Band 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.

Consider a noninteracting band model in which all states of one energy band would be occupied and the next energy band empty. Define the global indirect band gapIf ${\Delta }_{\mathrm{ind}}>0$, this filling can have a positive bulk excitation gap. If ${\Delta }_{\mathrm{ind}}<0$, transferring an electron from the highest occupied states of the lower energy band to sufficiently low states of the upper energy band lowers the total energy. The nominal complete-band filling is not the equilibrium state: electron and electron hole populations can occupy the overlapping bands, precluding a gapped band insulator at that filling.

The equality case requires care. At ${\Delta }_{\mathrm{ind}}=0$, the excitation gap closes, but contact only at isolated wavevectors need not create finite-volume carrier pockets or a finite carrier density at zero temperature. It is therefore incorrect to infer strictly energy-lowering redistribution or partially occupied pockets merely from a band touching. Electron interactions and symmetry breaking can alter this single-particle description.

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.