Symmetries in Multiband Hamiltonians for Semiconductor Quantum Dots

Our current understanding of the symmetries of multiband envelope function Hamiltonians for semiconductor quantum dots and their signatures in the energy level structure and wave function shapes is reviewed. We show how symmetry can be used to block-diagonalize the Hamiltonian matrix and consequently strongly reduce the computational effort. A detailed analysis of symmetries of several different model Hamiltonians reveals that the true symmetry of square-based pyramidal quantum dots is captured if either the interface effects are taken into account or additional higher energy bands are included in the multiband Hamiltonian. This indicates that multiband envelope function methods are fully capable of capturing the true atomistic symmetry of quantum dots in contrast to some widespread beliefs. In addition, we show that translational symmetry can be artificially introduced by the numerical method used, such as the plane wave method. Plane wave method introduces artificial quantum dot replica whose charges interact with charges in the real quantum dot and create an additional strain field in the real dot. This issue can be circumvented by the introduction of proper corrections in the procedure for calculation of Coulomb integrals and strain.