The electronic properties of graphene sheets are often understood by starting from a simple phenomenological $\pi$-band tight-binding model. We provide a perspective on these models that is based on a study of ab initio maximally localized Wannier wave functions centered at carbon sites. Hopping processes in graphene can be separated into intersublattice contributions responsible for band dispersion near the Dirac point, and intrasublattice contributions responsible for electron-hole symmetry breaking. Both types of corrections to the simplest near-neighbor model can be experimentally relevant. We find that distant neighbor hopping parameters increase the ratio of the full $\pi$-band width to the Dirac point velocity and flatten bands along the $KM$ Brillouin-zone edge. We propose a five-parameter model which achieves a good compromise between simplicity and accuracy, and an alternate 15-parameter model achieves better accuracy with some loss of simplicity.

• Received 6 February 2013

DOI:https://doi.org/10.1103/PhysRevB.87.195450