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Physical Review B

Quantum Monte Carlo study of few-electron concentric double quantum rings

Leonardo Colletti, Francesc Malet, Martí Pi, Francesco Pederiva

Published Mar 17, 2009• 14 citations
Publisher

Abstract

We consider few-electron concentric double quantum rings with parabolic confining potential and compare the ground-state energies calculated by exact diagonalization of the Hamiltonian, accurate quantum Monte Carlo, and local spin-density functional approaches. Electronic localization in one of the rings and the formation of rotating Wigner molecules is shown, respectively, from the one-body and the two-body density operators. As the confinement strength of the outer ring is finely increased, the circularly symmetric electron density exhibits a radial crossover from this ring to the inner one without altering the angular character of the system.

DOI
10.1103/physrevb.79.125315

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We consider few-electron concentric double quantum rings with parabolic confining potential and compare the ground-state energies calculated by exact diagonalization of the Hamiltonian, accurate quantum Monte Carlo, and local spin-density functional approaches. Electronic localization in one of the rings and the formation of rotating Wigner molecules is shown, respectively, from the one-body and the two-body density operators. As the confinement strength of the outer ring is finely increased, the circularly symmetric electron density exhibits a radial crossover from this ring to the inner one without altering the angular character of the system.
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We study a system of interacting electrons on a one-dimensional quantum ring using exact diagonalization and the variational quantum Monte Carlo method. We examine the accuracy of the Slater-Jastrow -type many-body wave function and compare energies and pair distribution functions obtained from the two approaches. Our results show that this wave function captures most correlation effects. We then study the smooth transition to a regime where the electrons localize in the rotating frame, which for the ultrathin quantum ring system happens at quite high electron density.
We study a system of interacting electrons on a one-dimensional (1D) quantum ring using exact diagonalization (ED) and the variational quantum Monte Carlo (VMC) method. We examine the accuracy of the Slater–Jastrow-type many-body wavefunction and compare energies and pair distribution functions obtained from the two approaches. Our results show that this wavefunction captures most correlation effects. We then study the smooth transition to a regime where the electrons localize in the rotating frame, which for the ultra-thin quantum ring system happens at quite high electron density.
We study a system of two electrons interacting with a Coulomb potential in a sphere of radius R, bounded by an infinite wall using exact diagonalization. We have also investigated the influence of an additional parabolic potential (of strength k)k)k) arising from a uniform background smeared throughout the sphere. The convergence of the ground state energy of the singlet spin state of the system is investigated as a function of sphere size (essentially rs,{r}_{s},rs​, the Wigner--Seitz density parameter) for cases where there is no background potential (k=0)(k=0)(k=0) and for when $k\ensuremath{\ne}0.$ With k=0k=0k=0 and small rs,{r}_{s},rs​, we observe a maximum in the ground state density at the origin of the sphere. At ${r}_{s}\ensuremath{\approx}8\mathrm{a}.\mathrm{u}.,$ the ground state density acquires a minimum at the origin. For this and larger systems we identify the formation of a ``Wigner'' molecule state. We further investigate the ground state density as a function of k and also the correlation hole density as a function of rs{r}_{s}rs​ and k. We invert the Kohn--Sham equation for a two electron system and calculate the local effective potential and correlation potential (to within an additive constant) as functions of the radial coordinate for a number of values of rs{r}_{s}rs​ and k.
The trial wavefunctions for few-electron quantum rings are presented to describe the spin-dependent rotating Wigner molecule states. The wavefunctions are constructed from the single-particle orbits which contain two variational parameters to describe the shape and size dependence of electron localization in the ring-like confinement. They can explicitly show the size dependence of single-particle orbital occupation to give an understanding of the spin rules of ground states without magnetic fields. They can also correctly describe the spin and angular momentum transitions in magnetic fields. By examining the von Neumann entropy, it is demonstrated that the wavefunctions can illustrate the entanglement between electrons in quantum rings, including the AB oscillations as well as the spin and size dependence of the entropy. Such trial wavefunctions will be useful in investigating spin-related quantum behaviors of a few electrons in quantum rings.
This paper presents a few case studies of finite electron systems for which strong correlations play a dominant role. In simple metal clusters, the valence electrons determine the stability and shape of the clusters. The ionic skeleton of alkali metals is soft, and cluster geometries are often solely determined by electron correlations. In quantum dots and rings, the electrons may be confined by an external electrostatic potential formed by a gated heterostructure. In the low-density limit, the electrons may form the so-called Wigner molecules, for which the many-body quantum spectra reveal the classical vibration modes. High rotational states increase the tendency for the electrons to localize. At low angular momenta, the electrons may form a quantum Hall liquid with vortices. In this case, the vortices act as quasi-particles with long-range effective interactions that localize in a vortex molecule, in much analogy with the electron localization at strong rotation.