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

The electronic structure of quantum wells, wires, and dots is conventionally described by the envelope-function eight-band k⋅p method (the ‘‘standard k⋅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-Γ bulk states is crucial for quantitative modeling of nanostructure. While multiband generalization of the k⋅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-Γ states. A previously neglected interband overlap matrix now appears in the k⋅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 Γ¯(Γ) energy level of (001) SL at small periods p; (ii) the type-II–type-I crossover at p≊8 SL, and (iii) the even-odd oscillation of the energies of the R¯/X¯(L) state of (001) SL and Γ¯(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 ∼250 000 atoms) shows a type-II–type-I crossover for a dot diameter of 70 Å, with an almost zero Γ-X repulsion at the crossing point. Such a calculation takes less than 30 min on an IBM/6000 workstation model 590. © 1996 The American Physical Society.