Electronic Structure of Multi-Electron Quantum Dots

Abstract

This article describes the use of the SACI package-a package for calculating the energy levels and wavefunctions of a multi-electron quantum dot modelled as a 2D harmonic well with electrons interacting through a Coulomb potential and under the influence of a perpendicular magnetic field. ‡ Introduction Quantum dots are artificially fabricated atoms, in which charge carriers are confined in all three dimensions just like electrons in real atoms. Consequently, they exhibit properties normally associated with real atoms such as quantised energy levels and shell structures. These properties are described by the electron wavefunctions whose evolution is governed by the Schrödinger equation and the Pauli exclusion principle.There are many methods available to solve the Schrödinger equation for multiple electrons. They roughly fall into the categories of the diagonalisation method, mean-field density-functional theory, and the self-consistent field approach. One of the first theoretical studies of quantum dots was by Pfannkuche et al. [1], who compared the results of Hartree-Fock self-consistent calculations and exact diagonalisation of the Hamiltonian for two electrons in a circularly symmetric parabolic potential. They found good agreement between the two methods for the triplet state but marked differences for the singlet state, indicating important spin correlations were not included properly in their HartreeFock model. This suggests that the proper treatment of electron spins is crucial for correctly obtaining the electronic structures in quantum dots.Examples of self-consistent field approaches in the literature include Yannouleas and Landman [2,3], who studied circularly symmetric quantum dots using an unrestricted spin-space Hartree-Fock approach, and McCarthy et al. [4], who developed a Hartree-Fock Mathematica package. Macucci et al. [5] studied quantum dots with up to 24 electrons using a mean-field local-density-functionalThe Mathematica Journal 10:2

Interacting electrons in a magnetic field in a center-of-mass free basis

“…We shall present in section 7 in particular for total spin S = 0, a complete analysis of these spurious states. Their intrusion of state space is demonstrated in the spectrum of determinantal eigenstates of the standard SACI approach [25,26,29]. Moreover, if determinantal eigenstates are variationally computed in a subspace defined by a fixed maximal oscillator excitation, we find that the admixture of spurious states distorts this subspace and the variational results.…”

“…The study of the four-electron quantum-dot with Coulomb interactions in the relativecoordinate basis elucidates and overcomes some of the problems inherent in the commonly used configuration interaction methods such as SACI [27,28]: If the cm excitations, as in determinantal state space, cannot be separated, a large cloud of spurious cm excited states, augmented by spurious angular momenta, intrudes the variational state space and obscures the variational energy spectrum reflecting the Coulomb interaction. Within SACI, higher eigenenergies are distorted by the Coulomb interaction and a loss in accuracy is seen, which can be avoided by the separation of the relative and cm coordinates.…”

“…If we require that the product of spatial and spin-part represents electrons (antisymmetric states), the basis set is decomposed into separate spin states. This results in spin-adapted (SA) configuration methods, for instance implemented in the SACI program code [25,26].…”

“…If we require that the product of spatial and spin-part represents electrons (antisymmetric states), the basis set is decomposed into separate spin states. This results in spin-adapted (SA) configuration methods, for instance implemented in the SACI program code [27,28]. At this point SACI methods used in condensed matter theory for more than 3 electrons stop and do not exploit additional properties of the basis set, such as a de-composition into relative and (interaction-free) center-of-mass (cm) states.…”

On the exact (whole) diagonalization of large semi-empirical configuration interaction matrices

“…Recently, Tanaka et al [5] developed a spin projected full configuration interaction (FCI) code adapted to parallel computers. Obtaining spin eigenfunctions [6][7][8][9][10][11] and eigenvectors of large matrices [12][13][14] are related topics of current interest. The importance of the diagonalization in the calculation of quantum dots has been reviewed by Reimann and Manninen [9].…”