It is shown that for monolayer graphene electrons are confined on a perfect two-dimensional surface. The implications for the electronic properties of corrugated graphene are discussed in view of a derivation of the constrained relativistic dynamics for the massless carriers in two dimensions in accord with the Heisenberg uncertainty principle. Surface curvature is related to a series of phenomena with practical applications such as curvature induced $p\text{−}n$ junctions, band-gap opening, and decoherence. We also establish a bending free energy by treating graphene as a soft electronic membrane.

• Received 28 January 2010

DOI:https://doi.org/10.1103/PhysRevB.81.205409