Exact symmetries of electron states and optical selection rules in wurtzite-based nanostructures

The crystal structure of wurtzite-based (hexagonal) quantum wells (QW’s), such as $(\mathrm{GaN}{)}_m/\mathrm{AlN}$ ones for example, is found to be described by the layer group $P3m1\mathrm{}$ (DG69) and does not depend on the number of atomic monolayers constituting the QW whereas the symmetry of wurtzite-based superlattices (SL’s), such as $(\mathrm{GaN}{)}_m{\mathrm{}\left(\mathrm{AlN}\right)\mathrm{}}_n$ ones for example, has been previously shown to be described by the space groups $C_{3v}^1$ or $C_{6v}^4$ depending on $m+n$ is even or odd. The $P3m1\mathrm{}$ (DG69) group is a factor group of the $C_{3v}^1$ group, the latter being the product of the $P3m1\mathrm{}$ group and the subgroup containing the translations along the z axis. Basing on these symmetries, we have determined the exact symmetries of Bloch states at the Γ and other symmetry points of the Brillouin zones of QW’s and SL’s and derived optical selection rules for carriers and excitons. The latters present large Rydberg values. We have shown that the built-in electric field, directed along the z axis due to the symmetry, breaks the translational invariance of the SL’s along this direction reducing their symmetry to that of a single QW. We have established that when one (several) phonon(s) is (are) involved in a radiative process, it is always possible to connect any initial state to any final one. The energy of the emitted photon depends on the nature of the phonon(s) if several channels are allowed for the transition. The symmetry of electron states in very thin QW’s and short-period SL’s is shown to be determined by their exact symmetry rather than that implied in envelope function approximation (EFA). Within the domain of validity of the EFA, i.e., for not too thin layers, a detailed analysis of the Bloch-state symmetry is performed on imposing the invariance of the structure under the change of z to $-z$ (the ${\sigma }_z$ symmetry operation). The correspondence is established between the symmetry of a Bloch state and the parity with respect to ${\sigma }_z$ of its associated envelope function. It is shown that EFA artificially induces a splitting of energy levels and appearance of new dark excitons.