We present here the minimal tight-binding model for a single layer of transition metal dichalcogenides (TMDCs) MX2 (M-metal, X-chalcogen) which illuminates the physics and captures band nesting, massive Dirac Fermions and Valley Lande and Zeeman magnetic field effects. TMDCs share the hexagonal lattice with graphene but their electronic bands require much more complex atomic orbitals. Using symmetry arguments, a minimal basis consisting of 3 metal d-orbitals and 3 chalcogen dimer p-orbitals is constructed. The tunneling matrix elements between nearest neighbor metal and chalcogen orbitals are explicitly derived at K, −K and Γ points of the Brillouin zone. The nearest neighbor tunneling matrix elements connect specific metal and sulfur orbitals yielding an effective 6 × 6 Hamiltonian giving correct composition of metal and chalcogen orbitals but not the direct gap at K points. The direct gap at K, correct masses and conduction band minima at Q points responsible for band nesting are obtained by inclusion of next neighbor Mo-Mo tunneling. The parameters of the next nearest neighbor model are successfully fitted to MX2 (M=Mo, X=S) density functional (DFT) ab-initio calculations of the highest valence and lowest conduction band dispersion along K −Γ line in the Brillouin zone. The effective two-band massive Dirac Hamiltonian for MoS2, Lande g-factors and valley Zeeman splitting are obtained.

We discuss here the effect of band nesting and topology on the spectrum of excitons in a single layer of MoS2, a prototype transition metal dichalcogenide material. We solve for the single particle states using the ab initio based tight-binding model containing metal d and sulfur p orbitals. The metal orbitals contribution evolving from K to Γ points results in conduction-valence band nesting and a set of second minima at Q points in the conduction band. There are three Q minima for each K valley. We accurately solve the Bethe-Salpeter equation including both K and Q points and obtain ground and excited exciton states. We determine the effects of the electron-hole single particle energies including band nesting, direct and exchange screened Coulomb electron-hole interactions and resulting topological magnetic moments on the exciton spectrum. The ability to control different contributions combined with accurate calculations of the ground and excited exciton states allows for the determination of the importance of different contributions and a comparison with effective mass and k · p massive Dirac fermion models.

We present here results of atomistic theory of electrons confined by metallic gates in a single layer of transition metal dichalcogenides. The electronic states are described by the tight-binding model and computed using a computational box including up to million atoms with periodic boundary conditions and parabolic confining potential due to external gates embedded in it. With this methodology applied to MoS2, we find a twofold degenerate energy spectrum of electrons confined in the two non-equivalent K-valleys by the metallic gates as well as six-fold degenerate spectrum associated with Q-valleys. We compare the electron spectrum with the energy levels of electrons confined in GaAs/GaAlAs and in self-assembled quantum dots. We discuss the role of spin splitting and topological moments on the K and Q valley electronic states in quantum dots with sizes comparable to experiment. arXiv:1907.09512v1 [cond-mat.mes-hall]

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We determine here the evolution of the bandgap energy with size in graphene quantum dots (GQDs). We find oscillatory behaviour of the bandgap and explain its origin in terms of armchair and zigzag edges. The electronic energy spectra of GQDs are computed using both the tight binding model and ab initio density functional methods. The results of the tight binding model are analyzed by dividing zigzag graphene quantum dots into concentric rings. For each ring, the energy spectra, the wave functions and the bandgap are obtained analytically. The effect of inter-ring tunneling on the energy gap is determined. The growth of zigzag terminated GQD into armchair GQD is shown to be associated with the addition of a one-dimensional Lieb lattice of carbon atoms with a shell of energy levels in the middle of the energy gap of the inner zigzag terminated GQD. This introduces a different structure of the energy levels at the bottom of the conduction and top of the valence band in zigzag and armchair GQD which manifests itself in the oscillation of the energy gap with increasing size. The evolution of the bandgap with the number of carbon atoms is compared with the notion of confined Dirac Fermions and tested against ab initio calculations of Kohn-Sham and TD-DFT energy gaps.

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We describe here recent work on the electronic properties, magnetoexcitons and valley polarised electron gas in 2D crystals. Among 2D crystals, monolayer M oS 2 has attracted significant attention as a direct-gap 2D semiconductor analogue of graphene. The crystal structure of monolayer M oS 2 breaks inversion symmetry and results in K valley selection rules allowing to address individual valleys optically. Additionally, the band nesting near Q points is responsible for enhancing the optical response of M oS 2 .We show that at low energies the electronic structure of M oS 2 is well approximated by the massive Dirac Fermion model. We focus on the effect of magnetic field on optical properties of M oS 2 .We discuss the Landau level structure of massive Dirac fermions in the two nonequivalent valleys and resulting valley Zeeman splitting. The effects of electronelectron interaction on the valley Zeeman splitting and on the magneto-exciton spectrum are described. We show the changes in the absorption spectrum as the self-energy, electron-hole exchange and correlation effects are included. Finally, we describe the valley-polarised electron gas in W S 2 and its optical signature in finite magnetic fields. arXiv:1810.12402v1 [cond-mat.mes-hall] 29 Oct 2018 20 2. Electronic structure of a monolayer of M oS 2We start with the electronic structure of a best known TMDC, M oS 2 . Fig. 1 shows the ab-initio band structure of a single-layer of M oS 2 obtained with the Abinit package [7,17,20]. M oS 2 has a layered structure formed by a triangular lattice of Mo atoms sandwiched between planes of triangularly arranged S 25 atoms, resembling honeycomb structure of graphene when viewed from above.Analogous to graphene the first Brillouin zone is hexagonal, with 6 K points at the six corners. Like in graphene, the 6 K points can be divided into two groups

Atomically thin semiconductors from the transition metal dichalcogenide family are materials in which the optical response is dominated by strongly bound excitonic complexes. Here, we present a theory of excitons in two-dimensional semiconductors using a tight-binding model of the electronic structure. In the first part, we review extensive literature on 2D van der Waals materials, with particular focus on their optical response from both experimental and theoretical points of view. In the second part, we discuss our ab initio calculations of the electronic structure of MoS2, representative of a wide class of materials, and review our minimal tight-binding model, which reproduces low-energy physics around the Fermi level and, at the same time, allows for the understanding of their electronic structure. Next, we describe how electron-hole pair excitations from the mean-field-level ground state are constructed. The electron–electron interactions mix the electron-hole pair excitations, resulting in excitonic wave functions and energies obtained by solving the Bethe–Salpeter equation. This is enabled by the efficient computation of the Coulomb matrix elements optimized for two-dimensional crystals. Next, we discuss non-local screening in various geometries usually used in experiments. We conclude with a discussion of the fine structure and excited excitonic spectra. In particular, we discuss the effect of band nesting on the exciton fine structure; Coulomb interactions; and the topology of the wave functions, screening and dielectric environment. Finally, we follow by adding another layer and discuss excitons in heterostructures built from two-dimensional semiconductors.