 Nearest-neighbour model of graphene

Take a planar honeycomb lattice with two sites $A,B$ per primitive cell and one localized atomic orbital per site. In this explicitly idealized tight-binding model, all three nearest-neighbour hopping matrix elements are $-t$, with $t>0$; there is no on-site offset or further-neighbour hopping. Ignore Electron interactions and treat physical spin as a separate degeneracy. Let the bond length be $b$, and choose the three vectors from an $A$ site to its neighbours asThe Bloch state amplitudes on $A,B$ form a two-component vector. A consistent phase convention gives the HamiltonianIts characteristic equation is $E^2=t^2|f{|}^2$, giving the two displayed energy bands. The two-component structure comes from the two sites, rather than two values of physical spin. This is a particularly small matrix model in which a crystal's geometry creates an unexpected low-energy law.

Historically, Philip Russell Wallace used a tight-binding model to investigate the electronic band structure of graphite in 1947. The material in that paper was layered graphite; the single-sheet model here isolates one simpler part of the problem.^[ref]