Electronic structures of [110]-faceted self-assembled pyramidal InAs/GaAs quantum dots

We calculate the electronic structures of pyramidal quantum dots with supercells containing 250 000 atoms, using spin-orbit-coupled, nonlocal, empirical pseudopotentials. We compare the results with previous theoretical calculations. Our calculation circumvents the approximations underlying the conventional effective-mass approach: we describe the potential, the strain and the wave functions using atomistic rather than continuum models. The potential is given by a superposition of screened atomic pseudopotentials, the strain is obtained from minimizing the atomistic strain energy, and the wave function is expanded using a plane-wave basis set. We find the following. (1) The conduction bands are formed essentially from single envelope functions, so they can be classified according to the nodal structure as $s,\hspace{0.5em}p,$ and d. However, due to strong multiband coupling, most notably light hole with heavy hole, the valence states cannot be classified in the language of single-band envelope functions. In fact, the hole states have no nodal planes. (2) There is a strong anisotropy in the polarization of the lowest valence state to conduction state optical transition. This is in contrast to the eight band $\mathbf{k}\cdot \mathbf{p}$ model, which finds essentially zero anisotropy. (3) There are at least four bound electron states for a 113-Å-based quantum dot. This number of bound states is larger than that found in eight band $\mathbf{k}\cdot \mathbf{p}$ calculations. (4) Since our atomistic description retains the correct $C_{2v}$ symmetry of a square-based pyramid made of zinc-blende solids, we find that the otherwise degenerate p states are split by about 25 meV. This splitting is underestimated in the eight-band $\mathbf{k}\cdot \mathbf{p}$ calculation.