Fundamentals of envelope function theory for electronic states and photonic modes in nanostructures

The increasing sophistication used in the fabrication of semiconductor nano structures and in the experiments performed on them requires more sophistic ated theoretical techniques than previously employed. The philosophy behind the author's exact envelope function representation method is clarified an d contrasted with that of the conventional method. The significance of glob ally slowly varying envelope functions is explained. The difference between the envelope functions that appear in the author's envelope function repre sentation and conventional envelope functions is highlighted and some erron eous statements made in the literature on the scope of envelope function me thods are corrected. A perceived conflict between the standard effective ma ss Hamiltonian and the uncertainty principle is resolved demonstrating the limited usefulness of this principle in determining effective Hamiltonians. A simple example showing how to obtain correct operator ordering in electr onic valence band Hamiltonians is worked out in detailed tutorial style. It is shown how the use of out of zone solutions to the author's approximate envelope function equations plays an essential role in their mathematically rigorous solution. In particular, a demonstration is given of the calculat ion of an approximate wavefunction for an electronic state in a one dimensi onal nanostructure with abrupt interfaces and disparate crystals using out of zone solutions alone. The author's work on the interband dipole matrix e lement for slowly varying envelope functions is extended to envelope functi ons without restriction. Exact envelope function equations are derived for multicomponent fields to emphasize that the author's method is a general on e for converting a microscopic description to a mesoscopic one, applicable to linear partial differential equations with piecewise or approximately pi ecewise periodic coefficients. As an example, the method is applied to the derivation of approximate envelope function equations from the Maxwell equa tions for photonic nanostructures.