DSpace Collection:https://dspace.lboro.ac.uk/2134/66232017-03-28T17:49:03Z2017-03-28T17:49:03ZPath integral calculation of the Wigner functionLindsey, Neilhttps://dspace.lboro.ac.uk/2134/245092017-03-23T14:53:43Z2008-01-01T00:00:00ZTitle: Path integral calculation of the Wigner function
Authors: Lindsey, Neil
Abstract: Elementary Wigner function calculations of the infinite square well and Schroedinger cat
states are presented as an introduction to the quasi-probability function. An entangled
cat state is calculated and the Wigner function of the state is found. Properties
of the entanglement of the state and the nature of its entanglement are found to be
distinguishable by this distribution.
This work is mostly concerned with obtaining the Wigner function via a path integral
method, following a previously published technique. The method approximates the
ground state Wigner function by finding the classical path associated with each point
in phase space, assuming the P-function of the Hamiltonian of the system is able to
be found. The imaginary part of action determines the phase of the path integral
and depends on the geometry of the path; specifically the area which it encloses. An
investigation into two systems, the Morse potential and the double well potential, was
performed to try and find classical paths enclosing area and thus recreating the negative
features of the exact Wigner function. The minimisation of the action found the classical
path for each phase space point. This was performed numerically using tools created in
Excel and Mathematica. In general, it was discovered that the classical paths did not
enclose any area and therefore the Wigner function approximations were everywhere
positive. The majority of those paths which were found to enclose some area produce a
phase which is not large enough to change the sign of the path integral.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2008-01-01T00:00:00ZEffects of magnetic field on electron transport in semiconductor superlatticesZhang, Lianghttps://dspace.lboro.ac.uk/2134/219222016-07-11T08:29:42Z2016-01-01T00:00:00ZTitle: Effects of magnetic field on electron transport in semiconductor superlattices
Authors: Zhang, Liang
Abstract: Quantum superlattice with a narrow energy band is an artificial semiconductor structure demonstrating both nonlinear and active high-frequency electromagnetic properties. These types of superlattices are used as key elements in various miniature electronic devices including frequency multipliers and quantum cascade lasers. Interaction between terahertz radiation and magnetic field in semiconductor superlattices has been the subject of growing research interest, both theoretical and experimental. In this thesis, we study the nonlinear dynamics of electrons in minibands of the semiconductor superlattices subjected to a terahertz electric field and a magnetic field.
Electron transport in a semiconductor superlattice with an electric field and a tilted magnetic field has been studied using semiclassical equations. In particular, we consider how dynamics of electron in superlattices evolve with changing the strength and the tilt of a magnetic field. In order to investigate the influence of a tilted magnetic field on electron transport, we calculate the drift velocity for different values of the magnetic field. Studies have shown that the resonance of Bloch oscillations and cyclotron oscillations produces additional peaks in drift velocity. We also found out that appearance of these resonances can promote amplification of a small ac signal applied to the superlattice.
In the presence of the electromagnetic field, the superlattice is expected to demonstrate the Hall effect, which however should have a number of very specific features due to an excitation of Bloch oscillations and a significant electric anisotropy. Here, we theoretically study the Hall effect in a semiconductor superlattice both for the steady electron transport and for the transient response. We studied the coherent Hall effect in an extraordinary configuration where the electric field is applied in the transverse direction of the superlattice growth direction. By mapping the momentum dynamics to the pendulum equivalent, we distinguished the two regimes of the oscillations from the viewpoint of the effective potentials. We discuss the experimental manifestation of the Hall effect in a realistic superlattice. We also made the numerical simulations of the polarized THz field and the time-resolved internal electro-optic sampling (TEOS) signals where we found the unusual shaped waveforms of the THz signals.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZEdge states, magnetisation and topological domain walls in grapheneLiu, Yanghttps://dspace.lboro.ac.uk/2134/218012016-07-05T10:28:04Z2016-01-01T00:00:00ZTitle: Edge states, magnetisation and topological domain walls in graphene
Authors: Liu, Yang
Abstract: We studied the edge states and their roles in conductivity
and magnetism of graphene nanoribbions and flakes. we studied the Aharonov-Bohm effect in graphene nanodisks and rings. We described the quantum oscillations of the magnetization of graphene flakes. we have examined the snake-like states of transport
electrons in the configurations of graphene ribbons with a domain wall in the centre.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZGraphene electronic devices in magnetic fieldBrada, Matejhttps://dspace.lboro.ac.uk/2134/218002016-07-05T10:17:42Z2016-01-01T00:00:00ZTitle: Graphene electronic devices in magnetic field
Authors: Brada, Matej
Abstract: This thesis discusses the two dimensional allotrope of carbon known as graphene in presence of magnetic field, with special focus on edge states. The structure of graphene is described in detail and from the structure, two models are formed. The Dirac equation is a good description of graphene for large samples, far away from edges, where the boundaries can be ignored. However, it causes problems with most types of edge and hard wall approximation has to be implemented.
The Dirac equation is described in detail and used to obtain an energy spectrum, wavefunction and density of states for graphene edge in a strong magnetic field. For comparison, a Bohr-Sommerfield approximation was used to find the dispersion relation and compare it to the results obtained numerically from the Dirac equation.
The second model, better fitting for nano-scale systems, is the tight binding model. This model was utilized to find Energy spectrum for graphene flakes in magnetic field, which resembles Hofstadter's butterfly spectrum. The spectrum was analyzed and periodic oscillations of magnetisation dependent on magnetic field (known as the de Haas-van Alphen effect) were described. The oscillation of magnetisation depends on the shape of the dot, even though the main properties remain the same: at low magnetic field, periodic oscillations due to Aharonov-Bohm effect, turning into more chaotic oscillations depending on the boundary conditions of the given quantum dot.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00Z