Exact symmetries of electron Bloch states and optical selection rules in [001] GaAs/AlAs quantum wells and superlattices

We determined the exact symmetries of conduction and valence Bloch states in type-I and type-II [001] $\left(\mathrm{GaAs}{)}_m\right(\mathrm{AlAs}{)}_n$ superlattices at the Γ point and at some other symmetry points of the Brillouin zone of the superlattices and derived optical selection rules. Contrary to a result widely accepted in the envelope-function approximation (EFA), $p_z$ atomic orbitals cannot mix with $p_x$ and $p_y$ orbitals to build Bloch states. The phonon-assisted transitions involving the Γ point as an initial or final state are allowed both without and with taking into account the spin-orbit interaction whatever are the symmetries of the initial and final states. The electron band structure of the superlattices is discussed. Within the domain of validity of EFA (i.e., for not too small values of m and n), a detailed analysis of the Bloch-state symmetry and selection rules is provided on imposing invariance of the superlattice structure under the change of z to -z (the ${\sigma }_z$ symmetry operation). It is shown that optical transitions between the conduction states arising from the Γ states of GaAs on one hand and the conduction states arising from the X states of AlAs on the other hand can be allowed from spin-orbit coupling only. The correspondence is provided between the symmetry of a Bloch state and the parity with respect to ${\sigma }_z$ of its associated envelope function. The effect of an electric field parallel to the growth axis is discussed. Quantum wells do not differ from superlattices with regard to Bloch-state and envelope-function symmetries or optical selection rules. All the above results are still valid for any pseudomorphic superlattice or quantum well made of two binary compounds with zinc-blend structure and identical cations or anions, such as, for example, in the GaN/AlN system.