Devising ways of opening a band gap in graphene to make charge-carrier masses finite is essential for many applications. Recent experiments with graphene on hexagonal boron nitride (h-BN) offer tantalizing hints that the weak interaction with the substrate is sufficient to open a gap, in contradiction of earlier findings. Using many-body perturbation theory, we find that the small observed gap is what remains after a much larger underlying quasiparticle gap is suppressed by incommensurability. The sensitivity of this suppression to a small modulation of the distance separating graphene from the substrate suggests ways of exposing the larger underlying gap.PACS numbers: 73.22.Pr, 73.20.At, 81.05.ue Introduction. The constant velocity of the charge carriers in graphene that results from the linear dispersion of the energy bands about the Dirac point gives rise to many of its intriguing properties [1-3] but also poses a serious limitation to its application in high-performance electronic devices [4,5]. For logic applications, transistors with on-off ratios of order 10 6 are needed, requiring band gaps of ∼ 0.4 eV [5]. Different approaches have been adopted to open a gap, the most promising of which is to use the interaction of graphene with a substrate to modify the linear dispersion of the bands.The current front runners to open a band gap using a sublattice symmetry-breaking interaction are SiC [6] and hexagonal boron nitride (h-BN) substrates [7,8]. Because of its flatness, similarity to graphene, and the development of practical methods for preparing single layers of graphene on hexagonal boron nitride substrates [8] there has been an explosion in the number of studies of this system. h-BN is a very suitable insulating substrate for making graphene-based devices [9] because it has dielectric characteristics similar to those of SiO 2 , but contains fewer charged impurities and is atomically flat [10,11]. These properties result in a higher charge carrier mobility for graphene on h-BN compared to graphene on SiO 2 [8], and in electron-hole puddles [12] that are larger in size and less deep [13,14].In this paper we will show that the recently observed gap of order 30 meV [15] results from a large many-body enhanced quasiparticle gap being canceled by the incommensurability of graphene and h-BN that is only partially restored by a lateral variation of the height of graphene above the h-BN substrate. The large size of the underlying band gap and the mechanism of its cancellation suggest ways of recovering the large bare bandgap.Graphene on top of h-BN experiences a perturbing potential comprising two components. First, the 1.8% lattice mismatch between the two honeycomb lattices and orientational misalignment give rise to a slowly varying component that has been observed as moiré patterns in scanning tunneling microscopy images [13,14]. Because

The semiconducting two-dimensional transition metal dichalcogenides MX 2 show an abundance of onedimensional 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.

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We present a low energy Hamiltonian generalized to describe how the energy bands of germanene (Ge) are modified by interaction with a substrate or a capping layer. The parameters that enter the Hamiltonian are determined from first-principles relativistic calculations for Ge|MoS2 bilayers and MoS2|Ge|MoS2 trilayers and are used to determine the topological nature of the system. For the lowest energy, buckled germanene structure, the gap depends strongly on how germanene is oriented with respect to the MoS2 layer(s). Topologically non-trivial gaps for bilayers and trilayers can be almost as large as for a free-standing germanene layer.

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