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.

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 moir\'e pattern of about 25% ($\sim${}0.4\,\,\AA{}) 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.

We explore the spatial variations of the unoccupied electronic states of graphene epitaxially grown on Ru(0001) and observed three unexpected features: the first graphene image state is split in energy; unlike all other image states, the split state does not follow the local work function modulation, and a new interfacial state at +3\,\,eV appears on some areas of the surface. First-principles calculations explain the observations and permit us to conclude that the system behaves as a self-organized periodic array of quantum dots.

It is shown that the surface of an epitaxial graphene monolayer grown on Ru(0001) could be used as a quite efficient external mirror for He-atom microscopy, with a specular reflectivity of 20% of the incident beam. Furthermore, the system is stable up to 1150 K, and the He reflectivity remains almost unchanged after exposure to air. Additionally, the high reflectivity for H2 molecules (11%) opens up the development of a H2 microprobe suitable for lithography. The Debye temperature for this epitaxial graphene monolayer has been determined from a study of the temperature dependence of the He specular intensity as a function of incident parameters. A value of 1045 K has been obtained, which is much higher than the 590 K reported for graphite under similar conditions, and close to the value of 1287 K calculated for isolated graphene.