Finite difference solutions of quantum wire and quantum dot systems

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.