Connection rules for envelope functions at semiconductor-heterostructure interfaces

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Abstract
Connection rules for envelope functions are formulated in terms of a transfer matrix, the elements of which are calculated using a scattering-matrix approach based on empirical pseudopotentials. We propose a 6×6 transfer matrix for the case when a four-band k⋅p model is used, contrary to the conventional 8×8 matrix. The transfer-matrix elements are calculated for a GaAs/AlAs interface, in which case the transfer matrix appears to be almost diagonal. The envelope functions are to a good approximation continuous at the GaAs/AlAs interface, whereas the first derivatives appear to be more or less discontinuous at the interfaces. However, we find no evidence for connection rules for the first derivatives of envelope functions involving effective-mass ratios, which is at variance with the rules as commonly used in the literature. For the InAs/GaSb interface the off-diagonal elements of the transfer matrix appear to be nonzero, whereas the diagonal elements are negligibly small.