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 semi-spherical 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-Schr\"{o}dinger perturbation theory is presented to obtain analytical expressions for the energy levels and wavefunctions 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 semi-circular geometry are broken for $b\neq a$, our perturbation approach allows tracking of the split states. Energy states and electronic wavefunctions with $m=0$ present the most pronounced influence on the reduction of the lens height. The analytical expressions presented here can be used to better parameterize the states in realistic self-assembled quantum dots.