Electronic states in a quantum lens

We present a model to find analytically the electronic states in self-assembled quantum dots with a truncated spherical cap (“lens”) geometry. A conformal analytical image is designed to map the quantum dot boundary into a dot with semispherical shape. The Hamiltonian for a carrier confined in the quantum lens is correspondingly mapped into an equivalent operator and its eigenvalues and eigenfunctions for the corresponding Dirichlet problem are analyzed. A modified Rayleigh-Schrödinger perturbation theory is presented to obtain analytical expressions for the energy levels and wave functions as a function of the spherical cap height b and radius a of the circular cross section. Calculations for a hard wall confinement potential are presented, and the effect of decreasing symmetry on the energy values and eigenfunctions of the lens-shape quantum dot is studied. As the degeneracies of a semicircular geometry are broken for $b\ne a,$ our perturbation approach allows tracking of the split states. Energy states and electronic wave functions with $m=0\mathrm{}$ present the most pronounced influence on the reduction of the lens height. The method and expressions presented here can be straightforwardly extended to deal with more general Hamiltonians, including strains and valence-band coupling effects in Group III–V and Group II–VI self-assembled quantum dots.