Finite difference solutions of quantum wire and quantum dot systems

• Main Author: El-Moghraby, Dureid
• Other Authors: Harrison, Paul ; Johnson, Rob
• Published: University of Leeds 2005
• Subjects: 537.50151853
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.414288

A new implementation of the finite difference method was developed, and discussed, for solving the time-independent, constant effective mass Schrodinger equation in three dimensions. The motivation behind this approach was to develop a computational technique which is fast to execute and requires a small memory footprint. To demonstrate its validity, this numerical finite difference method was utilised to calculate the electronic eigenenergies of an infinitely deep quantum wire (QWW), where the results were within 0.25 meV of the analytical values. The method was used to calculate energies of a triangular QWW of finite depth that was found in the literature [62]. The calculated energies showed very good agreement with that of Gangopahdhyay [62], with the difference in eigenenergies ranging between 1 and 10 meV. This difference is likely to arise from the simplified constant effective mass Hamiltonian. The case of a pyramidal quantum dot (QD) was then investigated. It was found that the calculated results were within 2 meV of the values found in the literature [5]. However, the advantages o{ this method become apparent as it requires a fraction of the memory needed by the eigenvalue method and the computational times also compare favourably. The effect of the inter-dot separation in a system of vertically aligned pyramidal QDs was then investigated. It was found that when the separation between the QDs was large enough, they behaved as if isolated. As the proximity increased, so did the interaction, which manifests itself as an increase in the peak value of the wave function of the higher energy dot and a reduction in the overall eigenenergies. The method was extended to incorporate the Poisson equation, and used to calculate the eigenenergies of a QD for a varying number of electrons. As would be expected the eigenenergies of the system rose as more electrons were added to the system. The effect of introducing a varying number of electrons into a system of vertically aligned QDs,for a number of inter-dot separations, showed that the eigenenergies for a single electron increased as the inter-dot separation was increased. However, for the case of multiple electrons, it was found that the eigenenergies initially decrease and then increase as the inter-dot separation is increased.