Dynamical screening of the Coulomb interaction for two confined electrons in a magnetic field

TL;DR

This paper investigates how vertical and lateral dynamics in a three-dimensional quantum dot system can be separated to analyze electron interactions via an effective two-dimensional model with a screened Coulomb potential, considering different confinement potentials.

Contribution

It introduces an adiabatic approximation approach to derive effective Coulomb interaction screening in confined electrons, providing analytical and numerical solutions for various potential shapes.

Findings

Analytical solution for parabolic confinement potential.

Numerical solutions for square and triangular wells.

Effective charge depends on the shape of the vertical confinement.

Abstract

We show that a difference in time scales of vertical and lateral dynamics permits one to analyze the problem of interacting electrons confined in an axially symmetric three-dimensional potential with a lateral oscillator confinement by means of the effective two-dimensional Hamiltonian with a screened Coulomb interaction. Using an adiabatic approximation based on action-angle variables, we present solutions for the effective charge of the Coulomb interaction (screening) for a vertical confinement potential simulated by parabolic, square, and triangular wells. While for the parabolic potential the solution for the effective charge is given in a closed anlytical form, for the other cases similar solutions can be easily calculated numerically.