  Density functional theory calculations are used to investigate the electronic
structures of localized states at reconstructed armchair graphene edges. We
consider graphene nanoribbons with two different edge types and obtain the
energy band structures and charge densities of the edge states. By examining
the imaginary part of the wavevector in the forbidden energy region, we reveal
the decay behavior of the wavefunctions in graphene. The complex band
structures of graphene in the armchair and zigzag directions are presented in
both tight-binding and first-principles frameworks.
