Ground state of the holes localized in II-VI quantum dots with Gaussian potential profiles

We report on the theoretical study of the hole states in II-IV quantum dots of a spherical and ellipsoidal shape, described by a smooth potential confinement profiles, that can be modelled by a Gaussian functions in all three dimensions. The universal dependencies of the hole energy, $g$-factor and localization length on a quantum dot barrier height, as well as the ratio of effective masses of the light and heavy holes are presented for the spherical quantum dots. The splitting of the four-fold degenerate ground state into two doublets is derived for anisotropic (oblate or prolate) quantum dots. Variational calculations are combined with numerical ones in the framework of the Luttinger Hamiltonian. Constructed trial functions are optimized by comparison with the numerical results. The effective hole $g$-factor is found to be independent on the quantum dot size and barrier height and is approximated by simple universal expression depending only on the effective mass parameters. The results can be used for interpreting and analyzing experimental spectra measured in various structures with the quantum dots of different semiconductor materials.