 Band filling

Band filling counts occupied one-particle states in an energy band. With spin degeneracy two, a crystal of $N$ primitive unit cells has $2N$ states in each orbital-derived energy band, as in state counting in a Bloch band. Full occupation therefore means two electrons per primitive cell in that band.

In the group velocity approximation, a fully occupied band contributes current densityThe integral vanishes because $E$ is periodic across opposite faces of the Brillouin zone; inversion symmetry is unnecessary.^[ref] A partially occupied band leaves nearby empty states available for ordinary longitudinal conduction. A filled set of bands with a positive band gap instead gives a band insulator at zero temperature.

The vanishing group velocity term does not exclude every electrical response. A filled Chern insulator band can have a transverse quantum Hall effect governed by Berry curvature.^[ref]