We present an analytic-numerical framework for carriers and excitons in lens-shaped (semi-ellipsoidal) quantum dots within the effective-mass and envelope-function approximations, assuming hard-wall confinement. Exploiting the geometry, we use an adiabatic separation of fast (axial) and slow (planar) motion to obtain closed-form single-particle states and energies. The exciton binding energy is evaluated numerically in first-order perturbation theory using the analytic envelopes. Interband absorption follows from bright-state selection rules, and photoluminescence is obtained from absorption via the van Roosbroeck–Shockley relation with Lorentzian broadening. It is shown that single-particle confinement energies are much more sensitive to the dot height than to the lateral size, states with higher axial quantum number lie well above the lowest branch, the Coulomb binding decreases with increasing size, exhibiting comparable fractional sensitivity to both the axial and planar semi-axes and grouping primarily by radial quantum number. The framework yields compact formulas, transparent scaling trends, and interpretable spectra for the design and analysis of lens-shaped quantum dots.
Gevorgyan et al. (Thu,) studied this question.