Charge inhomogeneities due to smooth ripples in graphene sheets

We study the effect of the curved ripples observed in the freestanding graphene samples on the electronic structure of the system. We model the ripples as smooth curved bumps and compute the Green's function of the Dirac fermions in the curved surface. Curved regions modify the Fermi velocity that becomes a function of the point on the graphene surface and induce energy dependent oscillations in the local density of states around the position of the bump. This effect is due to the change of the Pauli matrices with the position and is independent of the well known gauge potential generated by the spin connection. The corrections to the density of states are only due to the first effect and are estimated to be of a few percent of the flat density of states. Local probes such as scanning tunnel microscopy should be able to observe the predicted correlation of the morphology with the electronics. We discuss the connection of the present work with a recent observation of charge anisotropy in graphene and argue that the ripples can provide an alternative explanation.