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

Abstract

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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