Pseudopotential-based multibandk⋅pmethod for∼250 000-atom nanostructure systems

Abstract

The electronic structure of quantum wells, wires, and dots is conventionally described by the envelope-function eight-band k\ensuremath{\cdot}p method (the ``standard k\ensuremath{\cdot}p model'') whereby coupling with bands other than the highest valence and lowest conduction bands is neglected. There is now accumulated evidence that coupling with other bands and a correct description of far-from-\ensuremath{\Gamma} bulk states is crucial for quantitative modeling of nanostructure. While multiband generalization of the k\ensuremath{\cdot}p exists for bulk solids, such approaches for nanostructures are rare. Starting with a pseudopotential plane-wave representation, we develop an efficient method for electronic-structure calculations of nanostructures in which (i) multiband coupling is included throughout the Brillouin zone and (ii) the underlying bulk band structure is described correctly even for far-from-\ensuremath{\Gamma} states. A previously neglected interband overlap matrix now appears in the k\ensuremath{\cdot}p formalism, permitting correct intervalley couplings. The method can be applied either using self-consistent potentials taken from ab initio calculations on prototype small systems or from the empirical pseudopotential method. Application to both short- and long-period (GaAs${)}_{\mathit{p}}$/(AlAs${)}_{\mathit{p}}$ superlattices (SL) recovers (i) the bending down (``deconfinement'') of the \ensuremath{\Gamma}\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\Gamma}) energy level of (001) SL at small periods p; (ii) the type-II--type-I crossover at p\ensuremath{\approxeq}8 SL, and (iii) the even-odd oscillation of the energies of the R\ifmmode\bar\else\textasciimacron\fi{}/X\ifmmode\bar\else\textasciimacron\fi{}(L) state of (001) SL and \ensuremath{\Gamma}\ifmmode\bar\else\textasciimacron\fi{}(L) state of (111) SL. Introducing a few justified approximations, this method can be used to calculate the eigenstates of physical interest for large nanostructures. Application to spherical GaAs quantum dots embedded in an AlAs barrier (with \ensuremath{\sim}250 000 atoms) shows a type-II--type-I crossover for a dot diameter of 70 \AA{}, with an almost zero \ensuremath{\Gamma}-X repulsion at the crossing point. Such a calculation takes less than 30 min on an IBM/6000 workstation model 590. \textcopyright{} 1996 The American Physical Society.