We present the analytical solution of the wavefunction and energy dispersion of armchair graphene nanoribbons (GNRs) based on the tight-binding approximation. By imposing hard-wall boundary condition, we find that the wavevector in the confined direction is discretized. This discrete wavevector serves as the index of different subbands. Our analytical solutions of wavefunction and associated energy dispersion reproduce the numerical tight-binding results and the solutions based on the k · p approximation. In addition, we also find that all armchair GNRs with edge deformation have energy gaps, which agrees with recently reported first-principles calculations.

We report a first-principles study on the electronic structures of deformed graphene nanoribbons (GNRs). Our theoretical results show that the electronic properties of zigzag GNRs are not sensitive to uniaxial strain, while the energy gap modification of armchair GNRs (AGNRs) as a function of uniaxial strain displays a nonmonotonic relationship with a zigzag pattern. The subband spacings and spatial distributions of the AGNRs can be tuned by applying an external strain. Scanning tunneling microscopy dI/dV maps can be used to characterize the nature of the strain states, compressive or tensile, of AGNRs. In addition, we find that the nearest neighbor hopping integrals between pi-orbitals of carbon atoms are responsible for energy gap modification under uniaxial strain based on our tight binding approximation simulations.

Based on the tight-binding model, we investigate band structures of graphene nanohole ͑GNH͒ superlattices as a function of NH size and density. One common origin of band gaps for GNH superlattices with NHs of either armchair or zigzag edges is the quantum-confinement effect due to the periodic potential introduced by the NHs, which turns the semimetallic sheet into a direct-gap semiconductor. Additional band gaps also open for GNH superlattices with NHs of zigzag edges in a ferromagnetic ground state, arising from the staggered sublattice potential on the zigzag edges due to edge magnetization. Our calculations reveal a generic scaling relation that both types of band gaps increase linearly with the product of NH size and density.The electronic structure of nanoscale carbon materials such as fullerenes and carbon nanotubes has been intensively studied during the past two decades. 1 Among the carbon materials, graphene is a rapidly rising star showing a wealth of interesting unconventional electronic properties and a broad range of potential applications. 2 In addition to twodimensional ͑2D͒ graphene sheet, much attention has also been drawn to graphene-based low-dimensional nanostructures such as zero-dimensional nanodots 3-5 and nanoholes ͑antidots͒, 6-9 one-dimensional nanoribbons, 10-16 as well as 2D graphene nanohole superlattices. 6-8 These structures exhibit unique electronic structures different from the graphene sheet itself and hence offering other application potentials.There are two key effects in determining the electronic properties of these low-dimensional graphene nanostructures: the size effect and edge effect, the former induces quantum confinement converting the semimetal graphene into semiconductors and the latter induces edge magnetism as well as opening a small band gap. Both these effects have been extensively studied in graphene nanoribbons and show strong size and shape dependences. [10][11][12][13][14][15][16] For example, it has been shown that the band gap of nanoribbons scales inversely with their width 10-12 and nanoribbons with zigzag edges may have either a ferromagnetic ͑FM͒ or antiferromagnetic ͑AFM͒ ground state depending on their edge orientations. 8,10,16 On the other hand, less studies have been done for 2D GNH superlattices. Pedersen et al. 7 showed a linear scaling of the band gaps for circular GNH superlattice. However, it has been recently observed that graphene edges are generally straight stabilized with either armchair or zigzag edges. 17,18 Motivated by these new experiments, we have carried out tight-binding ͑TB͒ calculations and theoretical analyses to investigate the relationship between the band gap and the nanohole size/density for GNH superlattices with armchair and zigzag edges.Our numerical calculations show that both the armchairand zigzag-edged triangular GNH ͑TGNH͒ and rhombus GNH ͑RGNH͒ superlattices have nonzero and direct band gaps. The band gaps of armchair GNH superlattice originate solely from the quantum-confinement effect of 2D periodic potential cre...