Correlation-induced inhomogeneity in circular quantum dots

Abstract

Properties of the 'electron gas'-in which conduction electrons interact by means of Coulomb forces but ionic potentials are neglected-change dramatically depending on the balance between kinetic energy and Coulomb repulsion. The limits are well understood 1 . For very weak interactions (high density), the system behaves as a Fermi liquid, with delocalized electrons. In contrast, in the strongly interacting limit (low density), the electrons localize and order into a Wigner crystal phase. The physics at intermediate densities, however, remains a subject of fundamental research 2-8 . Here, we study the intermediate-density electron gas confined to a circular disc, where the degree of confinement can be tuned to control the density. Using accurate quantum Monte Carlo techniques 9 , we show that the electron-electron correlation induced by an increase of the interaction first smoothly causes rings, and then angular modulation, without any signature of a sharp transition in this density range. This suggests that inhomogeneities in a confined system, which exist even without interactions, are significantly enhanced by correlations.Quantum dots 10 -a nanoscale island containing a puddle of electrons-provide a highly tunable and simple setting to study the effects of large Coulomb interaction. They introduce level quantization and quantum interference in a controlled way, and can, in principle, be made in the very-low-density regime, where correlation effects are strong 11 . In addition, there are natural parallels between quantum dots and other confined systems of interacting particles, such as cold atoms in traps.Therefore, we consider a model quantum dot consisting of electrons moving in a two-dimensional (2D) plane, with kinetic energy (−(1/2) i ∇ with n being the density of electrons. For our confined system in which n(r) varies, we define r s in the same way using the mean densityn ≡ n 2 (r)dr/N. We have studied this system up to N = 20 electrons. The spring constant ω makes the oscillator potential narrow (for large ω) or shallow (for small ω); it thereby tunes the average density of electrons between high and low values, thus controlling r s . For example, for N = 20, varying ω between 3 and 0.0075 changes r s from 0.4 to 17.7. The radius of the dot grows significantly as r s increases, in an approximately linear fashion (see Fig. 1).In the bulk 2D electron gas, numerical work suggests a transition from a Fermi-liquid state to a Wigner crystal around r c s ≈ 30-35 (refs 2-4,8). On the other hand, experiments on the 2D electron gas (which include, of course, disorder and residual effects of the ions) show more-complex behaviour, including evidence for a metal-insulator transition 5 . Circular quantum dots have been studied previously using a variety of methods, yielding a largely inconclusive scenario. Many studies 12-14 have used density functional theory or the HartreeFock method. These typically predict charge or spin-density-wave order even for modest r s (unless the symmetry is restored after the fact 14 ), ...

Spin-projected unrestricted Hartree-Fock ground states for harmonic quantum dots

“…With these techniques, the shell structure, Hund's rules, Wigner crystallization and the occurrence of "magic" angular momenta have been investigated. 26,37,38,42,45,46,47,48 Most of the results for higher particle numbers have been restricted to zero magnetic field. It is believed that QMC provides better estimates for the energies of the ground states for larger electron numbers as compared to the "exact" methods.…”

“…With presently available computational technology, reliably converged "exact" results can be obtained only for electron numbers up to N ≈ 8 electrons. 35 For N ≤ 13, N = 16, 24, 48, Quantum Monte Carlo (QMC) 37,38,39,40,41,42,43,44 methods have been used. They can provide accurate estimates for ground and excited states energies.…”

Crystallization in Mass‐Asymmetric Electron‐Hole Bilayers

“…2 also clearly shows the effect of the mass ratio. With increasing M the holehole correlations increase leading to increased hole localiczation [5,6]. This is accompanied by a pronounced modulation of the radial density n(R) and the pair distribution (PDF) g hh , see Fig.…”

“…In particular, Wigner crystal formation is one of the most prominent correlation phenomena observed in ultracold ions [2], dusty plasmas [3,4], quantum dots, e.g. [5,6] and other confined (non-neutral) systems. Recently crystal formation in two-component (neutral) quantum plasmas was demonstrated by simulations [7] confirming early predictions of hole crystallization in semiconductors by Halperin and Rice [8], Abrikosov [9] and others.…”

Density-Functional Theory for Strongly Correlated Bosonic and Fermionic Ultracold Dipolar and Ionic Gases

“…Being an in principle exact theory (although in practice relying on approximations), DFT allows to go beyond the mean-field description by taking into account correlations between the interacting particles. The Kohn-Sham (KS) mapping of the many-body problem into a non-interacting one makes it possible to apply the method to particle numbers orders or magnitude larger than those accessible with wave-function methods [3,[9][10][11][12], as well as to capture a large part of the effects of quantum statistics on the kinetic energy. Initial efforts have already been made to generalize the formalism to bosonic and fermionic ultracold quantum gases [13][14][15][16][17].…”