2 spectra and the conduction properties of metallic AGNR. Finally, in Section IV we present the conclusions of our paper and we discuss it implicants. bor (Δ a ) we have: 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. t = t$_{0}$ + t$_{0}$ a α Δ a Δ a = √ a + ( z ( x ) − z ( x ′ )) 2 − a (2) 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: A. The model 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: z ( x ) = h$_{0}$sin ( 2 π τ x ) (1) H = − ∑ $^{l}$,$^{δ}$,σ t [ z$_{B}$ l + δ − z$_{A}$ l ]( $_{c}$† l $_{A,σ}$c$_{l}$$_{+}$$_{δ}$$_{B,σ}$ + h.c. ) (3) where l = ( l$_{x}$, l$_{y}$ ) denotes the integer coordi- nates in the hexagonal Bravais lattice and δ = { (0 , 0) , ( − 1 , 0) , ( − 1 , 1) } . 3 2 1 4 3 2 1 x H H$_{C}$ H$_{R}$ L x z(x) h$_{0}$ (b) (a) n$_{y}$ n x 3 n a FIG. 1: (a) Schematic view of an armchair grap hene nanorib- bon. Vertical lines separates the regions adopte d for the con- ductance calculation described in Sec.III B trou gh the Hamil- tonians displayed on top. The central region pos sesses n x = 4 and n y = 3, listed as indicated. (b) Side vie w in the x − z plane of the displacement of Carbon atoms due t o rippling. h 0 is the amplitude and τ the period. a is the lat tice constant of graphene. Vertical dotted lines from (a) to (b) g ives reference for the actual positions of Carbon atoms in t he corrugated nanoribbon. 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-