| Summary: | This thesis examines the topics of disorder and electron-electron interactions in three distinct quantum systems. Firstly, the Anderson transition is studied for the body centred cubic and face centred cubic lattices. We obtain high precision results for the critical disorder at the band centre and the critical exponent using the transfer-matrix method and finite size scaling. Comparing the critical disorder between the simple cubic, body centred cubic and face centred cubic lattices, an increase in the critical disorder is observed as a function of the coordination number of the lattice. The critical exponent is found to be v ≃ 1:5 in agreement with the value for the simple cubic lattice. Energy-disorder phase diagrams are plotted for both lattice types. Next, we consider the Aharonov-Bohm effect for an exciton in a 1D ring geometry. The aim is to determine how the addition of a constant electric field in the plane of the ring effects the Aharonov-Bohm oscillations, which occur as a function of the magnetic ux threading the ring. We develop a self consistent equation for the ground state energy, which is then solved numerically. Oscillations in the ground state energy have an increasing amplitude as a function of electric field strength until a critical electric field value. At this point, oscillations in the oscillator strength become inverted, with the oscillation minimum reaching zero at half a magnetic ux quantum. This suggests a possible process for controlling the formation and recombination of excitons through tuning the applied fields. The final and largest section of the thesis is concerned with collective excitations of graphene in a strong perpendicular magnetic field. The excitations, which are most strongly mixed are identified and used as a basis to diagonalise the Hamiltonian, which includes the Coulomb interaction between electrons and holes. In this way the oscillator strengths and energies of collective excitations are obtained. The good quantum numbers for collective excitations are identified. In particular, we study those arising from the SU(4) symmetry, which is due to two spin and two valley pseudospin projections. This enables us to determine the multiplet structure of the states. In addition to neutral collective excitations or excitons, we investigate the possible formation of charged collective excitations or trions from nearly full or nearly empty Landau levels. The localisation of neutral collective excitations upon a single Coulomb or δ-function impurity is also examined.
|
|---|
