The Mathematica Journal
Volume 10, Issue 2

Search

In This Issue
Articles
In and Out
Trott's Corner
Beyond Sudoku
New Products
New Publications
Calendar
News Bulletins
New Resources
Classifieds

Download This Issue 

About the Journal
Editorial Policy
Staff and Contributors
Submissions
Subscriptions
Advertising
Back Issues
Contact Information

Electronic Structure of Multi-Electron Quantum Dots
Ranga D. Muhandiramge
Jingbo Wang

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 Hartree-Fock 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-functional approach, in which the spin exchange-correlation potential was approximated by an empirical polynomial expression. Lee et al. [6] also studied an N-electron quantum dot using density-functional theory, where the generalised-gradient approximation was used for exchange-correlation potentials. Exchange interaction comes directly from the antisymmetrisation of wavefunctions as required by Pauli's exclusion principle. In density-functional theory this is a major problem since the mathematical object is the electron density rather than the electron wavefunction, making evaluation of the exchange interaction intrinsically difficult.

Diagonalisation approaches in the literature include Ezaki et al. [7, 8], Eto [9, 10], Reimann et al. [11], and Reimann and Manninen [12], who each applied a brute force approach by numerically diagonalising the N-electron Hamiltonian using Slater determinants composed of single-electron eigenstates as the basis functions. This approach, namely the configuration interaction (CI) method, takes into account the full interaction and correlation of the electrons in the system as long as the numerical results converge with an increasing number of basis functions. However, such an approach involves the calculation of a very large number of interaction integrals and the inversion of large matrices, which can be prohibitively expensive in terms of computer resources. Reimann et al. [11] employed matrices of dimensions up to 108,375 with 67,521,121 nonzero elements for a six-electron quantum dot. Calculations for any higher number of electrons were not considered numerically viable using the conventional CI formalism, even with state-of-the-art computing facilities.

We have recently developed a spin-adapted configuration interaction (SACI) method to study the electronic structure of multi-electron quantum dots [13]. This method is based on earlier work by quantum chemist R. Pauncz [14], which expands the multi-electron wavefunctions as linear combinations of antisymmetrised products of spatial wavefunctions and spin eigenfunctions. The SACI method has an advantage over using Slater determinants in that a smaller basis is used. This reduces the computational resources required, allowing calculations for dots with more than six electrons on a desktop computer.

After some theoretical background, we present results from a Mathematica package which employs the SACI method to calculate the energy levels and wavefunctions of multiple electrons confined by a 2D harmonic well and under the influence of a perpendicular magnetic field. Mathematica enables the exact calculation of interaction integrals which greatly improves the speed and accuracy of calculations.



     
About Mathematica | Download Mathematica Player 
© Wolfram Media, Inc. All rights reserved.