Electronic band structure

Electronic band structure describes the allowed electron energies in a material as functions of crystal wavevector, with forbidden ranges separating some bands. It helps determine a material’s electrical, optical, and magnetic behavior.

Bloch's theorem: A theorem stating that electron wavefunctions in a periodic potential are plane waves multiplied by functions with the lattice periodicity. It explains why crystal electrons can be labeled by wavevectors within the Brillouin zone.
Crystal lattice: A periodic arrangement of points representing the translational symmetry of a crystalline solid. Its repeating structure underlies the periodic potential used to calculate bands.
Semiconductor: A material whose electrical conductivity lies between that of a conductor and an insulator and can be controlled by doping or fields. Its valence and conduction bands make it useful for electronic switching.
Free electron model: A model of electrons moving through a solid without experiencing a periodic crystal potential. Unlike crystal band theory, it does not naturally produce gaps from lattice periodicity.
Brillouin zone: A primitive cell in reciprocal space that contains the distinct wavevectors of a periodic crystal. Band energies are plotted across this reciprocal-space region.
Reciprocal lattice: A lattice in reciprocal space that represents the periodicity of a crystal in wavevector space. It supplies the natural coordinates for describing crystal electron wavevectors.
Direct and indirect band gaps: Band gaps distinguished by whether the band-edge states share the same crystal wavevector. This distinction predicts how efficiently a semiconductor emits or absorbs light.
Nearly free electron model: A model treating electrons as nearly free particles weakly perturbed by a periodic crystal potential. It shows how a weak lattice potential opens gaps at Brillouin-zone boundaries.
Periodic potential: A potential energy landscape that repeats with the translation symmetry of a crystal lattice. Its periodicity splits free-electron energies into bands and opens gaps.
Fermi energy: The chemical potential of electrons at absolute zero, marking the highest occupied energy in a system. Its position relative to bands and gaps helps identify electronic behavior.

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