Successes and failures of the k

Abstract

The k\ensuremath{\cdot}p method combined with the envelope-function approximation is the tool most commonly used to predict electronic properties of semiconductor quantum wells and superlattices. We test this approach by comparing band energies, dispersion, and wave functions for GaAs/AlAs superlattices and quantum wells as computed directly from a pseudopotential band structure and using eight-band k\ensuremath{\cdot}p. To assure equivalent inputs, all parameters needed for the k\ensuremath{\cdot}p treatment are extracted from calculated bulk GaAs and AlAs pseudopotential band structures. Except for large exchange splittings in the in-plane dispersion for thin superlattices, present in pseudopotential calculations but absent from the k\ensuremath{\cdot}p results, we find generally good agreement for heterostructure hole bands within \ensuremath{\sim}200 meV of the GaAs valence-band maximum. There are systematic errors in band energies and dispersion for deeper hole bands (all other than hh1 and lh1) and significant qualitative and quantitative errors for the conduction bands. Errors for heterostructure conduction states which are derived from the zinc-blende \ensuremath{\Gamma} point diminish as length scales increase beyond \ensuremath{\sim}20 ML, while significant errors persist for L- and X-derived states.For bulk GaAs and AlAs, eight-band k\ensuremath{\cdot}p bands agree well with pseudopotential results very near the zinc-blende \ensuremath{\Gamma} point (where k\ensuremath{\cdot}p parameters are fit) but the first GaAs X point conduction band is \ensuremath{\simeq}26 eV too high with respect to the pseudopotential result. We show that this inadequate description of the bulk band dispersion is the principal source of k\ensuremath{\cdot}p errors in these heterostructures. A wave-function projection analysis shows that k\ensuremath{\cdot}p errors for heterostructures simply reflect corresponding errors for the bulk constituents, weighted by the amount that such bulk states participate in heterostructure states. \textcopyright{} 1996 The American Physical Society.