\) & 3.153 & 1.571 & 6.160 \\ WSe\({}_{2}\) & 3.260 & 1.657 & 6.422 \\ \hline \end{tabular}
\end{table} 
Table 1: Lattice parameters for the TMDs (transition metal dichalcogenides) considered in the text. \(a\) represents the \(M\)–\(M\) atomic distance, \(u\) the internal vertical separation between the \(M\) and the \(X\) planes, and \(c^{\prime}\) the distance between the metal \(M\) layers. In bulk systems, \(c=2c^{\prime}\) accounts for the \(z\)-axis lattice parameter. All values are in Å units. Figure 1: (**a**) Sketch of the atomic structure of _MX\({}_{2}\)_. The bulk compound has a 2H–_MX\({}_{2}\)_ structure with two _MX\({}_{2}\)_ layers per unit cell, each of them being built up from a trigonal prism coordination unit. The value of the lattice constants for each family is given in Table 1; (**b**) top view of monolayer _MX\({}_{2}\)_ lattice. **Green (blue)** circles indicate \(M\) (\(X\)) atoms. The nearest neighbors (\(\delta_{i}\)) and the next nearest neighbors (\(a_{i}\)) vector are shown in the figure. In order to study the electronic band structure of single-layer TMDs, we use the Density Functional Theory (DFT) calculations presented by some of the authors in Reference [28], in which the intrinsic spin-orbit interaction term for all atoms is included. Figure 3 shows the band structures for single-layer \(MX_{2}\) (black lines) together with the TB bands that will be discussed later (red lines) (DFT calculations were done using the Siesta code [39; 40], with the exchange-correlation potential of Ceperly-Alder [41] as parametrized by [PERSON] [42]. A split-valence double-\(\zeta\) basis set including polarization functions was considered [43]. The energy cutoff and the BZ sampling were chosen to converge the total energy with a value of 300 Ry and \(30\times 30\times 1\). The energy cutoff was set to \(10^{-6}\) eV. Spin-orbit interaction of the different compounds was considered by following the method developed in Reference [44]. The lattice parameters used in the calculation are set to their experimental values [45; 46; 47] and are given in Table 1). One of the main characteristics of TMDs is that, contrary to what happens in other 2D crystals like graphene or phosphorene, the valence and conduction bands of \(MX_{2}\) present a very rich orbital contribution. As explained in detail in Reference [35], they are made by hybridization of the \(d\) orbitals of the transition metal, and the \(p\) orbitals of the chalcogen.