**Green’s function approach to edge states in transition metal dichalcogenides**

Mojtaba Farmanbar, Taher Amlaki, and Geert Brocks Physical Review B**93**, 205444 (2016).

The semiconducting two-dimensional transition metal dichalcogenides MX_{2} show an abundance of one-dimensional metallic edges and grain boundaries. Standard techniques for calculating edge states typically model nanoribbons, and require the use of supercells. In this paper, we formulate a Green’s function technique for calculating edge states of (semi-)infinite two-dimensional systems with a single well-defined edge or grain boundary. We express Green’s functions in terms of Bloch matrices, constructed from the solutions of a quadratic eigenvalue equation. The technique can be applied to any localized basis representation of the Hamiltonian. Here, we use it to calculate edge states of MX_{2} monolayers by means of tight-binding models. Aside from the basic zigzag and armchair edges, we study edges with a more general orientation, structurally modifed edges, and grain boundaries. A simple three-band model captures an important part of the edge electronic structures. An 11-band model comprising all valence orbitals of the M and X atoms is required to obtain all edge states with energies in the MX_{2} band gap. Here, states of odd symmetry with respect to a mirror plane through the layer of M atoms have a dangling-bond character, and tend to pin the Fermi level.

Figure 1 (a) *k*-resolved density of states (DOS) per MoS_{2} unit of bulk MoS_{2} in the three-band tight-binding model, with the *k* vector parallel to the zigzag edge, and the zero of energy at the top of the valence band. The DOS is plotted on a logarithmic scale (right-hand side n denotes amplitude 10^{−n}) using a broadening parameter η = 0.05 eV. (b), (c), (d) *k*-resolved DOSs of the Mo edge, the S edge, and the armchair edge, respectively. (e), (f), (g), (h) *k*-integrated DOS per MoS_{2} unit of bulk MoS_{2}, the Mo edge, the S edge, and the armchair edge, all plotted on a linear scale (η = 0.05 eV). The red solid lines give the counting function and the green dashed lines indicate the charge-neutrality level (CNL).