An exact formulation of the envelope function method for the determination of electronic states in semiconductor microstructures

In a recent letter (see ibid., vol.2, p.460 (1987)) the author outlined the first exact formulation of the envelope function method. In this paper the details of this new exact formulation are given. For clarity the discussion is carried out for the one-dimensional case. In previous approximate treatments coupling between envelope functions is mediated only by the action of the momentum operator on the periodic functions. It is shown that in the exact treatment, the action of the Hamiltonian operator on the periodic functions also introduces coupling between envelope functions and that this coupling is non-local in general. The nature of these new coupling terms is examined for an abrupt heterojunction. The relation between the new exact envelope function equations and previous work is discussed extensively. In particular, by making reasonable approximations to the new coupling terms, the author has managed to derive the effective mass equation for multilayer structures (particle in a box model in the quantum well case); the derivation sheds light on the vexed question of boundary conditions to be used in conjunction with this equation. The paradox of obtaining non-local envelope function equations from a local Schrodinger equation is also resolved.