We present first-principles calculations of electronic properties of graphene under uniaxial and isotropic strains, respectively. The semi-metallic nature is shown to persist up to a very large uniaxial strain of 30% except a very narrow strain range where a tiny energy gap opens. As the uniaxial strain increases along a certain direction, the Fermi velocity parallel to it decreases quickly and vanishes eventually, whereas the Fermi velocity perpendicular to it increases by as much as 25%. Thus, the low energy properties with small uniaxial strains can be described by the generalized Weyl's equation while massless and massive electrons coexist with large ones. The work function is also predicted to increase substantially as both the uniaxial and isotropic strain increases. Hence, the homogeneous strain in graphene can be regarded as the effective electronic scalar potential. PACS numbers: 81.05.Uw, 73.90+f Mechanical strain often gives rise to surprising effects on electronic properties of carbon nanomaterials 1-5 . It can turn the metallic nanotube into semiconductor and vice versa 1-5 . Along with the uniquely strong mechanical properties of the sp 2 -and sp 3 -bonded carbon materials 6 , the interplays between mechanical and electronic properties may be useful in various applications 7 . A recent successful isolation of a new carbon allotrope 8 , graphene, offers a new opportunity to explore such interesting electromechanical properties in two dimensions.At low energies, graphene at equilibrium has two linear energy bands that intersect each other at the high symmetric points, K and K ′ , of the first Brillouin zone (BZ) and are isotropic with respect to the points 9 . Without strains, the density of states vanishes linearly at the Fermi energy (E F ) or the Dirac point (E D ), exhibiting a semi-metallic nature. Thus, charge carriers are well described by the Dirac's equation for a (2+1)D free massless fermion 9-11 . Electron states here have another quantum number called a pseudospin which is either parallel or antiparallel to the wavevector of the electron and is of central importance to various novel phenomena 9-13 . Mechanical strains can introduce new environments in studying such novel physics of graphene.Recently, several experiments have been performed to investigate the physical properties of graphene when its hexagonal lattice is stretched out of equilibrium 14-20 . Strain can be induced on graphene either intentionally or naturally. The uniaxial strain can be induced by bending the substrates on which graphene is elongated without slippage [14][15][16][17] . Elastic responses are measured by pushing a tip of atomic force microscopes on suspended graphene 18 . Graphene on top of SiO 2 19 or SiC surface 20 also experiences a moderate strain due to surface corrugations or lattice mismatch. Motivated by recent works 14,21-24 pointing to a remarkable stability of graphene with large strains, we have carried out firstprinciples calculations and theoretical analysis to explore the electronic structures ...

letters to nature 132 NATURE | VOL 399 | 13 MAY 1999 | www.nature.com where g is the collection of all sites connected to X through drainage directions. The theorem predicts that, for ef®cient drainage basins, a log±log plot of C versus A should have a slope of 3/2 because D 2 and river networks in nature are known to be ef®cient and directed 12 . Our observational data (Fig. 2) are found to agree with the predictions over ®ve decades of scales.Our general results should be applicable to a wide variety of distributed networks including the¯ow of water, blood, sewage, food, air and electrical currents. Even though the speci®c details vary signi®cantly, the novel behaviour built into ef®cient transportation networks provides a uni®ed framework 13 underlying the allometric scaling of diverse systems. M

The effect of oxygenation on the electronic properties of semiconducting carbon nanotubes is studied from first principles. The O2 is found to bind to a single-walled nanotube with an adsorption energy of about 0.25 eV and to dope semiconducting nanotubes with hole carriers. Weak hybridization between carbon and oxygen is predicted for the valence-band edge states. The calculated density of states shows that weak coupling leads to conducting states near the band gap. The oxygen-induced gap closing for large-diameter semiconducting tubes is discussed as well. The influence of oxygen on the magnetic property is also addressed through a spin-polarized calculation and compared to experiment.

The thermoelectric power (TEP) of single-walled carbon nanotubes (SWNTs) is extremely sensitive to gas exposure history. Samples exposed to air or oxygen have an always positive TEP, suggestive of holelike carriers. However, at fixed temperature the TEP crosses zero and becomes progressively more negative as the SWNTs are stripped of oxygen. The time constant for oxygen adsorption/desorption is strongly temperature dependent and ranges from seconds to many days, leading to apparently "variable" TEP for a given sample at a given temperature. The saturated TEP can be accounted for within a model of strong oxygen doping of the semiconducting nanotubes.