pressure, we refer to this 2D equivalent as the layer modulus (symbol γ ). This property has many analogies in other fields of study such as the "membrane stretching modulus" (also known as the "area-stretching elastic constant") used in the study of lipid bilayer membranes 21 and other soft materials. We use the EOS to extract fit parameters, including the layer modulus, for the monolayer systems of graphene (which we refer to as C) and boronitrene (also know as single-layer boron nitride, which we refer to as BN), and we also include results for Si, Ge, GeC, and SiC in the isostructural honeycomb structure for comparison. We consider four graphene allotropes to test the possibility of 2D phase transitions from graphene. We also consider bilayer, trilayer, and four-layered graphene (henceforth denoted as two-graphene, three-graphene, and four-graphene) to discover if the EOS can indicate any trends. In all cases, the elastic properties are calculated and the EOS is used to predict intrinsic strength.In Sec. II, we present the theoretical concepts and equation of state used to investigate the two-dimensional systems as well as the computational parameters. In Sec. III we apply our methods to various 2D systems, and we present and discuss our findings. Lastly, in Sec. IV, we give our conclusions and suggest possible future work.

Equation (10) for the two-dimensional bulk modulus γ of a general anisotropic medium should be replaced by [1] γ = c 11 c 22 − c 2 12 (c 11 + c 22 − 2c 12). Equation (10) was not used in our paper since all materials described were isotropic and since this expression is correct in the special case of isotropy. Thus, no results have been affected by this finding. We acknowledge Prof. D. Tomanek and Prof. A. Every for pointing out the error.

a b s t r a c tWe study Li on graphene using the Vienna ab initio Simulation Package employing the projector-augmented wave method within the generalized gradient approximation for the exchange-correlation potential. We give detailed structural and electronic results for various configurations involving Li on the (1 Â 1), (2 Â 1) and (2 Â 2) two-dimensional unit cells, and we consider the isolated Li dimer on graphene. We consider more detailed configurations than have been studied before, and our results compare favourably with previously calculated results where such results exist. For 100% coverage, we have new results for Li on the on-top site, which suggests a staggered configuration for the lowest energy structure for which the Li adatoms are alternately pushed into and pulled out of the graphene layer. For 50% coverage, Li favours the hollow site. We have discovered that a careful relaxation of the system also shows a staggered configuration, a result that has not been investigated before.

We present a comparative density functional study of the adsorption of hydrogen on bilayer graphene. Six different exchange-correlation functionals are employed to explore the possible configurations of hydrogen adsorption at 50% coverage. Using the four variants of the non-local van der Waals density functional, we identify three distinct competing configurations that retain the coupled bilayer structure at 0 K. One of the configurations undergoes a spontaneous transformation from hexagonal to tetrahedral structure, under hydrogenation, with heat of formation ranging between -0.03 eV (vdW-DF) and -0.37 eV (vdW-DFC09 x ). This configuration has a finite band gap of around 3 eV, whereas all other competing configurations are either semi-metallic or metallic. We also find two unique low-energy competing configurations of decoupled bilayer graphene, and therefore suggest the possibility of graphene exfoliation by hydrogen intercalation.

Density Functional Theory (DFT) calculations study of Cu doped {0001} and {01-12} surfaces of hematite for enhanced water splitting have been carried out. The doping was restricted to planes in the vicinity of the surface, specifically from the top most layers to the third inner layer of Fe atoms. Thermodynamic stabilities were evaluated based on surface energies and formation energies. The evaluation of thermodynamic stabilities (negative formation energy values) shows that the systems are thermodynamically stable which suggest that they can be synthesized in the laboratory under favorable conditions. Doping on the top most layer yields the energetically most favorable structure. The calculated charge density difference plots showed the concentration of charge mainly at the top of the surface (termination region), and this charge depleted from the Cu atom to the surrounding Fe and O atoms. This phenomenon (concentration of charge at the top of the surface) is likely to reduce the distance moved by the charge carriers, decrease in charge recombination leading to facile transfer of charge to the adsorbate and, suggesting improved photoelectrochemical water oxidation activity of hematite. The analysis of electron electronic structure reveals that Cu doped surface systems does not only decrease the band gap but also leads to the correct conduction band alignment for direct water splitting without external bias voltage.

Density functional theory calculations have been performed to study the thermodynamic stability, structural and electronic properties of various chromium (Cr) line-ordered alloy configurations in a molybdenum disulfide (MoS) hexagonal monolayer for band gap engineering. Only the molybdenum (Mo) sites were substituted at each concentration in this study. For comparison purposes, different Cr line-ordered alloy and random alloy configurations were studied and the most thermodynamically stable ones at each concentration were identified. The configurations formed by the nearest neighbor pair of Cr atoms are energetically most favorable. The line-ordered alloys are constantly lower in formation energy than the random alloys at each concentration. An increase in Cr concentration reduces the lattice constant of the MoS system following the Vegard's law. From density of states analysis, we found that the MoS band gap is tunable by both the Cr line-ordered alloys and random alloys with the same magnitudes. The reduction of the band gap is mainly due to the hybridization of the Cr 3d and Mo 4d orbitals at the vicinity of the band edges. The band gap engineering and magnitudes (1.65 eV to 0.86 eV) suggest that the Cr alloys in a MoS monolayer are good candidates for nanotechnology devices.

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First principles density functional theory (DFT) calculations have been performed to explore the stability, structural and electronic properties of Be and S codoped graphene sheets. The band-gap of graphene has been tuned by co-doping with beryllium and sulphur at different sites. The results show that by codoping graphene with Be and S, the band-gap increases from zero up to 0.58 eV depending on the doping sites. The cohesive and the formation energies of the systems were also determined. All the isomers formed by exploring different doping sites differ notably in stability, bond length and band-gap. Nevertheless, the planar structure of all the systems investigated was preserved even after geometry optimisation. Majority of the isomers that correspond to co-doping at non-equivalent sites favour higher band-gap opening, but lesser stability, than the other set of isomers with equivalent doping sites. Bader charge analysis was adopted to account for charges distribution in the systems. As a result of the difference in electronegativity among carbon atoms and the impurities, it was observed that electrons accumulation occurred more on the carbon atoms in the proximity of Be and S than at any other position in the graphitic systems investigated.