Boundary-condition problem in the Kane model

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Abstract
The boundary conditions for a multicomponent effective wave function are obtained in the eight-band Kane model. The relations are established between boundary-condition matrix elements and the Kane Hamiltonian parameters in constitutive semiconductors. In general, these relations are incompatible with component-by-component continuity of the effective wave function, traditionally employed in multiband models. We show that the boundary conditions in the isotropic eight-band Kane model involve three linearly independent phenomenological parameters. Neglecting the spin-orbit interaction at the heterointerface, only two parameters are required to completely describe the matching conditions. These parameters do not depend on the energy of the charge carrier state, hence the nonparabolic regime is described in the most natural way. The boundary condition matrices are derived also for the most important approximate limits: the six-band Kane model, describing the energy spectrum of narrow-gap semiconductors, and the four-band Luttinger model, describing the valence-band top energy region in zinc-blende and diamondlike semiconductors.