Tight-binding description of graphene

We investigate the tight-binding approximation for the dispersion of the $\ensuremath{\pi}$ and ${\ensuremath{\pi}}^{*}$ electronic bands in graphene and carbon nanotubes. The nearest-neighbor tight-binding approximation with a fixed ${\ensuremath{\gamma}}_{0}$ applies only to a very limited range of wave vectors. We derive an analytic expression for the tight-binding dispersion including up to third-nearest neighbors. Interaction with more distant neighbors qualitatively improves the tight-binding picture, as we show for graphene and three selected carbon nanotubes.