For spin-one-half electrons, one nondegenerate orbital band contains two one-particle states per primitive cell, one for each spin. A partially filled band has nearby empty states into which an electric field can move electrons and is ordinarily conducting. A completely filled band carries no net current under a small field because all crystal momenta are occupied symmetrically.
A band insulator has an integer number of completely filled energy bands, with the Fermi energy inside a positive band gap. At zero temperature, no arbitrarily low-energy charged excitation is then available.
If the minimum of the next energy band lies at or below the maximum of the nominally filled band, electrons can lower their energy by occupying the next band and leaving holes in the first. The resulting partially filled bands prevent a band-insulating state.
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