Distortion of Wigner molecules : pair function approach

TL;DR

This paper analyzes a three-electron quantum dot in a magnetic field, revealing how electron pair states and quantum numbers influence the geometric distortion of Wigner molecules, showing conditions for equilateral versus isosceles configurations.

Contribution

It introduces a pair function approach with a coordinate transformation to decouple the Hamiltonian, providing new insights into the geometric structure of Wigner molecules based on quantum states.

Findings

Ground state forms an equilateral triangle only for specific spin and angular momentum.

The shape of the electron triangle varies with total angular momentum, becoming isosceles otherwise.

Pair states and quantum numbers determine the geometric distortion of the electron configuration.

Abstract

We considered a two dimensional three electron quantum dot in a magnetic field in the Wigner limit. A unitary coordinate transformation decouples the Hamiltonian (with Coulomb interaction between the electrons included) into a sum of three independent pair Hamiltonians. The eigen-solutions of the pair Hamiltonian provide a spectrum of pair states. Each pair state defines the distance of the two electrons involved in this state. In the ground state for given pair angular momentum $m$, this distance increases with increasing $|m|$. The pair states have to be occupied under consideration of the Pauli exclusion principle, which differs from that for one-electron states and depends on the total spin $S$ and the total orbital angular momentum $M_L=\sum m_i$ (sum over all pair angular momenta). We have shown that the three electrons in the ground state of the Wigner molecule form an equilateral triangle (as might be expected) only, if the state is a quartet ($S=3/2$) and the orbital angular momentum is a magic quantum number ($M_L=3 m ; m=$ integer). Otherwise the triangle in the ground state is isosceles. For $M_L=3 m+1$ one of the sides is longer and for $M_L=3 m-1$ one of the sides is shorter than the other two.