  Theoretical progress in graphene physics has largely relied on the
application of a simple nearest-neighbor tight-binding model capable of
predicting many of the electronic properties of this material. However,
important features that include electron-hole asymmetry and the detailed
electronic bands of basic graphene nanostructures (e.g., nanoribbons with
different edge terminations) are beyond the capability of such simple model.
Here we show that a similarly simple plane-wave solution for the one-electron
states of an atom-based two-dimensional potential landscape, defined by a
single fitting parameter (the scattering potential), performs better than the
standard tight-binding model, and levels to density-functional theory in
correctly reproducing the detailed band structure of a variety of graphene
nanostructures. In particular, our approach identifies the three hierarchies of
nonmetallic armchair nanoribbons, as well as the doubly-degenerate flat bands
of free-standing zigzag nanoribbons with their energy splitting produced by
symmetry breaking. The present simple plane-wave approach holds great potential
for gaining insight into the electronic states and the electro-optical
properties of graphene nanostructures and other two-dimensional materials with
intact or gapped Dirac-like dispersions.
