Band filling

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.

With spin degeneracy two, one full square-zone band has density $Q^2/(2{\pi }^2)$. Thus $n_e-n_h=n-Q^2/(2{\pi }^2)$. If $Q=\alpha \sqrt{2\pi n}$, compensation requires $\alpha =\sqrt{\pi }$ and the real-space cell contains two electrons. A global gap is also necessary for an insulator.

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.

In an independent-electron crystal with two spin states and no spin splitting, each completely filled energy band contains two electrons per primitive cell. An odd electron count per primitive cell therefore cannot fill an integer number of such bands. Enlarging the actual primitive cell, breaking the relevant degeneracy or including correlated-electron physics changes the premises. Even filling is necessary in this model but not sufficient: bands can overlap.

A Kondo insulator has a bulk insulating state associated with mixing between relatively localized correlated electronic states and itinerant conduction states. A renormalized two-band picture captures a narrow hybridization gap, although an uncorrelated orbital-mixing model does not explain the formation and temperature dependence of the coherent low-energy state. Samarium hexaboride is an example.

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.

Suppose a model replaces a set of free-electron boundary energies $E^{(0)}$ by lower and upper edges $E^{(0)}-V$ and $E^{(0)}+V$, with $V>0$. If these are the relevant global band edges, the indirect band gap is $(E_{\min }^{(0)}+V)-(E_{\max }^{(0)}-V)$, giving the displayed expression. A positive direct gap $2V$ at each crossing need not be enough: a band insulator needs $2V>E_{\max }^{(0)}-E_{\min }^{(0)}$. This is a band-edge estimate, not a general three-dimensional nearly-free-electron theorem; multiple Bragg planes at a corner can require a larger coupling matrix.