Connection of envelope functions at semiconductor heterointerfaces. II. Mixings of Γ and X valleys in GaAs/${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As

The effective-mass approximation is extended so as to take into account mixings between Γ and X conduction-band valleys at heterointerfaces consisting of ${\mathrm{Al}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As with different x’s. Effects of the mixings are included by boundary conditions expressed in terms of a 6×6 interface matrix that gives a set of linear relations among envelope functions and their derivatives at the interface. The interface matrix is calculated in an ${\mathrm{sps}}^{\mathrm{*}}$ tight-binding model with only nearest-neighbor transfer integrals. The intervalley couplings, although not so large, can be represented by two off-diagonal elements of the interface matrix. The usefulness of the present formulation is demonstrated by actual calculations of transmissions and reflections across a single interface and a single barrier structure and also of energy levels in multiple quantum wells. The interface-matrix formalism is extended to treat tunnelings across high barriers to which the envelope-function approximation is not applicable.