content_image/407380.jpg content_image/407382.jpg content_image/407386.jpg content_image/407396.jpg content_image/407381.jpg content_image/407397.jpg content_image/407391.jpg content_image/407393.jpg
If we neglect other effect than the shifting of the DP, the translational symmetry is preserved and implies that Ψ$_{1(2)}$$_{,A}$$_{(}$$_{B}$$_{)}$( r ) could be written as: 

The solutions of the Dirac equation with the previous stated boundary conditions (BC) has now the form: 

with the energies given by: 

The values of k$_{n}$ fulfills sin[( k$_{n}$ + K$_{y}$ − k$_{0}$ ) L$_{y}$ ] = 0 and are given by: 

In absence of k$_{0}$ , k$_{n}$ could vanish giving rise to a gapless spectra as is seen from Eq. (16). This is the case when the width of the ribbons is L$_{y}$ = 3( n$_{y}$ − $_{1)}$√$_{3}$$_{a}$ with n$_{y}$ integer. However when k$_{0}$ is present the condition for gapless spectra could not be fulfilled in general. Even more for the L$_{y}$ and n$_{y}$ giving a gapless spectra in absence of the rippling we now have a dispersion of the form ϵ = ± v$_{F}$ √ k 2 x + k 2 0 and therefore a gap given by: 2 

Thus, from the previous consideration we expect a gap for AGNR due to rippling that scale as ( h$_{0}$ τ ) $^{2}$. A more accurate treatment of the rippling will correct the con- stant multiplying the scaling law and possible produce correction to this scaling for large enough rippling val- ues. 

To confirm the previous predictions we have calcu- lated numerically the eigenstates of corrugated armchair graphene nanoribbons. This was performed following the steps described at the end of Sec. II taken into account here the armchair ribbon geometry displayed on fig. 1. As we were interested in detecting the gap and in ana- lyzing its dependence with the corrugation, we have ex- plored a wide set of rippling parameters keeping fixed the width of the ribbons in the values in which they possesses metallic behavior being uncorrugated. Figure 4 shows the zero energy gap as a function of h$_{0}$/τ for the various sets of parameters displayed on the plot, for a width given by n$_{y}$ = 8. Note that for the small values of h$_{0}$/τ included in this figure, the gap does not depende on h$_{0}$ and τ in- dependently but on the ratio between these quantities. 

<img src='content_image/407380.jpg'>

<img src='content_image/407382.jpg'>

<img src='content_image/407386.jpg'>

<img src='content_image/407396.jpg'>

<img src='content_image/407381.jpg'>

We have fitted the calculated gap using the least squares method for a function of the type Δ = C ( h$_{0}$ τ ) 2 obtaining excellent agreement with correlation coefficients near one with a precision of 10 − $^{3}$. The obtained fitting is shown in Fig. 4 with dotted line. In accord to the stated above results, we have observed that by including higher values of h$_{0}$/τ the fitting become less precise. 

<img src='content_image/407397.jpg'>

Finally, we analyze how all the above findings affect the electronic transport properties of the armchair graphene nanoribbons. For this purpose, we employed the non- equilibrium Green’s function formalism 13,15 which pro- vides an useful method to study the quantum conduc- tance in nanoscopic systems. 

Before a discussion of the results let us present a briefly description of the method and the implementation adopted. We considered one corrugated unit cell, as cen- tral region H$_{C}$ , connected to two semi-infinite outstand- ing graphene nanoribbons flat leads, all having width length n$_{y}$ = 8 which corresponds to a zero band gap structure in the uncorrugated case. The hopping param- eter in both leads Hamiltonians, H$_{L}$ and H$_{R}$ , being flat, is t$_{0}$ . Hence, the whole system is then described by 

where h$_{LC}$ and h$_{LR}$ are the hopping terms from both leads to the central corrugated portion of nanoribbon, which we set equal to t$_{0}$ (see fig.1(a)). Within this for- malism, the Landauer conductance G ( E ) of the system, in a zero bias approximation, is expressed as 

## B. Quantum Conductance 

<img src='content_image/407391.jpg'>

<img src='content_image/407393.jpg'>

