Electronic Energy Levels in Quantum Dots with Weakly Modulated Spherical-Harmonic Surfaces

The precise control of quantum dot (QD) geometry is crucial for tailoring their optoelectronic properties, yet an exact analytical description of non-spherical nanoparticles remains mathematically challenging. In this work, we present a comprehensive theoretical framework to calculate the electronic energy spectra of QDs with arbitrary weakly modulated spherical-harmonic surfaces. Utilizing the perturbation theory up to the second order, we derive analytical expressions for the energy corrections. The important feature of our approach is the explicit incorporation of Wigner 3j-symbols, which establishes selection rules based on the spatial and axial symmetries of the surface corrugations. The model is validated through numerical calculations for a GaAs QD under various geometric configurations, including spheroidal, pear-shaped, and multi-harmonic shapes. The proposed analytical tool offers deep physical insights into the quantum confinement effect in realistically corrugated nanocrystals, providing a computationally efficient platform for the shape-engineering of semiconductor nanomaterials.