Energy-band theory is strictly applicable only to perfectly periodic crystals. This means, in particular, that it does not apply when macroscopic electric fields are present. Devices are not generally useful unless they contain such fields, so we need a formulation which can include them along with the crystal potential which produces the band structure. Such a formulation is provided by the effective-mass theorem [15,16,17]. This theorem provides a decomposition of the wavefunction into an atomic-scale part and a more slowly varying envelope function, and supplies a Schrödinger equation for the envelope function:
where is the envelope function, is the effective mass, is the energy at the edge of the nth band, and is the electrostatic potential. The effects of the band structure are incorporated in the material-dependent parameters and . The standard picture of freely moving electrons and holes with material-dependent masses follows from the effective-mass theorem via the quantum-mechanical correspondence principle.