Energy band

An energy band is a continuous branch $E_j(k)$ of the dispersion relation of a periodic quantum system.

In one dimension the band effective mass is defined by $1/m^{\ast }(k)=E^{\prime \prime }(k)/{\hslash }^2$. Together with $v=E^{\prime }(k)/\hslash$ and $\hslash \dot{k}=F$, this gives $\dot{v}=F/m^{\ast }$. Negative band curvature produces negative effective mass for the electron wave packet; near a filled-band maximum the equivalent missing-electron description uses positively responding holes. In several dimensions the inverse mass is the Hessian ${\partial }_i{\partial }_{j}E/{\hslash }^2$.

An allowed energy band is an interval of energies supporting extended Bloch states. Adjacent bands may be separated by a band gap.

A band gap is an energy interval between adjacent energy bands containing no allowed bulk state.

An avoided crossing occurs when two parameter-dependent eigenvalues approach one another but an interaction prevents equality. Near a two-level crossing, a nonzero off-diagonal matrix element opens a positive minimum gap.

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.