We grow epitaxial graphene monolayers on Ru(0001) that cover uniformly the substrate over lateral distances larger than several microns. The weakly coupled graphene monolayer is periodically rippled and it shows charge inhomogeneities in the charge distribution. Real space measurements by scanning tunneling spectroscopy reveal the existence of electron pockets at the higher parts of the ripples, as predicted by a simple theoretical model. We also visualize the geometric and electronic structure of edges of graphene nanoislands.

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Elaborate density functional theory (DFT) calculations that include the effect of van der Waals (vdW) interactions have been carried out for graphene epitaxially grown on Ru(0001). The calculations predict a reduction of structural corrugation in the observed moiré pattern of about 25% ($ 0:4 # A) with respect to DFT calculations without vdW corrections. The simulated STM topographies are close to the experimental ones in a wide range of bias voltage around the Fermi level. DOI: 10.1103/PhysRevLett.106.186102 PACS numbers: 73.22.Pr, 68.37.Ef, 68.55.Àa, 71.15.Mb The exceptional electronic properties of perfect twodimensional graphene, such as its high electron mobility or its anomalous quantum Hall effect [1], makes it a promising material for applications in microelectronics and sensing. In practice, synthesis of graphene can be achieved by epitaxial growth of carbon monolayers on transition metals [2][3][4][5]. This has stimulated a lot of experimental work on different metal substrates [4,[6][7][8][9][10][11][12][13][14], which has already led to the production of relatively large domains (a crucial aspect to reproduce the properties of infinitely planar graphene). Nevertheless, in most cases, the presence of the substrate leads to modifications of the graphene morphology. One of these is the appearance of moiré patterns [3], as it is the case for graphene grown on Ru(0001) (G=Ru for short). The presence of moiré patterns not only implies changes in the graphene morphology but also in its electron density, which is no longer uniform [4]. This non uniform electron density is responsible for, e.g., the formation of quantum dots [15,16] or the selective deposition of organic molecules on well defined regions of the graphene sheet [17,18].In spite of the efforts to determine the structural corrugation of the moiré observed in G=Ru, the actual value of such corrugation is still a subject of controversy. Low energy electron diffraction (LEED) experiments give a corrugation of 1.5 Å [19], while surface x-ray diffraction (SXRD) measurements suggest two possible values, 0.82 and 1.5 Å [6,20]. In scanning tunneling microscopy (STM) measurements, the apparent corrugation of the moiré ripples decreases from 1.1 to 0.5 Å when the tunneling bias voltage goes from À0:8 V to þ0:8 V [21]. Helium atom scattering (HAS) experiments, that are sensitive to the surface total charge corrugation, give 0.15-0.4 Å [22]. On the theoretical side, density functional theory (DFT) calculations [19,[23][24][25][26], in which the C and Ru high symmetry directions are aligned but the unit cell is large enough to account for the moiré pattern, predict a corrugation in the range $1:5-1:7 # A [23-25]. In the latter calculations, the effect of van der Waals (vdW) or dispersion forces could not be included. However, there is strong evidence that these forces play a crucial role in the adsorption of aromatic molecules on metal surfaces [27][28][29][30][31], leading, in comparison with nonvdW DFT calculations, to a significant increase of adsorpti...