The assumptions of conventional effective-mass theory, especially the one of continuity of the envelope function at an abrupt interface, are reviewed critically so that the need for a fresh approach becomes apparent. A new envelope-function method, developed by the author over the past few years, is reviewed. This new method is based on both a generalization and a novel application to microstructures of the Luttinger-Kohn envelope-function expansion. The differences between this new method and the conventional envelope-function method are emphasized. An alternative derivation of the new envelope-function equations, which are exact, to that already published is provided. A new and improved derivation of the author's effective-mass equation is given, in which the differences in the zone-centre eigenstates of the constituent crystals are taken into account. This derivation is valid for abrupt interfaces and relies only on the slow variation of the envelope function(s). Unambiguous boundary conditions that automatically conserve probability current are derived. The cause of the kinks in the conventional effective-mass envelope function, at abrupt effective-mass changes, is identified. The formal results are extensively illustrated with a numerical example. A plot of the author's exact envelope function shows that it has a soft kink at an effective-mass discontinuity. This soft kink is also seen in the exact wavefunction. It is this feature which is approximated in conventional models by the effective-mass-related kink. The reasons why conventional effective-mass theory works so well become dear. The distinction between Luttinger-Kohn and Wannier-Slater envelope functions is highlighted. It is demonstrated that the author's generalization and new application of the Luttinger-Kohn envelope-function expansion can also be carried out for the Wannier-Slater case. However, it is found that the derivation of effective-mass equations for the generalized Wannier-Slater envelope functions is not as straightforward as for the Luttinger-Kohn case. Some problems uncovered recently in trying to extend effective-mass theory to the non-parabolic regime are resolved.
