Correlation energy of anisotropic quantum dots

TL;DR

This paper investigates the high-density correlation energy of two electrons in anisotropic quantum dots across different dimensions, revealing limitations of the 2D approximation and providing a model for energy behavior over all anisotropies.

Contribution

It introduces a comprehensive analysis of correlation energy in anisotropic quantum dots, including a smooth dimensional interpolation and a simple model for energy behavior.

Findings

Correlation energy varies smoothly with anisotropy.

Two-dimensional models are inadequate for certain anisotropies.

A simple function accurately reproduces correlation energy across all anisotropies.

Abstract

We study the $D$-dimensional high-density correlation energy $\Ec$ of the singlet ground state of two electrons confined by a harmonic potential with Coulombic repulsion. We allow the harmonic potential to be anisotropic, and examine the behavior of $\Ec$ as a function of the anisotropy $\alpha^{-1}$. In particular, we are interested in the limit where the anisotropy goes to infinity ($\alpha\to0$) and the electrons are restricted to a lower-dimensional space. We show that tuning the value of $\alpha$ from 0 to 1 allows a smooth dimensional interpolation and we demonstrate that the usual model, in which a quantum dot is treated as a two-dimensional system, is inappropriate. Finally, we provide a simple function which reproduces the behavior of $\Ec$ over the entire range of $\alpha$.