Collective effects of the Coulomb interaction in anharmonic quantum dots

TL;DR

This paper investigates how Coulomb interactions and confining potential shapes influence the energy levels in quantum dots, revealing distinct spectral signatures and providing efficient computational methods for analysis.

Contribution

It introduces a detailed analysis of Coulomb effects in anharmonic quantum dots and derives simplified formulas for matrix elements, enhancing computational efficiency.

Findings

Level spacings vary with potential shape, affecting electron states.

Cone deformations influence near-ground state levels.

Spectral similarities occur between deformed and perfect oscillators with reduced Coulomb interaction.

Abstract

Deviations from the uniform oscillator spacing, related to the shape of the confining potential, have a strong influence on few-electron states in quantum dots when Coulomb effects are included. Distinct signatures are found for level spacings increasing (pot shape) and decreasing (cone shape) with energy. Cone deformations affect the levels near the ground state, such that observable effects are predicted. Pot deformations partially negate the effect of the Coulomb force, thus their spectra are similar to those of a perfect oscillator with a smaller Coulomb repulsion. The Coulomb force is treated by exact diagonalization, for which purpose efficient closed-form expressions for the matrix elements are derived. The Coulomb matrix element in relative coordinates is reduced to a single sum over four binomial coefficients, for which simple analytic approximations are found.