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spectra and the conduction properties of metallic AGNR. Finally, in Section IV we present the conclusions of our paper and we discuss it implicants. 

## II. THE EFFECT OF THE RIPPLING: THE GRAPHENE SHEET 

Let us start by studying the effects of the corruga- tion on the electronic states for the case of 2D graphene layer. This question was examined in a series of previ- ous papers $^{10,11}$. As it will be the starting point of our study of a corrugated nanoribbons in the next Section, we reanalyze this problem in the present Section. This will be presented after a detailed description of the model adopted. 

We describe the electronic properties of graphene layer by means of a tight-binding model with one π -orbital for each Carbon atom and nearest-neighbor hopping between them. Carbon atoms arrange in a honeycomb lattice. It is not a Bravais lattice but can be constructed from the hexagonal lattice by putting a basis of two atoms. The hexagonal lattice could be constructed as R l = l$_{x}$ a 1 + l$_{y}$ a 2 with l$_{x}$ and l$_{y}$ integers, a 1 = a $_{2}$(3 $_{,}$√$_{3) and}$ a 2 = a $_{2}$(3 , − √$_{3) are the primitive vectors. The basis is given}$ by the atom A sited at R l and B at R l + a $_{2}$(1 $_{,}$√$_{3).}$ a ∼ 1 . 41 ˚ A is the lattice constant and hereafter will be taken as unit of distances. 

The corrugation was included as a sinusoidal function which module the z coordinate of each lattice site. The x and y coordinates remain into its values in the hon- eycomb lattice. For simplicity, we assume that rippling only depend on the x direction (see fig. 1 for the case of a nanoribbon). Then, the adopted z ( x ) function reads: 

where h$_{0}$ is the amplitude and τ the period of the rip- pling. This simple model for the corrugation has been used in previous works $^{10,11}$. As a superposition of func- tions of the type (1) with different wave vectors and dif- ferent amplitudes could generate a quite general corruga- tion, our study should be taken as a starting point for the effect of a general corrugated function. Evermore as we point to the study of nanoribbons which by definition are much longer than wider the one-dimensional character of the rippling function is justified. 

The modulated structure induce an spacial variation of the hopping parameter t , being now dependent on the distance between the atoms. Developing t up to first or- der in the perturbed distance between the nearest neigh- 

## A. The model 

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bor (Δ a ) we have: 

where t$_{0}$ ≈ 2 . 66 eV is the hopping parameter of the un- deformed graphene which is taken as a unit of energy in the present work. α = ∂ log t ∂ log a ∼ 2, whose value is taken from Ref. (2). The rippling also produce a bending of the p-orbitals. However it has been shown that the rela- tive change of the hopping due to the bending is weaker than the one due to the change of the bond length $^{12}$. For simplicity, we neglect this effect in the present paper. 

The tight binding Hamiltonian in the presence of the corrugation becomes: 

<img src='content_image/408645.jpg'>

where l = ( l$_{x}$, l$_{y}$ ) denotes the integer coordi- nates in the hexagonal Bravais lattice and δ = { (0 , 0) , ( − 1 , 0) , ( − 1 , 1) } . 

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