The Kohn-Luttinger envelope-function method is generalized to the case of heterostructures with atomically sharp heterojunctions based on lattice-matched layers of related semiconductors with zinc-blende symmetry. For electron states near the Γ point in (001) heterostructures the single-band effective-mass equation is derived, taking into account both the spatial dependence of the effective mass and effects associated with the atomically sharp heterojunctions. A small parameter is identified, in powers of which it is possible to classify the various contributions to this equation. For hole states only the main contributions to the effective Hamiltonian, due to the sharpness of the heterojunctions, are taken into account. An expression is derived for the parameter governing mixing of states of heavy and light holes at the center of the 2D Brillouin zone.

The Kohn-Luttinger representation and canonical transformation of the all-band k p Hamiltonian have been used to derive an envelope-function equation with position-dependent effective mass for the conduction band of a structure with a graded heterojunction between related lattice-matched zinc-blende symmetry semiconductors. This equation should also include non-parabolicity correction terms and an interface spin-split-off term. Several forms of the effective-mass equation have been obtained that are at first sight different, but equivalent to one another. The boundary conditions for an envelope function at the heterointerface (in the case when the heterojunction might be treated as an abrupt one) are shown to depend on how the envelope functions are defined. Attention has also been paid to the problem of the influence of the position-dependent effective mass on the coefficient of optical absorption for transitions between conduction subbands in a quantum well. The absorption is shown to be negligibly small for the case of normal incidence of light on the quantum well.

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The k · p method is used to analyze the problem of intervalley Γ-X z interaction of conduction band states in (001) lattice-matched III-V semiconductor heterostructures. A convenient basis for expansion of the wave function is systematically selected and a multi-band system of equations is derived for the envelope functions which is then reduced to a system of three equations for three valleys (Γ 1 , X 1 , and X 3 ) by using a unitary transformation. Intervalley Γ-X z mixing is described by short-range potentials localized at heterojunctions. The expressions for the parameters determining Γ-X z mixing strength explicitly contain the chemical composition profile of the structure, so mixing is naturally stronger for abrupt heterojunctions than for structures with a continuously varying chemical composition. It is shown that the direct Γ 1 -X 1 interaction of comparable strength to Γ 1 -X 3 interaction exists. This must be taken into account when interpreting tunnel and optical experiments since X 1 valley is substantially lower in energy than X 3 valley.