Density-functional theory of the phase diagram of maximum-density droplets in two-dimensional quantum dots in a magnetic field

Abstract

We present a density-functional theory ͑DFT͒ approach to the study of the phase diagram of the maximumdensity droplet ͑MDD͒ in two-dimensional quantum dots in a magnetic field. Within the lowest Landau level ͑LLL͒ approximation, analytical expressions are derived for the values of the parameters N ͑number of electrons͒ and B ͑magnetic field͒ at which the transition from the MDD to a ''reconstructed'' phase takes place. The results are then compared with those of full Kohn-Sham calculations, giving thus information about both correlation and Landau level mixing effects. Our results are also contrasted with those of Hartree-Fock ͑HF͒ calculations, showing that DFT predicts a phase diagram, which is in better agreement with the experimental results and the result of exact diagonalizations in the LLL than the HF calculations. ͓S0163-1829͑97͒06443-6͔Two-dimensional quantum dot systems, at high magnetic fields, have been recently studied by various authors. 1 The theoretical interest in these systems arises largely from the fact that they provide a few-electron realization of physical states that, in the macroscopic limit, are responsible for the occurrence of the quantum Hall effect. 2 The simplest example of such a state is the so-called maximum-density droplet ͑MDD͒, which, in the limit of a high magnetic field, can be written as a Slater determinant of lowest Landau level ͑LLL͒ orbitals with angular momenta 0,1, . . . ,NϪ1, where N is the number of electrons. 3 In the limit of N→ϱ this coincides with the incompressible state of the quantum Hall effect at filling factor ϭ1. Because, within the LLL, the MDD is the only N-electron state of angular momentum N(NϪ1)/2 ͑and there is none with lower angular momentum͒ it follows that it must be an exact eigenstate of the Hamiltonianif the small Coulomb coupling between different Landau levels is neglected. Here 0 is the frequency of the external parabolic potential, A i is the external vector potential, k is the dielectric constant, m* is the electron effective mass, B is the Bohr magneton, g* is the effective g factor for the Zeeman splitting, and i is the spin component along the axis perpendicular to the plane of the electrons. The question is whether this exact eigenstate ͑or rather its continuation to a finite magnetic field͒ can actually be the ground state of the quantum dot in some range of magnetic fields. The basic physics is simple: If the magnetic field is too large, the MDD cannot be the ground state because the compact arrangement of the electrons costs too much electrostatic energy. The electrostatic stress is released through a rearrangement of the electrons leading to a state of higher angular momentum. If, on the other hand, the magnetic field is too weak, the confinement energy will cause the external electrons in the MDD to be transferred to the center of the quantum dot, even though, in so doing, a higher Landau level becomes populated at the center of the dot. The conclusion of these arguments is that there will exist, at most, a ''window'' of magneti...

Density-functional calculations of magnetoplasmons in quantum rings

“…NϪ1 26,27 is limited from the left by a line B f representing, for a given number number of electrons, the magnetic field at which 2S z ϭN, and from the right by a line B r at which edge reconstruction starts. [28][29][30] This is a rather narrow region, a few tenths of a tesla wide, 30 because after being fully polarized the magnetic field is very effective in promoting electrons from high to higher l sp levels, reconstructing the dot edge. In rings this is not quite so, because the existence of an electron depletion at the center and the consequent upward bending of the sp bands allows for an alternative mechanism to keep increasing L z while retaining the simplicity of the gs wave function, namely, a Slater determinant made of the lowest possible l sp states from a minimum l m to a maximum l M such that N ϭl M Ϫl m ϩ1.…”

Wigner crystalline edges in ν≲1 quantum dots

“…When these quantum dots are subjected to large perpendicular magnetic fields, a rich phenomenology may be observed, much of it due to the same physics underlying the bulk quantum Hall effect [1,2]. In particular, there has been a great deal of interest in the maximum density droplet (MDD) [3][4][5][6][7][8][9][10][11][12][13][14][15][16], which is the small N analog of the bulk ν = 1 state. Experimentally, the MDD has been identified in several studies, including those of Ashoori et al [3] , Klein et al [4,5], and Oosterkamp et al [6].…”

Maximum-density droplet to lower-density droplet transition in quantum dots

“…This is the maximum density droplet (MDD) state. As B is increased further, Hartree-Fock [21] and density functional calculations [22,23] predict an edge reconstruction due to some electrons jumping from low angular momentum states to higher angular momentum states because the Coulomb interaction overwhelms the parabolic external potential. These lower density droplet (LDD) states were observed experimentally [24] but quantitative discrepancies with experiments exist.…”