6 where T ( E ) is the transmission function given by T ( E ) = Tr(Γ$_{L}$( E ) G $_{C}$( E )Γ$_{R}$( E ) G † $_{C}$( E )), with Γ$_{ℓ}$ = − 2 I m (Σ$_{ℓ}$( E )) ( ℓ = L, R ) being the couplings of the corrugated graphene nanoribbon to the leads, and G $_{C}$( E ) = ( E − H$_{C}$ − Σ$_{L}$ − Σ$_{R}$) − 1 the total Green’s function including the leads self energies, Σ$_{L}$ and Σ$_{R}$. These self energies must be calculated through the leads surfaces Green func- tions (SGF). Although it was reported a very time effi- cient way to compute the SGF of graphene nanoribbons leads (See Fig. 2 of Ref. 18), for the sake of simplicity we implemented the recursive iteration procedure, given by g$_{ℓ}$ = ( E − H$_{ℓ}$ − $_{T}$† $_{ℓ}$g$_{ℓ}$ T$_{ℓ}$ ) − $^{1}$, where H$_{ℓ}$ is the unit cell lead Hamiltonian and T$_{ℓ}$ is the interlayer coupling in the semi-infinite lead. For the small width nanoribbons stud- ied here we have checked that a fast convergence is ob- tained. In the iterative procedure, we set the tolerance as 10 − $^{6}$. Figures 5 (a), (b) and (c), show the transmission co- efficients obtained for armchair border type nanoribbons whose period τ is 72 a and with amplitudes h$_{0}$ = 0 . 0 , 3 . 0 and 6 . 0 in units of a , respectively. In the left panels of the plots we included the corresponding low energy bands structure computed numerically following the description given at the end of Section II suited for the nanoribbon case. The band reflexion at the first Brillouin zone can clearly be seen in fig. 5 (a) which correspond to the un- corrugated case. Here, and such as is expectable, the transmission coefficient is equals to one at the Fermi en- ergy. In the fig. 5 (b) the band gap at the Fermi en- ergy appears, and accordingly to this the transmission becomes reduced. The fact that it do not reach zero is produced because the central region possesses only one corrugated unit cell. Increasing the amplitude of the rip- pling the gap becomes broader as is observed in fig. 5 (c) and the transmission goes to zero around the Fermi energy. These results are consistent with the obtained above by solving the Dirac equation. Here also, a second null conductance region appears above a thin perfect con- ductance interval at higher energies. In addition to these results, obtained keeping fixed τ$_{x}$ and varying h$_{0}$ , we cal- culated T ( E ) for a wide range of corrugation periods and amplitudes, finding in all cases the appearance of a gap at low energies around E$_{F}$ . lated by the NEGF formalism, vanish around the Fermi energy of undoped corrugated samples. These results was preceded by an analysis of the cor- rugation effects over the electronic spectra of graphene layers. Therein, we have found that the corrugation just shift the Dirac points and renormalizes the Fermi veloc- ity. These findings agree with the results of numerical calculations. In view of this, we conclude that the corrugations are an important source of gap in the spectra in addition to the electronic correlations to which typically is assigned this effect $^{15}$. Although our study was performed model- ing the corrugations by single sinusoidal functions, since any lattice deformation could be approximated quite well by an adequate sum of sins, we strongly believe that the results obtained here are extensible to general cor- After having analyzed the Dirac equation includ- ing the pseudo-magnetic field induced by the corruga- tion, and having explored numerically a wide realis- tic range of rippling parameters sets, we have found that the corrugations produces a gap in the low en- ergy electronic spectrum of otherwise conducting arm- chair graphene nanoribbons. We have found that this gap scale quadratic with the rate between the high and the wavelength of the deformation. Accordingly to this, the quantum conductance G ( E ) of the ribbons, calcu- IV. CONCLUSIONS FIG. 5: Transmission coefficient of corrugated armchair graphene nanoribbons. n x = 24, n y = 8 and h 0 = 0 (a), h 0 = 3 (b) and h 0 = 6 (c), in units of a . In each left panel we included the corresponding low energy band structure which serve to a better analysis of the behavior of T ( E ).