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

The increasing sophistication used in the fabrication of semiconductor nanostructures and in the experiments performed on them requires more sophisticated theoretical techniques than previously employed. The philosophy behind the author's exact envelope function representation method is clarified and contrasted with that of the conventional method. The significance of globally slowly varying envelope functions is explained. The difference between the envelope functions that appear in the author's envelope function representation and conventional envelope functions is highlighted and some erroneous statements made in the literature on the scope of envelope function methods are corrected. A perceived conflict between the standard effective mass 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 electronic 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 calculation of an approximate wavefunction for an electronic state in a one dimensional nanostructure with abrupt interfaces and disparate crystals using out of zone solutions alone. The author's work on the interband dipole matrix element for slowly varying envelope functions is extended to envelope functions without restriction. Exact envelope function equations are derived for multicomponent fields to emphasize that the author's method is a general one for converting a microscopic description to a mesoscopic one, applicable to linear partial differential equations with piecewise or approximately piecewise periodic coefficients. As an example, the method is applied to the derivation of approximate envelope function equations from the Maxwell equations for photonic nanostructures.