Condensed matter physics

Band theory and semiconductors

In crystals, electron states form energy bands; the band gap and Fermi level shape electrical conduction.

In crystals, electron states form energy bands; the band gap and Fermi level shape electrical conduction.

Physical model

A periodic potential creates allowed bands separated by gaps. The Fermi level is the equilibrium electron chemical potential, not necessarily an occupied state.

Semiconductors have a moderate gap; doping changes carrier density. Near graphene’s K point, two bands meet in a linear Dirac cone, gapless in the ideal model.

Quick check

Bloch’s theorem states that eigenstates in a periodic potential have the form ψₙₖ(r)=e^{ik·r}uₙₖ(r), where u has lattice periodicity. The first Brillouin zone contains independent wave vectors; Bragg reflection at its boundary can open an energy gap. The curvature of Eₙ(k) near an extremum determines effective mass, so carrier mobility need not reflect the free-electron mass. A Fermi level in a gap suggests an insulator, whereas a band crossing it usually indicates metallic behavior.

The effective-mass approximation is useful near a band extremum, where E(k) can be expanded to quadratic order. At a band maximum the electron curvature corresponds to negative effective mass; describing the missing electron as a positively charged hole with positive mass is often more convenient. Doping shifts the chemical potential and changes carrier density. For optical transitions, crystal-momentum conservation and selection rules determine whether absorption is strong.

In an intrinsic semiconductor, thermal excitation creates conduction electrons and valence-band holes in equal densities. Donor or acceptor doping shifts the Fermi level and makes one carrier type dominant. The gap and effective masses influence optical absorption and mobility.