Solution to spurious bands and spurious real solutions in the envelope-function approximation

摘要

The envelope-function approximation (EFA) can suffer from the appearance of spurious bands and below-barrier propagating (nonnormalizable) wave functions whose origin are the spurious real solutions k_S of the inverse k_z vs E(k_‖,k_z) problem in the k·p theory for coupled valence-conduction band manifolds. However, in the standard EFA, k_S solutions are required mathematically in the construction of the envelope function in order to satisfy boundary conditions (BC's). Here, it is reasoned that the effect of spurious real solutions should not depend on variations in interface positions smaller than the distance between interface atoms. Away from interfaces, coherent superpositions of spurious real solutions that satisfy BC's for interface boundaries varying on the scale of k_S~(-1) (about 1 A—i.e., less than a typical interface width or the physical precision of defining an interface) are shown to suffer a total destructive interference. In the interfacial region, the rapid oscillations of - spurious solutions carry no physical information, so they can be replaced with exponentials that decay within the oscillation period of the spurious real solutions so that spurious bands and nonnormalizability are eliminated. The resulting envelope functions remain continuous across interfaces The present solution holds for any number of bands, any order of k·p theory, and is also applicable to the EFA with generalized BC's.