Addition spectra of quantum dots in strong magnetic fields

Abstract

We consider the magnetic field dependence of the chemical potential for parabolically confined quantum dots in a strong magnetic field. Approximate expressions based on the notion that the size of a dot is determined by a competition between confinement and interaction energies are shown to be consistent with exact diagonalization studies for small quantum dots. Fine structure is present in the magnetic field dependence which cannot be explained without a full many-body description and is associated with ground-state level crossings as a function of confinement strength or Zeeman interaction strength. Some of this fine structure is associated with precursors of the bulk incompressible states responsible for the fractional quantum Hall effect.PACS numbers: 73.20.Dx, 73.20.Mf Advances in nanofabrication technology have made it possible to manufacture "quantum dots" in which electrons are confined to a small area within a twodimensional (2D) electron gas [1]. Interest in these systems has grown as a result of recently developed techniques [2,3] which probe them spectroscopically. The quantity which is measured [4] in these experiments is the magnetic field dependence of the "addition spectrum," i.e., the energy to add one electron to a dot. This is given by fi^ = E% -E%_ 11 where E^ is the ground state energy of an Af-electron dot. Addition spectrum measurements have generally been interpreted in terms of "constant interaction" models in which electron-electron interactions are accounted for by including a charging energy which is characterized by a fixed self-capacitance; or, when this fails, by using Hartree or Hartree-Fock approximations. However, especially at strong magnetic fields, quantum dots can have strongly correlated [5,6] ground states, some of which are precursors of the bulk incompressible states responsible for the fractional quantum Hall effect. In this regime a complete interpretation of addition spectra measurements requires an exact treatment of the Coulombic electron-electron interactions.In this Letter we report on numerical exact diagonalization calculations of the addition spectrum for quantum dots in a strong magnetic field. We find that the addition spectrum has a surprisingly rich magnetic field dependence, showing a large number of sharp features superimposed on a smooth background. The background can be accounted for using a simple Hartree approximation. The sharp features are associated with energy-level crossings at fixed N, often between strongly correlated states. The spin degree of freedom has a nontrivial role, in general not consistent with expectations based on the Hartree-Fock approximation. The constant interaction model fails qualitatively for strong magnetic fields.We consider a system of 2D electrons confined by a parabolic potential [7], V(r) = mQ 2 r 2 /2.We confine our attention here to the strong magnetic field limit [8],where B± is the component of the magnetic field perpendicular to the 2D layer.) In this limit [1] the symmetric gauge single-particle eigenstates are...

From a few to many electrons in quantum dots under strong magnetic fields: Properties of rotating electron molecules with multiple rings

“…The groundstate arrangements of the electrons become structurally more complex as the number of electrons in the dot increases. Using the notation (n 1 , n 2 , n 3 , ...) for the number of electrons located on concentric polygonal rings (see section II.A), the ground-state arrangements are: (0,3) for N = 3, (0,4) for N = 4, (1,5) for N = 6, (2,7) for N = 9, (3,8) for N = 11, and (1,6,10) for N = 17.…”

“…10, we display the CPD for the REM wave function of N = 17 electrons. This case has a nontrivial three-ring structure (1,6,10), 25 which is sufficiently complex to allow generalizations for larger numbers of particles. The remarkable combined character (partly crystalline and partly fluid leading to a non-classical rotational inertia, see section VI) of the REM is illustrated in the CPDs of Fig.…”

“…We describe here a widely used approximation 4,10,31 for calculating the ground state at a given B, which takes advantage of the simplifications at the B → ∞ limit, i.e., when the relevant Hilbert space can be restricted to the lowest Landau level [thenhω 0 <<hω c /2 (for B → ∞) and the confinement can be neglected at a first step]. Then, the many-body hamiltonian [see Eq.…”

“…Naturally, QDs with a small number of electrons are also most attractive for theoretical investigations, since their ground-state properties and excitation spectra can be analyzed 4,5,6,7,8,10,11,13 through exact-diagonalization (EXD) solutions of the many-body Schrödinger equation. In particular, in combination with certain approximate methods, which are less demanding computationally while providing highly accurate results and a transparent physical picture (e.g., the method of successive hierarchical approximations, 7,9 see below), EXD calculations confirmed the spontaneous formation of finite rotating electron molecules (REMs) and the description of the excited states with magic angular momenta as yrast rotational bands of these REMs 7 (sometime the REMs are referred to as "rotating Wigner molecules," RWMs).…”

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.…”