We study the effects of the structural corrugation or rippling on the electronic properties of undoped armchair graphene nanoribbons (AGNR). First, reanalyzing the single corrugated graphene layer we find that the two inequivalent Dirac points (DP),
move away one from the other. Otherwise, the Fermi velocity decrease by increasing rippling. Regarding the AGNRs, whose metallic behavior depends on their width, we analyze in particular the case of the zero gap band-structure AGNRs. By solving the Dirac equation with the adequate boundary condition we show that due to the shifting of the DP a gap opens in the spectra. This gap scale with the square of the rate between the high and the wavelength of the deformation. We confirm this prediction by exact numerical solution of the finite width rippled AGNR. Moreover, we find that the quantum conductance, calculated by the non equilibrium Greens function technique vanish when the gap open. The main conclusion of our results is that a conductance gap should appear for all undoped corrugated AGNR independent of their width.
