Development of an eight-band theory for quantum dot heterostructures

We derive a nonsymmetrized eight-band effective-mass Hamiltonian for quantum dot heterostructures (QDH's) in Burt's envelope-function representation. The $8\ifmmode\times\else\texttimes\fi{}8$ radial Hamiltonian and the boundary conditions for the Schr\"odinger equation are obtained for spherical QDH's. Boundary conditions for symmetrized and nonsymmetrized radial Hamiltonians are compared with each other and with connection rules that are commonly used to match the wave functions found from the bulk $\mathit{k}\ensuremath{\cdot}\mathit{p}$ Hamiltonians of two adjacent materials. Electron and hole energy spectra in three spherical QDH's, HgS/CdS, InAs/GaAs, and GaAs/AlAs, are calculated as a function of the quantum dot radius within the approximate symmetrized and exact nonsymmetrized $8\ifmmode\times\else\texttimes\fi{}8$ models. The parameters of dissymmetry are shown to influence the energy levels and the wave functions of an electron and a hole and, consequently, the energies of both intraband and interband transitions.