Tight Binding Density of States

Here are plots of densities of states for the tight-binding Hamiltonian for ``cubic'' lattices in several dimensions. In three dimensions the energy is given by

$$\epsilon(k) = t [ 6 -2 (\cos k_x a + \cos k_y a + \cos k_z a ) ],$$ (1)

with analogous expressions for other dimensions. Note the Van Hove singularities.

Also shown is the dispersion relation in some directions for the honeycomb lattice, along with the density of states. The honeycomb lattice has two bands (because there are two atoms in the unit cell) and is unusual in that the band gap goes to zero at the corners of the Brillouin zone.