We review the tight-binding model of bilayer graphene which describes four low-energy electronic bands near the corner of the first Brillouin zone. The model takes into account terms arising from nearest and next-nearest neighbour hopping within each layer, non-orthogonality of atomic orbitals, various inter-layer couplings, as well as three independent parameters that describe differences between the on-site energies of the four atoms in the unit cell. We generalize the derivation of the two-component effective Hamiltonian that describes the behaviour of chiral quasiparticles at very low energy, taking these terms into account. Then, we explain how the various terms produce features in the electronic band structure, focussing on electron-hole asymmetry and the opening of an energy gap between the conduction and valence bands.
Mucha-Kruczynski, M., McCann, E., & Fal'Ko, V. I. (2010). Electron-hole asymmetry and energy gaps in bilayer graphene. Semiconductor Science and Technology, 25(3), . https://doi.org/10.1088/0268-1242/25/3/033001