PHYSICAL REVIEW B 84, 165425 (2011) Edge and passivation effects in armchair graphene nanoribbons Hassan Raza Department of Electrical and Computer Engineering, University of Iowa, Iowa City, Iowa 52242, USA (Received 6 July 2011; revised manuscript received 2 September 2011; published 14 October 2011) We report the electronic structure of armchair graphene nanoribbons (acGNRs) with periodic edge roughness. For the unpassivated edges, due to the dangling-bond states, we find, for both pristine and periodically rough acGNRs, the width dependence does not exhibit the hyperbolic trend in the band gap, which is otherwise a characteristic of the passivated acGNRs. The effective mass also does not follow the hyperbolic dependence for the passivated acGNRs with edge roughness. Furthermore, the distinct nonpristine edges, discussed in this paper, have a band-gap opening with and without passivation, since the probability of crossing the quantized wave vector and one of the two Dirac points becomes quite small due to the mixed boundary conditions. We finally report the electronic-structure modulation by applying an external electric field to decrease the band gap to a few meV. DOI: 10.1103/PhysRevB.84.165425 PACS number(s): 73.22.-f, 73.20.-r, 72.80.Rj I. INTRODUCTION Graphene is a unique monolayer membrane of carbon atoms with excellent electronic properties. 1–5 It has a zero band gap at the two Dirac points, which are related by the time-reversal symmetry, making it a semimetallic material system. This turns out to be a severe bottleneck for semiconducting applications. The solution to overcome this problem lies in the nanos- tructured graphene. One such possibility is a carbon nanotube (CNT), 2 which can conceptually be thought of as taking a one-dimensional nanoribbon out of graphene and rolling the edges over itself to form a tube. The second possibility is, of course, that of a graphene nanoribbon (GNR). In this context, the two most important GNRs (Refs. 6–14) are the armchair GNR (acGNR) and the zigzag GNR (zzGNR) with armchair and zigzag edges, respectively. When an acGNR is rolled, the resulting CNT has zigzag open ends—hence, the name zigzag CNT and vice versa. The electronic structure of GNRs or CNTs can be thought of as a one-dimensional subset of the two-dimensional graphene band structure depending upon the boundary conditions, i.e., periodic or rigid for CNTs or GNRs, respectively. To have an all-semiconducting material, the quantized wave vector, due to these boundary conditions, should not pass through the two Dirac points or their vicinity over the two-dimensional Brillouin zone of graphene. Unfortunately, this cannot be avoided in nature. With the periodic boundary conditions for CNTs, statistically one-third of the CNT arrangements lead to the crossing of the quantized wave vector with either of the Dirac points, resulting in semimetallic (or, in more popular terms, metallic) behavior. 2 This leads to a poor CNT yield for semiconducting applications. The situation for GNRs is no different. In fact, it is more disappointing, at least, theoretically. For example, the stable configuration of zzGNRs with closed benzene rings has a zero band gap. 6,7 Additionally, the band structure has localized edge states around the chemical potential at the Brillouin-zone boundary. On the other hand, within a p z -orbital tight-binding theory, one-third acGNRs are semimetallic 2,14 depending on the width of the nanoribbon. Although, one finds that these semimetallic acGNRs actually have a small band gap of a few tens of meV by using a more sophisticated theory. 14–16 These still, however, are not useful as a transistor material for logic and memory applications. In short, GNRs also suffer from the same bottleneck as that of CNTs, unless an exact control of the ribbon width or the nanotube chirality to the atomic precision is made possible. Fortunately, the GNRs have an additional degree of freedom of edge structure that CNTs do not possess. The simplest situ- ation of nonpristine edges could be that of the edge roughness, essentially by perturbing the boundary conditions of otherwise pristine edges. We have proposed earlier that one may obtain an all-semiconducting material by introducing periodic edge roughness in acGNRs. In this paper, our objective is to study the electronic structure of these mixed acGNRs with and without passivated edges. The nanoribbon fabrication with pristine and oscillatory edges would be challenging, and the current state of the art is far from achieving such an edge control. 18–28 However, a theoretical study of such ribbons is still interesting for exploring novel properties. Reference 29 discusses similar roughness effects in zzGNRs. Earlier studies have focused on the role of edge roughness in mobility degradation and mobility edge structure. 30–38 Detailed device analysis, in the presence of edge roughness, has also been performed. 39–41 In this paper, apart from studying the electronic structure of one such class of oscillatory edged acGNRs with and without hydrogen passivation, we also present a detailed analysis of the electronic-structure modulation of the band gap due to an external electric field. Such electric fields may be due to extrinsic sources by applying a gate voltage or intrinsic ones due to stray charges, charged impurities, or work function differences in the surrounding materials. For the electronic structure, we use a semiempirical extended H¨ uckel theory (EHT) model with nonorthogonal basis. EHT has successfully been applied to a wide range of nanoscale materials and systems. 42,43 The detailed model for graphene nanostructures has been reported in Ref. 10. In a previous paper, we show that structural relaxation at the edges has a minimal effect on the electronic spectrum. 14 Thus, we ignore its effect in this paper. 165425-1 1098-0121/2011/84(16)/165425(5) ©2011 American Physical Society