4 III. THE EFFECT OF THE RIPPLING: ARMCHAIR GRAPHENE NANORIBBONS to obtain a total flattening of the band. This could be interpreted as that the predicted zero LL appears only when the rippling is strong enough. This conclusion is consistent with the results of Ref. 11. In this work it was shown that as a result of the relaxation of the structure, the corrugation decrease and the flat band disappears from the spectra. If we had studied the behavior near K ′ by analyzing the second term in (4) we would have obtained instead of eq. (11) the result: E$_{2}$ ( k$_{x}$, k$_{y}$ ) = ± ˆ v$_{F}$ √ k 2 x + ( k$_{y}$ − k$_{0}$ ) 2 (12) Therefore K ′ moves opposite than K . As k$_{0}$ is a negative quantity the two DP away each other in presence of the rippling. In Fig. 2 we show the relationship | k$_{0}$ | K$_{y}$ and ˆ v$_{F}$ v$_{F}$ as a function of h$_{0}$ τ 2. ˆ v$_{F}$ decrease by increasing rippling. Let us now turn into the main subject of this work which is the study of the effects of corrugations on the spectra of graphene nanoribbons. As we have mentioned in the Introduction, a tight binding based description of the electronic properties of the ribbons predict that, de- pending on their width and border edge type, they can be metallic, with zero band gap, or insulators $^{8}$. For ex- ample, for armchair edge (fig.1) there are specific values of the width has for which metallic behavior is expected. Otherwise they should be insulators. In spite of that, the experiments shown that graphene nanoribbons are insulators independent of their width $^{9}$. Though device fabrication process does not give atomically precise con- trol on the type of edges, this seems to show that a pure tight binding model is not enough to describe the GNR as it is for the sheets. Both, electronic correlation 15 and edge disorder $^{16,17}$, have been proposed as possibles mech- anisms that give rise to this behavior. In this Section we show that rippling could give an additional source for the electronic gap, at least for the armchair edge type. A. Dirac equation with boundary conditions FIG. 3: Low energy band structure of uncorrugated (blue) and corrugated (red) graphene layers. The corrugation pa- rameters are τ = 72 a and h 0 equals 0 . 0, which correspond to the Dirac cone, and 6 . 0 in units of a , respectively. This re- sult qualitatively agree with the obtained by means the Dirac equation including the pseudo-magnetic field. As for the infinite graphene sheet, it is possible to study the low energy electronic structure of graphene nanorib- bons by a Dirac like equation, but in this case adequate boundary conditions must be imposed $^{8}$. For armchair edges, the valley states near the DPs, K and K ′ , get ad- mixture and metallic behaviors are obtained for certain widths. We have described in the previous Section that the rippling produce a shift of the two inequivalent DPs. As we will shown below, to include only this shifting is enough to obtain a drastic change in the electronic be- havior of AGNRs due to corrugations. The wavefunction for sub-lattices A (B) are therefore given by: For completeness, let us provide a comparison, just at a qualitative level, of the above findings with the band structure of corrugated graphene layers. Since we are try- ing to detect the displacement of the Dirac points, first we must compute the band structure of the uncorrugated case. Note here that since the adopted ripple develops in the x axis direction with a period equal to τ , as can be seen in fig.1 (b), then the unit cell must be extended up to include the whole rippled region from where the hop- ping becomes repeated again. This enlargement of the unit cell produces thus a reduction of the k$_{x}$ components of the Dirac points K and K ′ due to band reflexion at the border of the original first Brilloun zone. As can be seen in the fig.3, where we show the low energy spectrum ob- tained for corrugations whose period is τ = 72 a and with amplitudes h$_{0}$ 0.0 and 6 . 0 in units of a , the corrugation effectively shifts the Dirac point, located originally at K , in the k$_{y}$ direction. Also, in addition to this shift and in accord with the earlier predictions, the observed small slope of the dispersion bands of corrugated graphene im- plies a lower Fermi velocity of electrons near the Fermi level. It should be mention here that this result is related with the appearance of Landau like levels (LL) induced by rippling and studied in related works $^{10}$. Note that from Fig. 2 we see that large values of h$_{0}$ τ are required where Ψ$_{1}$$_{,A}$$_{(}$$_{B}$$_{)}$ are the components of the spinor wave function for states near K and Ψ$_{2}$$_{,A}$$_{(}$$_{B}$$_{)}$ the ones near K $^{′}$. For AGNR the wave function should vanish at y = 0 and y = L$_{y}$ . Φ$_{A}$$_{(}$$_{B}$$_{)}$( r ) = e i ( K − k$_{0}$ ) · $^{r}$Ψ$_{1}$$_{,A}$$_{(}$$_{B}$$_{)}$ ( r ) + e i ( K $^{′}$+ k$_{0}$ ) · $^{r}$Ψ$_{2}$$_{,A}$$_{(}$$_{B}$$_{)}$ ( r ) (13)