Modeling of strained quantum wires using eight-band k⋅p theory

We have calculated numerically the one-dimensional band structure and densities of states of a V-shaped ${\mathrm{In}}_{0.2}$${\mathrm{Ga}}_{0.8}$As/${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As single quantum wire using eight-band k⋅p theory. A finite-difference scheme is used for the calculations. The model includes the realistic orientation, shape, material composition, strain distribution, and piezoelectric charging of the wire. We find a dominant impact of the piezoelectric potential on the band structure and a marked spin splitting of the valence bands. Also, the conduction band is strongly nonparabolic. We propose an efficient procedure to calculate interior eigenvectors from Hamiltonians including conduction-band–valence-band interactions. This algorithm is 20–90 times faster than the best prevailing method and also applies to other Hamiltonians for the modeling of nanostructures, including those occurring in tight-binding or pseudopotential theory.