Spectral gap induced by structural corrugation in armchair graphene nanoribbons arXiv:1001.1179v1 [cond-mat.mes-hall] 7 Jan 2010 I. INTRODUCTION S. Costamagna, O. Hernandez and A. Dobry Instituto de F´ısica Rosario, Consejo Nacional de Investigaciones Cient´ıficas y T´ ecnicas, Universidad Nacional de Rosario, Rosario, Argentina (Dated: October 31, 2018) We study the effects of the structural corrugation or rippling on the electronic properties of un- doped armchair graphene nanoribbons (AGNR). First, reanalyzing the single corrugated graphene layer we find that the two inequivalent Dirac points (DP), move away one from the other. Other- wise, the Fermi velocity v F decrease by increasing rippling. Regarding the AGNRs, whose metallic behavior depends on their width, we analyze in particular the case of the zero gap band-structure AGNRs. By solving the Dirac equation with the adequate boundary condition we show that due to the shifting of the DP a gap opens in the spectra. This gap scale with the square of the rate between the high and the wavelength of the deformation. We confirm this prediction by exact numerical solution of the finite width rippled AGNR. Moreover, we find that the quantum conductance, cal- culated by the non equilibrium Green’s function technique vanish when the gap open. The main conclusion of our results is that a conductance gap should appear for all undoped corrugated AGNR independent of their width. PACS numbers: 73.22.Pr, 73.23.Ad, 72.10.Fk Between the research areas of graphene, a very impor- tant one is the study of nanoribbons where the sheets are catted with a particular pattern to manage the elec- trical properties. Depending on the type of the border edges they can be either in, namely, zigzag (ZGNR) or armchair configurations. It is known that a tight bind- ing model predict that ZGNRs are always metallic while The paper is organized as follows. In Section II we provide a detailed explanation of the corrugation model adopted and we study the effect of the corrugation on the electronic spectra of a graphene sheet. Then, in Section III we study the effect of the rippling on the electronic The spectacular interest that have been raised from the recent isolation of an atomic Carbon layer, the graphene $^{1}$, is based on the unusual dynamics the elec- trons have in this material. In graphene, the electrons behaves as massless relativistic particles giving rise, for example, to a sequence of Hall plateau quite different than the one observed in 2D electrons systems confined in semiconductor heterojuntions. Also the transport properties are remarkable, the mobility of electrons in suspended graphene could be even higher than in any known semiconductor $^{2}$. Even the existing of graphene as a two-dimensional atomic crystal and its stability un- der ambient conditions is an surprising fact. Accord- ing to the Mermin-Wagner theorem, there is not should be long-range crystalline order in two dimensions at fi- nite temperature. Even more, flexible membrane em- bedded in three-dimensional space should be crumpled because of long-wavelength bending fluctuations. How- ever these fluctuations can be suppressed by anharmonic coupling between bending and stretching modes. As a result, single-crystalline membranes can exist but should be rippled $^{3}$. Indeed, ripples were observed in graphene$^{4}$. It has been proposed that they should play an important role in its electronic properties $^{5,6}$. In particular, intrinsic rippling has been proposed as one of the possible mech- anism for electron scattering to explain the variation of the resistivity with the number of charge carrier experi- mentally seen in graphene $^{7}$. AGNRs can be either metallic or semiconductor, depend- ing on their width $^{8}$. However, experiments suggest that AGNR are always insulators with a gap that scale with the inverse of its width $^{9}$. The effect of the corrugation on the electronic properties of graphene has been studied in previous works by a tight binding type model 10 and by ab-initio LDA calculation $^{11}$. The rippling induce a nonuniform gauge field whose effects has been analyzed by analogy with an applied magnetic field. In this sens a pseudo Landau levels (LL) were predicted. Regarding nanoribbons the effects of the rippling on the conduc- tivity were analyzed in a model which include also the effect of the disorder produced by charged impurities $^{12}$. Although this approach is quite realistic it can not iso- late only the effect of the rippling in order to known the contribution of each perturbation separately. Therefore, in this work, our mainly purpose is to analyze the ef- fect of the corrugation on the electronic structure and the transport properties of AGNRs. First, as a necessary previous step we study the electronic properties of cor- rugated graphene layers. We reanalyze the condition for the appearance of a flat band associated with the zero LL. We show that an strong rippling on the sheet is nec- essary to produce such a flat band. Then, we focus on the armchair border type nanoribbons. Particularly we analyze the case of the zero gap band-structure AGNRs. We show that due to rippling an spectral gap is open in otherwise conducting ribbons. We subsequent analyze the quantum conductance by the non equilibrium green’s function technique 13 (NEGF) and show that the opening of the gap manifest in a insulating behavior of the un- doped samples.