Phase transitions in quantum dots

Müller, H.; Koonin, S. E.

doi:10.1103/physrevb.54.14532

Cited by 103 publications

(134 citation statements)

References 8 publications

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“…We find three phase boundaries: the polarization transition with the subsequent formation of the MDD, the broken-symmetry Chamon-Wen edge phase and finally the localization phase [21] where the MDD completely disappears. The range of magnetic fields where the MDD is a stable ground-state becomes much more narrow with increasing N , as it has been noted in Refs.…”

Section: Figmentioning

confidence: 99%

Quantum Dots in Magnetic Fields: Phase Diagram and Broken Symmetry at the Maximum-Density-Droplet Edge

et al. 1999

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Quantum dots in magnetic fields are studied within the current spin density functional formalism avoiding any spatial symmetry restrictions of the solutions. We find that the maximum density droplet reconstructs into states with broken internal symmetry: The Chamon-Wen edge co-exists with a modulation of the charge density along the edge. The phase boundaries between the polarization transition, the maximum density droplet and its reconstruction are in agreement with recent experimental results. PACS 73.20.Dx, 85.30.Vw Quantum dots are small electron islands made by laterally confining the two-dimensional electron gas in a semiconductor heterostructure. Such nano-sized systems attracted much interest since the techniques in their fabrication developed far beyond mesoscopic dimensions [1]. Vertical quantum dots can nowadays be made so small that they show atom-like behavior [2]: shell structure and Hund's rules determine the electronic properties.Much experimental effort focussed on systematically mapping the magnetic field dependence of the chemical potential obtained from single-electron capacitance spectroscopy [3]. As a bias is applied to the gates, single electrons tunnel into the quantum dot when its chemical potential µ(N, B) (which depends on the number of confined electrons N and the magnetic field strength B) equals the Fermi energy in one gate electrode. First experiments along these lines were performed by Ashoori et al. [3] and later Klein et al. [4]. Recently, Oosterkamp et al. [5] systematically extended the measurements to stronger fields B and larger sizes N . Cusps and steps in µ(N, B) were found to clearly separate different ranges of magnetic fields. From a comparison to results of exact diagonalization studies [6] these patterns were identified with phase transitions in the droplet: they occur at magnetic fields for which the ground-state charge distribution of the dot changes, defining sharp phase boundaries. The points at which a complete polarization of the electrons occurs mark the beginning of the so-called Maximum Density Droplet (MDD) phase. This new state suggested by McDonald, Yang and Johnson [7] is a homogeneous droplet in which the density is approximately constant at the maximum value ρ 0 = (2πl 2 B ) −1 that can be reached in the lowest Landau level. (ℓ B = h/eB is the magnetic length). In the spin-polarized MDD the electrons occupy adjacent orbitals with consecutive angular momentum. This compact occupation of states maximizes the electron density. The stability of the MDD is determined by a competition between the kinetic and external confinement contributions to the total energy, and the Coulomb repulsion of the electrons. The former would favor the MDD structure up to infinite fields: with increasing B, the droplet would decrease in radius such that close to the dot center it could maintain a density corresponding to filling factor one in the bulk limit [8].This, however, is inhibited by Coulomb repulsion: At a sharply defined transition point, the charge density distrib...

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Section: Figmentioning

confidence: 99%

Quantum Dots in Magnetic Fields: Phase Diagram and Broken Symmetry at the Maximum-Density-Droplet Edge

et al. 1999

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“…For QDM's with R W < 1, WC does not develop and instead EP's may form (SB of type II). We note here that while certain quantum-mechanical studies of electron localization (WC at high B) in single QD's have been discussed previously [8,14], this is the first study to explore, using a self-consistent quantum-mechanical treatment, broken symmetry WC and EP states in the B = 0 and small B regimes for both circular single QD's and lateral QDM's, thus providing new insights into the nature of these systems. Additionally, for single QD's with R W < 1 and N ≤ 20, we find that in the majority of cases the ground states exhibit CMF behavior without symmetry breaking; however, in several instances (e.g., N = 14), a pure SDW (SB of type III) develops.…”

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“…Indeed, using symmetry-restricted variational wave functions, we have reproduced the results of these studies, while with the sS-UHF, with no such restrictions, broken-symmetry solutions with lower energy were obtained as described here. We further note here that employing a Space-UHF, but only for fully polarized single QD's (i.e., under high magnetic fields where the spin unrestriction is not at play), Wigner crystallization has been investigated [8]. LSD calculations [9][10][11] where there are only two effective potentials (associated with the two spin directions) cannot yield in general crystallized solutions (except for N = 2 in a deformed single QD and in a QDM [11]).…”

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Erratum: Spontaneous Symmetry Breaking in Single and Molecular Quantum Dots [Phys. Rev. Lett. 82, 5325 (1999)]

Yannouleas¹,

Landman²

2000

Phys. Rev. Lett.

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Classes of spontaneous symmetry breaking at zero and low magnetic fields in single quantum dots (QD's) and quantum dot molecules (QDM's) are discussed in relation to the ratio RW between the interelectron Coulomb repulsion and the harmonic confinement, using spin-and-Space unrestricted Hartree-Fock calculations. These include: Wigner crystallization for RW > 1, and formation of non-crystallized electron puddles localized on the individual dots in QDM's, as well as spin-density waves in single QD's, for RW < 1.Pacs Numbers: 73.20.Dx, 71.45.Lr Two-dimensional (2D) electron gases have provided (e.g., the fractional quantum Hall effect [1,2]), and continue to provide (e.g., a charge-density wave at higher Landau levels [3]) a source of discovery of remarkable many-body phenomena. Recently, 2D artificial quantum dots (QD's) and quantum dot molecules (QDM's) have become available, with the capability of controlling the dots' size, shape, and number N of electrons [4,5].Single QD's are commonly referred to as "artificial atoms", since interpretations of transport and capacitance experiments draw often on analogies between such artificial structures and natural atoms [4,5]. Underlying these analogies is an effective (circular) central mean field (CMF) picture, with the electronic spectra exhibiting (at zero magnetic field) shell closures and following Hund's rules for open shells. Indeed, in experiments on single QD's, the addition energy (AE) spectra [4] exhibit maxima at the expected closed shells (N = 2, 6, 12), and at the mid-shells (N = 4, 9, and 16) in accordance with Hund's rule.Here, using the self-consistent spin-and-Space unrestricted Hartree-Fock (sS-UHF) [6,7] method, we discuss, for zero and low magnetic fields (B), three types of spontaneous symmetry breakings (SB) in circular single QD's and in lateral QDM's (i.e., formation of ground states of lower symmetry than that of the confining potentials [12]). These include: (I) Wigner crystallization (WC) [13] in both QD's and QDM's, i.e., (spatial) localization of individual electrons, (II) formation of electron puddles (EP's) in QDM's, that is localization of the electrons on each of the individual dots comprising the QDM, but without crystallization within each dot, and (III) pure spin-density waves (SDW's) which are not accompanied by spatial localization of the electrons [9]). Furthermore, we show that CMF descriptions at zero and low magnetic fields may apply only for low values of the parameter R W ≡ Q/hω 0 , where Q is the Coulomb interaction strength andhω 0 is the parabolic confinement; Q = e 2 /κl 0 , with κ being the dielectric constant, l 0 = (h/m * ω 0 ) 1/2 the spatial extension of the lowest state's wave function in the parabolic confinement, and m * the effective electron mass. With the sS-UHF, we find that WC occurs (SB of type I) in both QD's and QDM's for R W > 1. For QDM's with R W < 1, WC does not develop and instead EP's may form (SB of type II). We note here that while certain quantum-mechanical studies of electron localization (WC a...

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“…In quantum dots, they have been used for obtaining wave functions corresponding to specific angular momenta. 67,68,69,70,71 Recently, the random phase approximation has been used to restore the rotational symmetry of wave functions obtained by UHF.…”

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

Giovannini¹,

Cavaliere²,

Cenni

et al. 2008

Phys. Rev. B

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We report results for the ground state energies and wave functions obtained by projecting spatially unrestricted Hartree Fock states to eigenstates of the total spin and the angular momentum for harmonic quantum dots with N ≤ 12 interacting electrons including a magnetic field. The ground states with the correct spatial and spin symmetries have lower energies than those obtained by the unrestricted method. The chemical potential as a function of a perpendicular magnetic field is obtained. Signature of an intrinsic spin blockade effect is found.

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Phase transitions in quantum dots

Cited by 103 publications

References 8 publications

Quantum Dots in Magnetic Fields: Phase Diagram and Broken Symmetry at the Maximum-Density-Droplet Edge

Quantum Dots in Magnetic Fields: Phase Diagram and Broken Symmetry at the Maximum-Density-Droplet Edge

Erratum: Spontaneous Symmetry Breaking in Single and Molecular Quantum Dots [Phys. Rev. Lett. 82, 5325 (1999)]

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

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