Symmetry of $k∙p$ Hamiltonian in pyramidal $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots: Application to the calculation of electronic structure

Symmetry of $k ∙ p$ Hamiltonian in pyramidal $In As ∕ Ga As$ quantum dots: Application to the calculation of electronic structure

Nenad Vukmirović, Dragan Indjin, Vladimir D. Jovanović, Zoran Ikonić, and Paul Harrison

Phys. Rev. B 72, 075356 – Published 24 August 2005

Abstract

A method for the calculation of the electronic structure of pyramidal self-assembled $In As ∕ Ga As$ quantum dots is presented. The method is based on exploiting the ${\bar{C}}_{4}$ symmetry of the 8-band $k ∙ p$ Hamiltonian with the strain taken into account via the continuum mechanical model. The operators representing symmetry group elements were represented in the plane wave basis and the group projectors were used to find the symmetry adapted basis in which the corresponding Hamiltonian matrix is block diagonal with four blocks of approximately equal size. The quantum number of total quasiangular momentum is introduced and the states are classified according to its value. Selection rules for interaction with electromagnetic field in the dipole approximation are derived. The method was applied to calculate electron and hole quasibound states in a periodic array of vertically stacked pyramidal self-assembled $In As ∕ Ga As$ quantum dots for different values of the distance between the dots and external axial magnetic field. As the distance between the dots in an array is varied, an interesting effect of simultaneous change of ground hole state symmetry, type, and the sign of miniband effective mass is predicted. This effect is explained in terms of the change of biaxial strain. It is also found that the magnetic field splitting of Kramer’s double degenerate states is most prominent for the first and second excited state in the conduction band and that the magnetic field can both separate otherwise overlapping minibands and concatenate otherwise nonoverlapping minibands.