School of Physics - Theses
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ItemAstigmatic phase retrieval of lightfields with helical wavefronts( 2012)The controlled use of coherent radiation has led to the development of a wide range of imaging methods in which aspects of the phase are enhanced through diffraction and propagation. A mathematical description of the propagation of light allows us to determine the properties of an optical wavefield in any plane. When a sample is illuminated with coherent planar illumination and its diffracted wavefield is recorded in the far-field of propagation, a direct inverse calculation of the phase can be quickly performed through computational means – the fast Fourier transform. Algorithmic processing is required, however, because only the intensity of the diffracted wavefield can be recorded. To determine structural information about the sample, some other information must be known about the experimental system. What is known, and how it is processed computationally, has led to the development and successful application of a broad spectrum of phase reconstruction iterative algorithms. Vortices in lightfields have a helical structure to their wavefront, at the core of which exists, necessarily, a screw-discontinuity to their phase. They have a characteristic intensity distribution comprising a radially symmetric bright ring around a dark core which, for either handedness of the rotation of the vortex, appears identical. Observation of a vortex is, therefore, ambiguous in its ability to determine its true direction of rotation. The ubiquitous presence of vortices in all lightfields hinder the success of phase reconstruction methods based on planar illumination and, if successful, render any reconstruction of the phase non-unique, due to the ambiguity associated to their helicity. The presence of a controlled spherical phase distortion can break the symmetry of the appearance of the vortices and, hence, remove the ambiguity from the system and drive algorithms to a solution. For the pathological case of an on-axis vortex, however, spherical distortion will not break the radial symmetry. The astigmatic phase retrieval method separates the spherical distortion into cylindrical distortion in two orthogonal directions. This form of phase distortion breaks the symmetry of a vortex allowing a unique determination of the phase. The incorporation of such use of cylindrical distortion into an iterative phase reconstruction algorithm forms the basis for the astigmatic phase retrieval (APR) method. Presented in this thesis is the creation and propagation of lightfields with helical wavefronts, produced through simulation and experiment. Observation of the effects of cylindrical distortion on vortices is explored in detail, particularly for split high-charge vortices where their positions can inform the type and strength of the applied phase distortion. Experimentally, onaxis vortices are created and distorted for the purposes of astigmatic phase retrieval in both visible light and X-ray wavefields. This thesis presents the first experimental demonstration of the astigmatic phase retrieval (APR) method, successfully applied optically with a simple test sample. The method is also applied to lightfields with helical wavefronts. The successful unambiguous reconstruction of on-axis chargeone and charge-two visible light vortices are presented, which is the first experimental demonstration on the unique phase reconstruction of an on-axis vortex from intensity measurements alone. Experiments are then performed to apply the method to vortices created in X-ray wavefields. The parameters of the experiment and the data have not, however, allowed for a successful reconstruction in this case. It is demonstrated through extensive simulation analysis that the APR method is a fast and robust imaging method. It is also shown that, through observation of the error metric, experimental parameters can be corrected or even determined, making the method successful even if there is no a priori knowledge of the experimental system. The application of the APR method as a general imaging technique for use in high-resolution X-ray diffraction experiments is, therefore, is a logical extension of the work of this thesis.
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ItemPhysics of low-dimensional nanostructures( 2012)Nanoscale constructs are offering access to the quantum mechanical regime due to their constrained size. The unusual, and often counterintuitive, behaviours of such constructs are of considerable interest to those developing new devices across several fields, including (but not limited to) quantum computing, communications, in vivo applications such as the bionic eye and bio-sensors, standard electronics and computing, and magnetometry. The physics of zero-, one-, and two-dimensional nanostructures comprised of various dopants or arrays of dopants in either diamond or silicon are presented and discussed. In particular, the zero-dimensional Xe-related defects in diamond are considered theoretically, via density functional theory, lattice dynamics, and thermodynamics. Xe defects have also been characterised experimentally via the probe-enhanced Ra- man spectroscopy (PERS) technique. In silicon, a one-dimensional nanowire consisting of P donors is studied with density functional theory. This wire is monatomically thin in one direction, and two donors wide in the other, with the donors spaced at the currently realisable sheet density of 25%. The two-dimensional case of infinite monatomically thin sheets of P donors is considered, both with effective mass theory and density functional theory (which is again undertaken for the most common experimental sheet density, 25%). The effective mass theory model has been applied to several sheet densities, agrees well with literature calculations of sheets with in-plane disorder, is far more rapid in execution, and offers an analytic scaling theory to describe the dependence of several key results on the sheet density. The density functional theory approach is then extended to the quasi-two dimensional case of bilayers of monatomically thin P sheets, in order to address the approach to minimal two-dimensional confinement.
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ItemNovel applications of condensed matter theory: solid-light, quantum metamaterials, and quantum graphity( 2012)Condensed matter research is in essence the study of many-body interactions and the properties emergent from this complex process. Because of this, the condensed matter framework provides versatile concepts and techniques that can be applied to other strongly interacting systems that traditionally have no relation with condensed matter physics. This thesis embodies such as connection by borrowing from the condensed matter framework to study quantum optical systems of uniform lattice structures, extending this analysis to the fields of quantum emulation and metamaterials to study non-uniform lattice systems, and finally building on these results to study dynamical lattice models of quantum graphity. The conjunction of atom-cavity physics and photonic structures (``solid-light'' systems) offers new opportunities in terms of more device functionality and the probing of designed emulators of condensed matter systems. By analogy to the one-electron approximation of condensed matter physics, we introduce the one-polariton approximation to study a solid-light system. Using this approximation we apply Bloch states to the Jaynes-Cummings-Hubbard model of optical cavities to analytically determine the energy band structure. We demonstrate how the band structure can inform the phases of the system. We further apply this approach to systems with detuning impurities to design a solid-light based semiconductor emulator. Conventional metamaterials represent one of the most important frontiers in optical design, with applications in diverse fields ranging from medicine to aerospace. Up until now however, metamaterials have themselves been classical structures and interact only with the classical properties of light. Here we describe a class of dynamic metamaterial, based on the quantum properties of coupled cavity arrays, which are intrinsically lossless, reconfigurable, and operate fundamentally at the quantum level. Termed cavity array metamaterials (CAMs), we show that, combined with atom-cavity couplings, they can be used to create a reconfigurable quantum superlens possessing a negative index gradient for single photon imaging. Transformation optics is an important methodology for the control of electromagnetic trajectories in metamatarials. However since the size of the constituent elements of CAMs is commensurate with the operating wavelength, classical transformation optics cannot be applied on this platform. By directly transforming the internal geometry of the system, we provide an alternative framework suitable for tight-binding implementation of metamaterials. We investigate CAM-based homeomorphic cloaking as a case study. In a more general sense, with the inherent features of quantum superposition and entanglement of metamaterial properties, CAMs promise to open a new vista for quantum science and technology. Quantum graphity (QG), a background independent condensed matter model of quantum gravity, offers the intriguing notion that space emerges in the low energy states of the spatial degrees of freedom of a dynamical lattice. We explore metastable domain structures which are likely to exists in the low energy phase of QG lattice evolution. Specifically, through a simulated annealing process we investigate the formation of metastable defects at domain boundaries and the effects of domain structures on the propagation of bosons. We show that these structures should have observable consequences including scattering, double imaging, and gravitational lensing-like effects. Importantly the results serve as a framework that may be used to test QG.
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ItemThe fundamental plane and peculiar velocities from the 6dF galaxy survey( 2012)Early-type galaxies (ellipticals and lenticulars) are observed to populate the relation known as the Fundamental Plane that links their effective radius, R_e, stellar velocity dispersion, σ, and mean surface brightness, I_e. We have measured Fundamental Plane parameters in the near-infrared J, H and K passbands for ~10^4 of the brightest early-type galaxies in the 6dF Galaxy Survey (6dFGS). We improve upon previous regression techniques used to derive the Fundamental Plane by developing a robust maximum likelihood algorithm for fitting the galaxy distribution in Fundamental Plane space with a 3D Gaussian model. We exploit this large near-infrared-selected sample of galaxies to investigate trends in the Fundamental Plane with stellar population, morphology and environment. The 6dFGS galaxies exhibit clear stellar population trends in Fundamental Plane space, with age varying most strongly orthogonal to the plane. Remarkably, none of the stellar population parameters vary along the long axis of the plane, which corresponds to luminosity density. The Fundamental Plane slopes show little variation with either morphology or environment, but the Fundamental Plane size zeropoint is systematically larger for galaxies in lower density environments and for early-type spiral bulges. We speculate that age drives all the trends with residuals about the plane through its correlation with environment, morphology and metallicity. Using the Fundamental Plane, we measure distances and peculiar velocities for ~10^4 6dFGS galaxies to form the largest and most homogeneous peculiar velocity sample to date. Using a maximum-likelihood approach, we measure the overall bulk galaxy motions from the 6dFGS velocity field for the local volume of the universe, finding broad agreement with the predicted velocity field constructed from the 2MASS Redshift Survey. The local volume out to 16 120 km/s is found to have a bulk motion of 337 km/s in the direction (l,b) = (313°±9°,14°±10°), in good agreement with the results of other recent studies. A comparison of the observed and predicted fields is used to constrain parameters relating the distribution of galaxies and matter. We obtain a redshift-space distortion parameter β = 0.29±0.06 and a bias parameter for the 6dFGS velocity sample of b = 1.69±0.36. The 6dFGS velocity field provides an independent probe of cosmological parameters defining models of large-scale structure formation. Next steps include: (i) combining the 6dFGS sample in the south with the SDSS sample in the north to obtain an all-sky velocity field; (ii) deriving additional constraints on cosmological parameters from the velocity power spectrum analysis; and (iii) comparing the Fundamental Plane distances and velocities for early-type galaxies with the Tully-Fisher distances and velocities for spiral galaxies that will be obtained with the WALLABY survey on the Australian SKA Pathfinder.
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ItemFluctuation X-ray microscopy using curved beam illumination( 2012)Characterisation of the structural order of amorphous materials is a challenging problem. The presence, or absence, of structural order at any length scale can effect the properties of a material and hence its applications. Fluctuation electron microscopy is a well established technique that characterises structural order at intermediate length scales, referred to as medium range order. In this work the development of fluctuation X-ray microscopy using a probe with phase curvature has been investigated. The numerous commonalities between fluctuation microscopy and the iterative phase retrieval method of coherent diffractive imaging are explored. A hybrid method, ptychographic fluctuation microscopy, has been developed. Diffraction data from a film of densely packed polystyrene latex spheres was collected using the curved beam past the focus of a Fresnel zone plate at several different focus-to-sample distances. Several methods of characterising medium range order in the sample were investigated using these data, leading to the development of ptychographic fluctuation microscopy. This method reconstructs an extended complex sample function, which is then analysed using a fluctuation microscopy algorithm. A highly detailed `fluctuation map', detailing the behaviour of the diffraction patterns as a function of probe diameter and scattering vector has been produced. Analysis of this map indicates the presence of structural order within the sample at two distinct length scales, corresponding to order in the dense random packing of the spheres. Ptychographic fluctuation microscopy allows the characterisation of structural order in amorphous materials using a single data scan, unlike fluctuation microscopy which requires several scans. The method has been developed using X-rays, with potential application using next generation sources such as X-ray free electron lasers, but is fully compatible with electron radiation and could be applied using such sources with little change of approach. Approved by Bernadine 08/10/2012
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ItemFabrication, characterization and application of anodic aluminum oxide( 2012)A great many scientists have begun the long march towards the routine production of high quality nanomaterials, by focusing on simple and efficient methods for their bulk production. These high quality nanomaterials must exhibit uniform distribution and size properties if they are to be successfully applied to various technologies. Following the successful growth of carbon nanotubes, great interest has also been sparked in the field of low-dimensional materials, including nanowires, nanotubes and nanodots, an interest which will be further enhanced if these materials can be controllably produced over large surface areas. This is because highly ordered nanostructure arrays not only produce favorable electrical properties, they can also exhibit properties useful for many optical applications. Due to this large range of promising applications, there is significant motivation for improvements in the fabrication methods of these highly-ordered nanomaterials, including aspects such as cost minimization and ease-of-fabrication issues. To-date one of the most efficient and effective fabrication techniques is to employ template assisted methods, with different periodic template structures, since these techniques are relatively easy to implement and are capable of producing large area materials for a low cost. In this area, one of the most popular of these template materials is anodic aluminum oxide (AAO). AAO templates have been used to fabricate different nanowires, tubes and rods, using techniques such as evaporation, Sol-gel deposition and chemical vapor deposition. These templates have also been used as nanomasks for heavy metal-ion implantation, in order to generate highly ordered array structures, using very high energy ions. Since AAO itself is a transparent dielectric its optical properties have also been studied. For example, large pore sized (200 nm - 500 nm) AAO has been reported to exhibit an optical bandgap in the visible and near infrared region, under normal transmission measurements. Further experiments have also investigated the optical properties of AAO compound materials, including metal-filled metamaterial templates. A cursory review of the above-described work would seem to portray AAO as a material already comprehensively investigated. However, the precise formation mechanism for its self-assembled and highly ordered honeycomb structure is still not clear. In order to design and prepare different orientations and distributions of nanoporous alumina, for particular applications in the future, the mechanisms involved in formation of AAO templates by electrochemical reactions still need to be understood and controlled. Moreover, the internal structure of AAO templates are of considerable interest, with evidence that the stoichiometric ratio of Al and O is not the simple 2:3. It is widely recognized that the real Achilles heel of AAO is its thermal instability. Researchers who have made efforts to fully realize this material's potential for a range of devices have found that at high temperatures above 700 °C it cracks or distorts and loses its highly ordered structure. This material weakness has inhibited the wider application of AAO and overcoming it will require more information about its formation and properties. It is also desirable to improve AAO fabrication methods, making them simpler, and explore more novel applications for this promising material. With all of these motivations in mind it is important to find ways to successfully fabricate AAO using simple equipment and methods, and also to develop rapid methods for the characterization of the material. In this thesis, using a purpose-built electrochemical system, AAO templates are successfully fabricated. The first part of this thesis reports the investigations of the effects of the processing conditions on pore size, interpore distance and thickness of AAO are reported. Specific AAO etching and aluminum removal processes are also carefully investigated, utilizing a range of characterization techniques, such as AFM, SEM, FIB, TEM, EDX and XRD. TEM is found to be an especially important tool for the investigation of crystalline changes during the anodization of aluminum into AAO, whilst also being useful for exploration of its extremely straight, high aspect ratio channels. FIB techniques are adopted for the preparation of cross-sectional TEM samples, ultimately providing very important information on the inner structure of AAO, suitable for nano-characterization with HR-TEM and EDX analysis. The finding of inner/outer layer wall structures in these AAO channels is understood through comparison with COMSOL simulations of AAO formation models from literature. Whilst highlighting new aspects of AAO templates, such as the existence of aluminium rich and poor regions within the film, these results reveal first hand evidence of the mechanism of AAO formation, which is further complemented by direct exploration of specific etching mechanisms of these materials in the thesis. With the aid of image processing, pore sizes, inter pore distance distributions, pore circularizes and film porosities are investigated. Domain structures and defects across large AAO film areas are also studied in this way. The extent of 2D array ordering is qualitatively studied using Radial distribution functions. In the second part of the thesis AAO formation is investigated under a set of different conditions. •The fabrication of thin AAO materials onto quartz is reported using direct evaporation of aluminum on quartz and fabricating AAO in the electrochemical cell, aimed at conquering its Achilles heel , particularly in high temperature atmospheres and for optical and fabricating other nanostructure applications. •By taking advantage of the relationship between electric field currents and channel growth during AAO synthesis, multilayered porous AAO templates are also produced, by introducing a novel geometric bridge resistance during anodization, which induces spatial variations in the pore sizes being produced. •AAO templates were fabricated on curved surfaces. It is found that channels always grow into the film oriented normal to the original aluminium surface, whilst at the same time maintaining the shape of the arbitrary surface. The results of these studies were used to formulate an integrated model for the formation mechanism of the AAO templates. In the third part of the thesis, potential application areas are reported. •AAO templates and their periodic channels were used as a nanoscale caketin to create novel nanostructures. Simply by using the electroless deposition method with silver mirror test (Tollens reaction), 5 nm uniform silver nanoparticles and highly ordered silver nanowire arrays are fabricated, under different concentrations of silver reactant solution and annealing temperature. •We report the discovery of periodic triangular dot arrays on the top of these AAO templates. According to extensive characterization of the AAO template inner structure, these tips are composed of aluminum rich regions, which are themselves used for the production of aluminum droplets and applied as seeds for ultrathin SiC\AlSiC nanowire growths. The wires are of the order a few microns but with a uniform diameter of less than 10 nm. •AAO is investigated specifically as a periodic mask for ion implantation and ion beam exposure applications, including the implantation of phosphorous into semiconductor materials. In this way, with the aid of a larger pore size (200 nm) of AAO nanomasks, phosphorous doping is used to modify the conductivity of a silicon surface. Other similar investigations reported here include evaporation of gold through a tilted AAO nanomask, forming ellipse-shaped gold nanodot arrays, which can be controlled without changes to the periodic constant of the mask being used. This angle-dependent tuning is also applied in ion beam exposure investigations, where for the first time a white light source continuum-energy ion beam is generated. A theoretical model, Geant 4 simulation is utilized to further understand these experimental results. •The third major AAO application presented here relates to the template itself. As a periodic dielectric material, it can form a natural photonic crystal. In contrast to many published research studies about AAO as a photonic crystal, here it is studied for the first time using a dark field scattering mode to investigate the optical properties of AAO, and it is found that 60 nm pore-size thin AAO templates, grown on quartz, show partial bandgaps in the visible range of light. Software simulation and measurement results are matched to each other from these studies which are described in Chapter 9. k-space (back focal plane image) ring structures are also found using light propagation through the thicker (>20 µm) freestanding AAO templates, and are found to be polarization independent. Moreover, the measurements find that the thick, freestanding AAO changes the light propagation by using cross-polarization measurement, which indicated that thick AAO templates may be act as optical waveguides. In summary, via detailed structural and optical investigations of the properties of AAO templates we have elucidated the formation mechanism and extended the range of applications that have previously been identified. The techniques developed would appear to be scalable, at least to sizes of the order of meters, thus motivating further effort to exploit this material for large scale, and cost effective nanoscale fabrication.
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ItemThe theory of the nitrogen-vacancy colour centre in diamond( 2012)The nitrogen-vacancy (NV) colour centre in diamond is a model system for many quantum technologies including, metrology, information processing and communications. The NV centre is also highly suitable for employment in various nanotechnology applications, such as biological and sub-diffraction limit imaging, and in tests of fundamental physics, such as cavity quantum electrodynamics and the quantum entanglement of mesoscopic systems. The remarkable properties of the centre are however, not currently fully understood, with several unresolved issues limiting the performance of the centre in its many important applications. As the unresolved issues are interrelated and concern different aspects of the centre's properties, they may only be resolved by the development of a single self-consistent theory of the NV centre. The aim of this work has been to develop such a theory. The theory has been developed using a combination of the molecular model of deep level defects in semiconductors, group theoretical methods and ab initio calculations. The highly structured nature of the theory will enable its future use in the systematic identification of other colour centres that possess properties that exceed those of the NV centre.
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ItemNanoscale quantum sensing using nitrogen-vacancy centres in diamond( 2012)Devices that detect and spatially image magnetic fields are important in many areas of study including chemistry, electronics, materials science and biology. By extending the boundaries of what is currently achievable we may begin to explore areas previously inaccessible to science, such as wide-field imaging of neuron signaling or structural determination of single molecules. Here we present experimental progress towards the development of a nanoscale magnetic sensor operating under ambient conditions using the nitrogen-vacancy (NV) centre in diamond. This thesis describes the construction of a lab-built, confocal microscope capable of detecting single NV centres, with additional microwave control for coherent manipulation of the NV spin. The feasibility of using NV-diamond for real-time detection of the action potential generated by a neuron, and with high spatial resolution is experimentally demonstrated. The quantum coherence of single NV spins is monitored in various chemical solutions, and detection of nanoscale magnetic environments external to the diamond are demonstrated. This work sets the experimental foundation for using manufactured single quantum systems as sensitive probes of external chemical environments. In addition, the first measurements of single quantum coherent spins inside living cells are performed with NV centres in nanodiamonds. These studies on NV centres inside living cells demonstrate their promise as magnetic sensors for biology. Furthermore, alternative quantum sensing technologies emerge including rotational tracking of nanodiamonds and enhanced identification of fluorescent nanoparticles.
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ItemThe broad emission line region of quasars and gravitational lensing by early-type galaxies( 2012)This thesis has focused on predicting emission line flux ratios from the broad emission line region of quasars under different physical conditions, and measuring the dark matter fraction and total mass density slope within early-type galaxies using gravitational lensing. Quasars are the energetic cores of distant galaxies, and they reside in some of the oldest, most massive objects formed in the universe. Due to their incredible luminosity (as much as $10^5$ times greater than a typical galaxy), quasars can be observed at extremely large distances. Quasars have a unique spectrum, with bright, broad emission lines that are produced by photoionised gas that is close to the central super-massive black hole. Despite the prominence of these broad emission features, the gas physical conditions and the geometry of the emission region are poorly understood. Due to its small scale and large distance, the emission line region cannot be resolved directly — even with the most powerful telescopes — and simulations are required to understand the mechanism that produces the unique quasar spectrum. Using simulations of micro-physical processes, including photoionisation, the broad emission line flux ratios can be calculated for a range of gas densities and distances from the central black hole. Using the photoionisation code, Cloudy, hydrogen and helium line emission was over the range of possible broad emission line region conditions. The hydrogen and helium lines are of particular interest because the line emission has strong dependence on the gas number density and incident ionising flux, whilst having only a negligible dependence on several other free parameters of the model. These simulations were then used to find a set of interesting ratios that can be used to determine the limits on the upper limit on the gas number density, and outer radius of the emission region. This thesis demonstrates a new technique for determining the physical conditions of the broad line emitting gas in quasars, using optical and near-infrared hydrogen and helium emission lines. Near-infrared line ratios are advantageous, as they have a negligible dependence on the amount of internal dust. A locally optimally emitting cloud model of the broad emission line region was applied to four nearby (z $\sim$ 0.2) quasars from the Glikman et al. (2006) sample. By comparing simulated emission line ratios to measured ratios from optical and near-infrared spectroscopy, the physical conditions required to produce the observed emission lines were inferred. The model provides a good fit to three of the objects, and a fair fit to the fourth object. We find that low incident ionising fluxes ($phi <10^{18}$cm^-2 s^-1), and high gas densities (n>10^{12} cm^-3) are required to reproduce the observed line ratios. This analysis demonstrates that the use of composite spectra in photoionisation modelling is inappropriate; models must be fitted to the individual spectra of quasars. This thesis also derives properties of early-type galaxies using a joint gravitational lensing and stellar-dynamics analysis. The sample consists of 11 early-type galaxies from the Strong Lenses in the Legacy Survey (SL2S). The median deflector redshift is 0.5, making it the largest sample of intermediate redshift lenses that have been studied using a joint lensing and dynamics analysis. By combining measured redshifts and stellar velocity dispersions from Keck spectroscopy with lens models from Gavazzi et al. (2012, submitted), the total mass density slope inside the Einstein radius for each of the 11 lenses was derived. The average total density slope was found to be 2.16$\pm$0.9, with an intrinsic scatter of 0.25. The dark matter fraction for each lens within half the effective radius was also determined. The average projected dark matter mass fraction was found to be 0.42$\pm$0.08 with a scatter of 0.25 for a Salpeter initial mass function. By combining the SL2S results with those from previous studies, a mild trend in the cosmic evolution of the total mass density slope was found. This suggests that the total density profile of massive galaxies has become slightly steeper over cosmic time. If this result is confirmed by larger samples, it would indicate that either dissipative processes or off-axis major mergers play an important role in the growth of massive galaxies since a redshift of 1.
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ItemThick-brane worlds( 2012)We examine 4 + 1-dimensional field theories exhibiting both 3 + 1-dimensional standard model phenomenology and type 2 Randall-Sundrum gravity on a dynamically-generated thick brane. We consider problems with finding and guaranteeing the stability of thick brane solutions, and with the non-renormalizability of Yang-Mills gauge theory in 4 + 1-dimensions. This motivates us to take a more systematic approach to gauge theories and to study a richer variety of field theories beyond standard Yang-Mills gauge theories. This thesis comes in three parts. In the first part we explicitly construct a model featuring a single infinite extra dimension and an SO(10) grand unified theory, where standard model phenomenology is recovered on a 3 + 1-dimensional domain-wall brane topological defect. A 3 + 1-dimensional graviton and all standard model particles are dynamically localized to the brane. To localize gauge fields we invoke an analogue of the Dvali-Shifman mechanism. Similarly 3 + 1-dimensional left chiral fermions and effective 3 + 1-dimensional general relativity are recovered via the split fermion mechanism, and a Randall-Sundrum type 2 warped metric, respectively. We prove that this model is stable, respectively unstable, for two different regimes of free parameters in the dynamical equations. In the second part we use Lifshitz anisotropic scaling to fix inherent problems with the non-renormalizability of 4 + 1-dimensional Yang-Mills gauge theories. Here we address problems with the Dvali-Shifman mechanism by paving the way to a 4 + 1-dimensional domain-wall brane model in a Lifshitz field theory with critical exponent z = 2. We extend our analysis to look at specific examples of finite energy, stable, static topological defects in 3 + 1-dimensional Lifshitz field theories with critical exponent z = 2. These defects are forbidden in standard relativistic field theories by Derrick’s theorem. As such the observation of cosmic relics would be a hallmark signature of the breakdown of Lorentz invariance at short distances. In the final part of this thesis we consider grand unified theories (GUTs). These theories are important to a wide range of models in this thesis. We consider the set of vacuum expectation values which break the GUT to differently embedded isomorphic copies of a subgroup H. We define and characterize the relationship between these vacuum expectation values.