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Two-dimensional (2D) hexagonal boron nitride (hBN) is a wide-bandgap van der Waals crystal with a unique combination of properties, including exceptional strength, large oxidation resistance at high temperatures and optical functionalities. Furthermore, in recent years hBN crystals have become the material of choice for encapsulating other 2D crystals in a variety of technological applications, from optoelectronic and tunnelling devices to composites. Monolayer hBN, which has no center of symmetry, has been predicted to exhibit piezoelectric properties, yet experimental evidence is lacking. Here, by using electrostatic force microscopy, we observed this effect as a strain-induced change in the local electric field around bubbles and creases, in agreement with theoretical calculations. No piezoelectricity was found in bilayer and bulk hBN, where the centre of symmetry is restored. These results add piezoelectricity to the known properties of monolayer hBN, which makes it a desirable candidate for novel electromechanical and stretchable optoelectronic devices, and pave a way to control the local electric field and carrier concentration in van der Waals heterostructures via strain. The experimental approach used here also shows a way to investigate the piezoelectric properties of other materials on the nanoscale by using electrostatic scanning probe techniques.

We theoretically study the formation of single-particle bound states confined by strain at the center of bubbles in monolayers of transition-metal dichalcogenides (TMDs). Bubbles ubiquitously form in two-dimensional crystals on top of a substrate by the competition between van der Waals forces and the hydrostatic pressure exerted by trapped fluid. This leads to strong strain at the center of the bubble that reduces the bangap locally, creating potential wells for the electrons that confine states inside. We simulate the spectrum versus the bubble radius for the four semiconducting group VI TMDs, MoS$_2$, WSe$_2$, WS$_2$ and MoSe$_2$, and find an overall Fock-Darwin spectrum of bubble bound states, characterised by small deviations compatible with Berry curvature effects. We analyse the density of states, the state degeneracies, orbital structure and optical transition rules. Our results show that elastic bubbles in these materials are remarkably efficient at confining photocarriers., Comment: 8 pages, 6 figures

Analysis of the band structure of TiS$_3$ single-layers suggests the possibility of changing their physical behaviour by injecting electron carriers. The anisotropy of the valence and conduction bands is explained in terms of their complex orbital composition. The nature of the Fermi surface and Lindhard response function for different doping concentrations is studied by means of first-principles DFT calculations. It is suggested that for electron doping levels $x$ (number of electrons per unit cell) $\sim$ 0.18-0.30$e^-$ the system could exhibit incommensurate charge or spin modulations which, however, would keep the metallic state whereas systems doped with smaller $x$ would be 2D metals without any electronic instability. The effect of spin-orbit coupling in the band dispersion is analysed. The DFT effective masses are used to study the plasmon spectrum from an effective low energy model. We find that this material supports highly anisotropic plasmons, with opposite anisotropy for the electron and hole bands., Comment: Accepted for publication in 2D Materials

The electronic properties of single-layer antimony are studied by a combination of first-principles and tight-binding methods. The band structure obtained from relativistic density functional theory is used to derive an analytic tight-binding model that offers an efficient and accurate description of single-particle electronic states in a wide spectral region up to the mid-UV. The strong ($\lambda=0.34$ eV) intra-atomic spin-orbit interaction plays a fundamental role in the band structure, leading to splitting of the valence band edge and to a significant reduction of the effective mass of the hole carriers. To obtain an effective many-body model of two-dimensional Sb we calculate the screened Coulomb interaction and provide numerical values for the on-site $\bar{V}_{00}$ (Hubbard) and intersite $\bar{V}_{ij}$ interactions. We find that the screening effects originate predominantly from the 5$p$ states, and are thus fully captured within the proposed tight-binding model. The leading kinetic and Coulomb energies are shown to be comparable in magnitude, $|t_{01}|/(\bar{V}_{00}-\bar{V}_{01}) \sim 1.6$, which suggests a strongly correlated character of 5$p$ electrons in Sb. The results presented here provide an essential step toward the understanding and rational description of a variety of electronic properties of this two-dimensional material., Comment: 6 pages, 5 figures, 3 tables

Semiconducting transition metal dichalcogenides present a complex electronic band structure with a rich orbital contribution to their valence and conduction bands. The possibility to consider the electronic states from a tight-binding model is highly useful for the calculation of many physical properties, for which first principle calculations are more demanding in computational terms when having a large number of atoms. Here, we present a set of Slater-Koster parameters for a tight-binding model that accurately reproduce the structure and the orbital character of the valence and conduction bands of single layer MX$_2$, where M = Mo,Wand X = S, Se. The fit of the analytical tight-binding Hamiltonian is done based on band structure from ab initio calculations. The model is used to calculate the optical conductivity of the different compounds from the Kubo formula.

The analysis of the electronic properties of strained or lattice deformed graphene combines ideas from classical condensed matter physics, soft matter, and geometrical aspects of quantum field theory (QFT) in curved spaces. Recent theoretical and experimental work shows the influence of strains in many properties of graphene not considered before, such as electronic transport, spin-orbit coupling, the formation of Moir\'e patterns, optics, ... There is also significant evidence of anharmonic effects, which can modify the structural properties of graphene. These phenomena are not restricted to graphene, and they are being intensively studied in other two dimensional materials, such as the metallic dichalcogenides. We review here recent developments related to the role of strains in the structural and electronic properties of graphene and other two dimensional compounds., Comment: 75 pages, 15 figures, review article

In this note, we reply to the comment made by E.I.Kats and V.V.Lebedev [arXiv:1407.4298] on our recent work "Thermodynamics of quantum crystalline membranes" [Phys. Rev. B 89, 224307 (2014)]. Kats and Lebedev question the validity of the calculation presented in our work, in particular on the use of a Debye momentum as a ultra-violet regulator for the theory. We address and counter argue the criticisms made by Kats and Lebedev to our work., Comment: 5 pages, 4 figures

Single- and few-layer transition metal dichalcogenides have recently emerged as a new family of layered crystals with great interest, not only from the fundamental point of view, but also because of their potential application in ultrathin devices. Here we review the electronic properties of semiconducting $MX_2$, where $M=$Mo or W and $X=$ S or Se. Based on of density functional theory calculations, which include the effect of spin-orbit interaction, we discuss the band structure of single-layer, bilayer and bulk compounds. The band structure of these compounds is highly sensitive to elastic deformations, and we review how strain engineering can be used to manipulate and tune the electronic and optical properties of those materials. We further discuss the effect of disorder and imperfections in the lattice structure and their effect on the optical and transport properties of $MX_2$. The superconducting transition in these compounds, which has been observed experimentally, is analyzed, as well as the different mechanisms proposed so far to explain the pairing. Finally, we include a discussion on the excitonic effects which are present in these systems., Comment: 9 pages, 4 figures. Short review article for special issue of Ann. Phys. on "Two-dimensional materials"

Imperfections in the crystal structure, such as point defects, can strongly modify the optical and transport properties of materials. Here, we study the effect of point defects on the optical and DC conductivities of single layers of semiconducting transition metal dichalcogenides with the form $M$S$_2$, where $M$=Mo or W. The electronic structure is considered within a six bands tight-binding model, which accounts for the relevant combination of $d$ orbitals of the metal $M$ and $p$ orbitals of the chalcogen $S$. We use the Kubo formula for the calculation of the conductivity in samples with different distributions of disorder. We find that $M$ and/or S defects create mid-gap states that localize charge carriers around the defects and which modify the optical and transport properties of the material, in agreement with recent experiments. Furthermore, our results indicate a much higher mobility for $p$-doped WS$_2$ in comparison to MoS$_2$.

We investigate the thermodynamic properties and the lattice stability of two-dimensional crystalline membranes, such as graphene and related compounds, in the low temperature quantum regime $T\rightarrow0$. A key role is played by the anharmonic coupling between in-plane and out-of plane lattice modes that, in the quantum limit, has very different consequences than in the classical regime. The role of retardation, namely of the frequency dependence, in the effective anharmonic interactions turns out to be crucial in the quantum regime. We identify a crossover temperature, $T^{*}$, between classical and quantum regimes, which is $\sim 70 - 90$ K for graphene. Below $T^{*}$, the heat capacity and thermal expansion coefficient decrease as power laws with decreasing temperature, tending to zero for $T\rightarrow0$ as required by the third law of thermodynamics., Comment: 13 pages, 1 figure

One of the main characteristics of the new family of two-dimensional crystals of semiconducting transition metal dichalcogenides (TMD) is the strong spin-orbit interaction, which makes them very promising for future applications in spintronics and valleytronics devices. Here we present a detailed study of the effect of spin-orbit coupling (SOC) on the band structure of single-layer and bulk TMDs, including explicitly the role of the chalcogen orbitals and their hybridization with the transition metal atoms. To this aim, we combine density functional theory (DFT) calculations with a Slater-Koster tight-binding model. Whereas most of the previous tight-binding models have been restricted to the K and K' points of the Brillouin zone (BZ), here we consider the effect of SOC in the whole BZ, and the results are compared to the band structure obtained by DFT methods. The tight-binding model is used to analyze the effect of SOC in the band structure, considering separately the contributions from the transition metal and the chalcogen atoms. Finally, we present a scenario where, in the case of strong SOC, the spin/orbital/valley entanglement at the minimum of the conduction band at Q can be probed and be of experimental interest in the most common cases of electron-doping reported for this family of compounds., Comment: 12 pages, 6 figures

The electrodynamics of a two-dimensional gas of massless fermions in graphene is studied by a collisionless hydrodynamic approach. A low-energy dispersion relation for the collective modes (plasmons) is derived both in the absence and in the presence of a perpendicular magnetic field. The results for graphene are compared to those for a standard two-dimensional gas of massive electrons. We further compare the results within the classical hydrodynamic approach to the full quantum mechanical calculation in the random phase approximation. The low-energy dispersion relation is shown to be a good approximation at small wave vectors. The limitations of this approach at higher order is also discussed., Comment: 7 pages, 1 figure

The recent spectacular progress in the experimental and theoretical understanding of graphene, the basic constituent of graphite, is applied here to compute, from first principles, the UV extinction of nano-particles made of stacks of graphene layers. The theory also covers cases where graphene is affected by structural, chemical or orientation disorder, each disorder type being quantitatively defined by a single parameter. The extinction bumps carried by such model materials are found to have positions and widths falling in the same range as the known astronomical 2175 \AA features: as the disorder parameter increases, the bump width increases from 0.85 to 2.5 $\mu$m$^{-1}$, while its peak position shifts from 4.65 to 4.75 $\mu$m$^{-1}$. Moderate degrees of disorder are enough to cover the range of widths of the vast majority of observed bumps (0.75 to 1.3 $\mu$m$^{-1}$). Higher degrees account for outliers, also observed in the sky. The introduction of structural or chemical disorder amounts to changing the initial $sp^{2}$ bondings into $sp^{3}$ or $sp^{1}$, so the optical properties of the model material become similar to those of the more or less amorphous carbon-rich materials studied in the laboratory: a-C, a-C:H, HAC, ACH, coals etc. The present treatment thus bridges gaps between physically different model materials., Comment: 14 pages, 16 figures, to appear in MNRAS

In this paper we present a paradigmatic tight-binding model for single-layer as well as for multi-layered 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 modelling of the single-layer hopping terms by simply adding the proper interlayer hoppings ruled by the chalcogenide atoms. We show that such tight-binding model permits to understand and control in a natural way the transition between a direct-gap band structure, in single-layer systems, to 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.

We study the intra-valley spin-orbit mediated spin relaxation in monolayers of MoS2 within a two bands effective Hamiltonian. The intrinsic spin splitting of the valence band as well as a Rashba-like coupling due to the breaking of the out-of-plane inversion symmetry are considered. We show that, in the hole doped regime, the out-of-plane spin relaxation is not very efficient since the spin splitting of the valence band tends to stabilize the spin polarization in this direction. We obtain spin lifetimes larger than nanoseconds, in agreement with recent valley polarization experiments., Comment: final version, 9 pages, 5 figures

We analyze the microscopic origin and the physical properties of the superconducting phase recently observed in MoS$_2$. We show how the combination of the valley structure of the conduction band, the density dependence of the screening of the long range Coulomb interactions, the short range electronic repulsion, and the relative weakness of the electron-phonon interactions, makes possible the existence of a phase where the superconducting order parameter has opposite signs in different valleys, resembling the superconductivity found in the pnictides and cuprates.

We study the instability of the metallic state towards the formation of a new ground state in graphene doped near the van Hove singularity. The system is described by the Hubbard model and a field theoretical approach is used to calculate the charge and spin susceptibility. We find that for repulsive interactions, within the random phase approximation, there is a competition between ferromagnetism and spin-density wave (SDW). It turns out that a SDW with a triangular geometry is more favorable when the Hubbard parameter is above the critical value U_c(T), which depends on the temperature T, even if there are small variations in the doping. Our results can be verified by ARPES or neutron scattering experiments in highly doped graphene., Comment: 5 pages, 5 figures, 1 table

We present a theoretical description of Bernstein modes that arise as a result of the coupling between plasmon-like collective excitations (upper-hybrid mode) and inter-Landau-level excitations, in graphene in a perpendicular magnetic field. These modes, which are apparent as avoided level crossings in the spectral function obtained in the random-phase approximation, are described to great accuracy in a phenomenological model. Bernstein modes, which may be measured in inelastic light-scattering experiments or in photo-conductivity spectroscopy, are a manifestation of the Coulomb interaction between the electrons and may be used for a high-precision measurement of the upper-hybrid mode at small non-zero wave vectors., Comment: 5 pages. 2 figures; slightly extended version, title changed; accepted for publication in Phys. Rev. B

We study collective electronic excitations in graphene in the integer quantum Hall regime, concentrating mainly on excitations with spin reversal such as spin-flip and spin-wave excitations. We show that these excitations are correctly accounted for in the time-dependent Hartree-Fock and strong magnetic field approximations, in contrast to spin-conserving (magneto-exciton) modes which involve a strong Landau-level mixing at non-zero wave vectors. The collective excitations are discussed in view of prominent theorems, such as Kohn's and Larmor's. Whereas the latter remains valid in graphene and yields insight into the understanding of spin-dependent modes, Kohn's theorem does not apply to relativistic electrons in graphene. We finally calculate the exchange correction to the chemical potential in the weak magnetic field limit., Comment: Replaced with published version. 13 pages, 4 figures

Crystalline membranes at finite temperatures have an anomalous behavior of the bending rigidity that makes them more rigid in the long wavelength limit. This issue is particularly relevant for applications of graphene in nano- and micro-electromechanical systems. We calculate numerically the height-height correlation function $G(q)$ of crystalline two-dimensional membranes, determining the renormalized bending rigidity, in the range of wavevectors $q$ from $10^{-7}$ \AA$^{-1}$ till 10 \AA$^{-1}$ in the self-consistent screening approximation (SCSA). For parameters appropriate to graphene, the calculated correlation function agrees reasonably with the results of atomistic Monte Carlo simulations for this material within the range of $q$ from $10^{-2}$ \AA$^{-1}$ till 1 \AA$^{-1}$. In the limit $q\rightarrow 0$ our data for the exponent $\eta$ of the renormalized bending rigidity $\kappa_R(q)\propto q^{-\eta}$ is compatible with the previously known analytical results for the SCSA $\eta\simeq 0.82$. However, this limit appears to be reached only for $q<10^{-5}$ \AA$^{-1}$ whereas at intermediate $q$ the behavior of $G(q)$ cannot be described by a single exponent., Comment: 5 pages, 4 figures

The particle-hole excitation spectrum for doped graphene is calculated from the dynamical polarizability. We study the zero and finite magnetic field cases and compare them to the standard two-dimensional electron gas. The effects of electron-electron interaction are included within the random phase approximation. From the obtained polarizability, we study the screening effects and the collective excitations (plasmon, magneto-excitons, upper-hybrid mode and linear magneto-plasmons). We stress the differences with the usual 2DEG., Comment: Invited review article in Semicond. Sci. Technol.; accepted for publication

A doped graphene layer in the integer quantum Hall regime reveals a highly unusual particle-hole excitation spectrum, which is calculated from the dynamical polarizability in the random phase approximation. We find that the elementary neutral excitations in graphene in a magnetic field are unlike those of a standard two-dimensional electron gas (2DEG): in addition to the upper-hybrid mode, the particle-hole spectrum is reorganized in linear magneto-plasmons that disperse roughly parallel to $\omega=v_F q$, instead of the usual horizontal (almost dispersionless) magneto-excitons. These modes could be detected in an inelastic light scattering experiment., Comment: 8 pages, 3 figures. Version accepted for publication in Phys. Rev. B

The electron-electron interactions effects on the shape of the Fermi surface of doped graphene are investigated. The actual discrete nature of the lattice is fully taken into account. A $\pi$-band tight-binding model, with nearest-neighbor hopping integrals, is considered. We calculate the self-energy corrections at zero temperature. Long and short range Coulomb interactions are included. The exchange self-energy corrections for graphene preserve the trigonal warping of the Fermi surface topology, although rounding the triangular shape. The band velocity is renormalized to higher value. Corrections induced by a local Coulomb interaction, calculated by second order perturbation theory, do deform anisotropically the Fermi surface shape. Results are compared to experimental observations and to other theoretical results., Comment: 10 pages, 6 figures

The electron-electron interactions affect the low-energy excitations of an electronic system and induce deformations of the Fermi surface. These effects are especially important in anisotropic materials with strong correlations, such as copper oxides superconductors or ruthenates. Here we analyze the deformations produced by electronic correlations in the Fermi surface of anisotropic two-dimensional systems, treating the regular and singular regions of the Fermi surface on the same footing. Simple analytical expressions are obtained for the corrections, based on local features of the Fermi surface. It is shown that, even for weak local interactions, the behavior of the self-energy is non trivial, showing a momentum dependence and a self-consistent interplay with the Fermi surface topology. Results are compared to experimental observations and to other theoretical results., Comment: 13 pages, 10 figures