Crossover from Fermi Liquid to Wigner Molecule Behavior in Quantum Dots

Abstract

The crossover from weak to strong correlations in parabolic quantum dots at zero magnetic field is studied by numerically exact path-integral Monte Carlo simulations for up to eight electrons. By the use of a multilevel blocking algorithm, the simulations are carried out free of the fermion sign problem. We obtain a universal crossover governed only by the density parameter r s . For r s . r c , the data are consistent with a Wigner molecule description, while, for r s , r c , Fermi liquid behavior is recovered. The crossover value r c ഠ 4 is surprisingly small. [S0031-9007 (99)08929-2] PACS numbers: 73.20.Dx, 71.10.Ay, 71.10.CaQuantum dots can be considered as solid-state artificial atoms with tunable properties. Confining a small number of electrons N in a two-dimensional electron gas in semiconductor heterostructures, a number of interesting effects arising from the interplay between confinement and the Coulomb interaction between the electrons can be observed [1,2]. Since the confinement potential is usually quite shallow, the long-ranged Coulomb interaction among the electrons plays a prominent role, and in contrast to conventional atoms effective single-particle approximations quickly become unreliable. In the low-density (stronginteraction) limit, r s !`, classical considerations suggest a Wigner crystal-like phase with electrons spatially arranged in shells [3]. With quantum fluctuations, such a phase is best described as a Wigner molecule. In contrast, for high densities (weak interactions), r s ! 0, a Fermi liquidlike description is expected to be valid, where it is more appropriate to think of the behavior as resulting from the single-particle orbitals being filled. The noninteracting limit [4] is then typically used as a starting point for the theoretical description of quantum dots.To date, no reliable information exists for the crossover between these two limits. This is mainly due to a complete lack of sufficiently accurate methods that are able to cover the full range of r s , especially when no magnetic field is present. Exact diagonalization techniques are limited to very small particle numbers and small r s ; otherwise, a large error due to the truncation of the Hilbert space arises [5]. Hartree-Fock calculations become increasingly unreliable for large r s and are known to incorrectly favor spin-polarized states [6]. Similarly, density functional calculations [7] introduce uncontrolled approximations in the absence of exact reference data. In principle, the quantum Monte Carlo (QMC) method is the best candidate for producing reliable data for quantum dots. Unfortunately, the notorious fermion sign problem makes direct QMC simulations almost impossible [8]. To avoid the sign problem, the fixed-node approximation and a related variational approach have been employed in Ref. [9], but the results are no longer exact.In this Letter, we adopt a radically different approach to fermion QMC simulations, based on the recently developed multilevel blocking (MLB) algorithm [10,11]. The MLB algorithm is ab...

Manifestation of unconventional biexciton states in quantum dots

“…The height of the QD can additionally influence this critical diameter, as a smaller electronhole separation leads to a stronger lateral confinement of the holes by the attraction of the electrons. Prior two-dimensional studies of electron interactions in QDs [36][37][38][39][40] raised the question whether a Wigner molecule (of electrons) is reliably described within the unrestricted Hartree-Fock method, allowing for symmetry-broken solutions 41 . Our study uses such a (here three-dimensional) Hartree-Fock method (see the Simulation Methods subsection), which predicts a Wigner molecule-like hole separation within the hybrid-XX model.…”

Rapidly rotating boson molecules with long- or short-range repulsion: An exact diagonalization study

“…Motivated by the discovery in the case of electrons of REMs at high B (and from the fact that Wigner molecules form also at zero magnetic field [28,29,30,31,32,33]) some theoretical studies have most recently shown that analogous molecular patterns of localized bosons do form in the case of a small number of particles inside a static or rotating harmonic trap [34,35,36,37,38]. In analogy with the electron case, the bosonic molecular structures can be referred to [36] as rotating boson molecules (RBMs); a description of RBMs via a variational wave function [39] built from symmetry-breaking displaced Gaussian orbitals with subsequent restoration of the rotational symmetry was presented in Refs.…”

Spin and Conductance-Peak-Spacing Distributions in Large Quantum Dots: A Density-Functional Theory Study

“…We close with a caveat and a comment: First, this work is based on the 2D local spin density approximation whose validity, though already verified for small parabolic QD's [29,30], is not well tested for large non-parabolic quantum dots as studied here. Our result highlights the need to go beyond RPA-RMT and perform a real Fermi liquid theory study.…”

“…We use the standard local spin density approximation (LSDA) for E xc which works well in various semiconductors; in particular, we use Tanatar and Ceperley's 2D parametrization [28]. For 2D clean QDs, comparisons with quantum Monte-Carlo calculations for N ≤ 8 and interaction strength parameter r s ≤ 8.0 have shown that LSDA works well for both the ground state spin and energy [29,30].…”