Band structure methods

Band structure methods calculate the allowed electron energies and electronic states in periodic crystals, usually by solving approximate forms of the Schrödinger equation.

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Bloch theorem: Bloch’s theorem states that electron wavefunctions in a periodic potential are plane waves multiplied by functions with the crystal’s periodicity. It converts a crystal’s translational symmetry into the mathematical structure used by most band calculations.
Brillouin zone: The Brillouin zone is the primitive cell of a crystal’s reciprocal lattice. Band energies are sampled and plotted across this fundamental region of wavevector space.
Many-body perturbation theory: Many-body perturbation theory calculates interacting-particle effects through expansions involving the system’s response and self-energy. Methods such as GW improve quasiparticle energies beyond standard density-functional approximations.
Semiconductor: A semiconductor is a material whose electrical conductivity lies between that of a conductor and an insulator and can be controlled. Band gaps and band-edge states help predict its optical and electrical behavior.
Band-gap problem: The band-gap problem is the systematic underestimation of many semiconductor and insulator gaps by common density-functional approximations. It exposes a prominent accuracy limitation in widely used band calculations.
Density functional theory: Density functional theory approximates a many-electron system through its electron density and an effective one-electron problem. It supplies the electronic structure framework behind many practical calculations of crystal bands.
Crystal momentum: Crystal momentum is the wavevector label associated with translational symmetry in a periodic solid. It is the horizontal coordinate used to describe how each band energy varies through reciprocal space.
GW approximation: The GW approximation estimates electronic quasiparticle energies using a self-energy built from the Green’s function and screened Coulomb interaction. It often corrects the band-gap errors of semilocal density functional theory.
Topological insulator: A topological insulator has an insulating bulk and conducting boundary states protected by its band topology and symmetries. Band calculations identify the inversions and topological indices associated with these boundary states.
Strongly correlated material: A strongly correlated material is one whose electronic properties depend on interactions that cannot be treated as small corrections. Simple one-electron band pictures can fail when electron interactions dominate its behavior.