Cylindrically shaped zinc-blende semiconductor quantum dots do not have cylindrical symmetry: Atomistic symmetry, atomic relaxation, and piezoelectric effects

Self-assembled quantum dots are often modeled by continuum models (effective mass or k.p) that assume the symmetry of the dot to be that of its overall geometric shape. Lens-shaped or conical dots are thus assumed to have continuous cylindrical symmetry C-infinityv, whereas pyramidal dots are assumed to have C-4v symmetry. However, considering that the III-V dots are made of atoms arranged on the (relaxed) positions of a zinc-blende lattice, one would expect the highest possible symmetry in these structures to be C-2v. In this symmetry group all states are singly degenerate and there are no a priori reason to expect, e.g., the electron P states (usually the second and third electron levels of dominant orbital P character) to be degenerate. Continuum models, however, predict these states to be energetically degenerate unless an irregular shape is postulated. We show that, in fact, the true (atomistic) symmetry of the dots is revealed when the effects of (i) interfacial symmetry, (ii) atomistic strain, and (iii) piezoelectricity are taken into account. We quantify the contributions of each of these effects separately by calculating the splitting of electron P levels for different dot shapes at different levels of theory. We find that for an ideal square-based pyramidal InAs/GaAs dot the interfacial symmetry of the unrelaxed dot splits the P level by 3.9 meV, atomistic relaxation adds a splitting of 18.3 meV (zero if continuum elasticity is used to calculate strain) and piezoelectricity reduces the splitting by -8.4 meV, for a total splitting of 13.8 meV. We further show that the atomistic effects (i) and (ii) favor an orientation of the electron wave functions along the [1 (1) over bar0] direction while effect (iii) favors the [110] direction. Whereas effects (i) + (ii) prevail for a pyramidal dot, for a lens shaped dot, effect (iii) is dominant. We show that the 8-band k.p method, applied to pyramidal InAs/GaAs dots describes incorrectly the splitting and order of P levels (-9 meV instead of 14 meV splitting) and yields the orientation [110] instead of [1 (1) over bar0].