Controlling the bandstructure through local-strain engineering is an exciting avenue for tailoring optoelectronic properties of materials at the nanoscale. Atomically thin materials are particularly well suited for this purpose because they can withstand extreme non-homogeneous deformations before rupture. Here, we study the effect of large localized strain in the electronic bandstructure of atomically thin MoS 2 . Using photoluminescence imaging, we observe a strain-induced reduction of the direct bandgap, and funneling of photogenerated excitons towards regions of higher strain.To understand these results, we develop a non-uniform tight-binding model to calculate the electronic properties of MoS 2 nanolayers with complex and realistic local strain geometries, finding good agreement with our experimental results.

A striking feature of bilayer graphene is the induction of a significant band gap in the electronic states by the application of a perpendicular electric field [1][2][3][4][5][6][7] . Thicker graphene layers are also highly attractive materials. The ability to produce a band gap in these systems is of great fundamental and practical interest. Both experimental 8 and theoretical [9][10][11][12][13][14][15][16] investigations of graphene trilayers with the typical ABA layer stacking have, however, revealed the lack of any appreciable induced gap. Here we contrast this behaviour with that exhibited by graphene trilayers with ABC crystallographic stacking. The symmetry of this structure is similar to that of AB-stacked graphene bilayers and, as shown by infrared conductivity measurements, permits a large band gap to be formed by an applied electric field. Our results demonstrate the critical and hitherto neglected role of the crystallographic stacking sequence on the induction of a band gap in few-layer graphene.Producing a controlled and tunable band gap in graphene is a topic of central importance [1][2][3][4][5][6][7]17,18 . In addition to the intrinsic interest of altering the electronic properties of materials, the availability of an adjustable band gap opens up the possibility of a much wider range of applications for graphene in electronics and photonics. Both single-and few-layer graphene in their unperturbed state lack a band gap 19,20 . However, few-layer graphene materials under the application of a symmetry-lowering perpendicular electric field may exhibit an induced gap [9][10][11][12][13][14][15][16]21,22 . In this regard, trilayer graphene is an attractive material system. Unlike bilayer graphene, however, trilayers, which typically exhibit Bernal (ABA) stacking order and the associated mirror symmetry (Fig. 1a), have been shown both theoretically 9-16 and experimentally 8 not to support the induction of a significant band gap when subjected to a perpendicular electric field. As discussed below, this behaviour follows from the mirror symmetry of the unperturbed ABA trilayer 10,23 . Recent research 24,25 has, however, reported the existence of a new type of trilayer graphene, one with ABC (rhombohedral) stacking order between the graphene sheets (Fig. 1b). This crystal structure, like that of the bilayer, possesses inversion symmetry, but lacks mirror symmetry (Fig. 1b). The low-energy electronic structure of the ABC trilayer 20,22 is accordingly more similar to that of the AB-stacked bilayer graphene. In particular, the undoped ABC trilayer has only two-fold degeneracy 20 at the Fermi energy, like the graphene bilayer, rather than the four-fold degeneracy found in the ABA trilayer 20,23 . The two-fold degeneracy in the ABC trilayer band structure can be readily lifted by imposing different potentials on the top and bottom graphene layers by an applied electric field, which leads to the opening of a band gap 9,10,[13][14][15][16]21,22 . Although theory has predicted the induction of a large band gap fo...

In this paper we present a paradigmatic tight-binding model for single-layer as well as multilayered semiconducting MoS 2 and similar transition metal dichalcogenides. We show that the electronic properties of multilayer systems can be reproduced in terms of a tight-binding modeling of the single-layer hopping terms by simply adding the proper interlayer hoppings ruled by the chalcogenide atoms. We show that such a tight-binding model makes it possible to understand and control in a natural way the transition between a direct-gap band structure, in single-layer systems, and an indirect gap in multilayer compounds in terms of a momentum/orbital selective interlayer splitting of the relevant valence and conduction bands. The model represents also a suitable playground to investigate in an analytical way strain and finite-size effects.

One of the fascinating properties of the new families of two-dimensional crystals is their high stretchability and the possibility to use external strain to manipulate, in a controlled manner, their optical and electronic properties. Strain engineering, understood as the field that study how the physical properties of materials can be tuned by controlling the elastic strain fields applied to it, has a perfect platform for its implementation in the atomically thin semiconducting materials. The object of this review is to give an overview of the recent progress to control the optical and electronics properties of 2D crystals, by means of strain engineering. We will concentrate on semiconducting layered materials, with especial emphasis in transition metal dichalcogenides (MoS2, WS2, MoSe2 and WSe2). The effect of strain in other atomically thin materials like black phosphorus, silicene, etc., is also considered. The benefits of strain engineering in 2D crystals for applications in nanoelectronics and optoelectronics will be revised, and the open problems in the field will be discussed. Contents

We observe a giant increase of the infrared intensity and a softening of the in-plane antisymmetric phonon mode E(u) ( approximately 0.2 eV) in bilayer graphene as a function of the gate-induced doping. The phonon peak has a pronounced Fano-like asymmetry. We suggest that the intensity growth and the softening originate from the coupling of the phonon mode to the narrow electronic transition between parallel bands of the same character, while the asymmetry is due to the interaction with the continuum of transitions between the lowest hole and electron bands. The growth of the peak can be interpreted as a "charged-phonon" effect observed previously in organic chain conductors and doped fullerenes, which can be tuned in graphene with the gate voltage.

Strain engineering has emerged as a powerful tool to modify the optical and electronic properties of two-dimensional crystals. Here we perform a systematic study of strained semiconducting transition metal dichalcogenides. The effect of strain is considered within a full Slater-Koster tight-binding model, which provides us with the band structure in the whole Brillouin zone. From this, we derive an effective low-energy model valid around the K point of the BZ, which includes terms up to second order in momentum and strain. For a generic profile of strain, we show that the solutions for this model can be expressed in terms of the harmonic oscillator and double quantum well models, for the valence and conduction bands respectively. We further study the shift of the position of the electron and hole band edges due to uniform strain. Finally, we discuss the importance of spin-strain coupling in these 2D semiconducting materials.

The electrostatic screening in single and few‐layer MoS2 sheets is studied. Electrostatic force microscopy is used to probe the electric field generated by charge impurities in the substrate and incompletely screened by MoS2 sheets. A non‐linear Thomas–Fermi theory is employed to interpret the experimental results, demonstrating the important role of the interlayer coupling in the screening of MoS2.

We study the phase diagram of the t-J model using a mean field type approximation within the BaymKadanoff perturbation expansion for Hubbard X operators. The line separating the normal state from a d-wave flux or bond-order state starts near optimal doping at Tϭ0 and rises quickly with decreasing doping. The transition temperature T c for d-wave superconductivity increases monotonically in the overdoped region towards optimal doping. Near optimal doping a strong competition between the two d-wave order parameters sets in leading to a strong suppression of T c in the underdoped region. Treating for simplicity the flux phase as commensurate the superconducting and flux phases coexist in the underdoped region below T c , whereas a pure flux phase exists above T c with a pseudogap of d-wave symmetry in the excitation spectrum. We also find that incommensurate charge-density-wave ground states due to Coulomb interactions do not modify strongly the above phase diagram near the superconducting phase, at least, as long as the latter exists at all. ͓S0163-1829͑99͒01609-4͔