I have developed and improved a coherent anti-Stokes Raman scattering (CARS) microscope based on the spectral focusing (SF) technique. The CARS microscope uses an 800 nm oscillator and a photonic crystal fibre module to generate the supercontinuum Stokes. The photonic crystal fibre was originally designed to generate light beyond 945 nm which is useful for CARS microscopy in the CH/OH frequencies but essentially prevents access to the important fingerprint region at lower frequencies. With expert and nontraditional approaches to generating supercontinuum with sufficient power at wavelengths below 945 nm, I substantially extend the usefulness of the module for SF-CARS microscopy deep into the fingerprint region. Moreover, with the invention of a dynamic supercontinuum generation scheme we call "spectral surfing," I improve both the brightness of the CARS signal and extend the accessible CARS frequency range to frequencies as low as 350 cm$^{-1}$ and as high as 3500 cm$^{-1}$---all in a single scan-window. I demonstrate the capabilities of our broadband SF-CARS system through CARS and four-wave mixing hyperspectroscopy on samples such as astaxanthin, lily pollen and glass; liquid chemicals such as benzonitrile, nitrobenzene and dimethyl sulfoxide; and on pharmaceutical samples such as acetaminophen, ibuprofen, and cetirizine. Furthermore, In search of more useful Stokes supercontinuum sources, I compare the performance of two commercial photonic crystal fibre modules for use in SF-CARS applications, ultimately finding that one module provides better spectral characteristics for static supercontinuum use, while the other provides improved characteristics when spectral surfing is implemented. Author Keywords: coherent anti-Stokes Raman scattering, nonlinear microscopy, scanning microscopy, spectroscopy, supercontinuum generation, vibrational spectroscopy
Modelling the Lanthanum Aluminate-Strontium Titanate Interface with a Modified Transverse Ising Model
In 2004 it was discovered that a two-dimensional electron gas (2DEG) forms at the interface between lanthanum aluminate (LAO) and strontium titanate (STO). This 2DEG exhibits a variety of electronic and magnetic phenomena, motivating intense research into its applicability to electronic devices. Over the years several models have been developed in theoretical exploration of this system. Here, the transverse Ising model is applied to the LAO/STO interface for the first time. It is shown that the model as it is traditionally formulated cannot accurately predict the structure of the electron density at the interface. I show that this can be fixed with a simple modification of the model, and discuss how this modification affects both the polarization distribution in ferroelectric thin films and the electron density at the LAO/STO interface. The importance of including the depolarizing field when modelling spatially inhomogeneous ferroelectric systems is also explored. Author Keywords: ferroelectric thin film, lanthanum aluminate, strontium titanate, transverse Ising model, two-dimensional electron gas
Cluster Approach Applied to the One-Dimensional Anderson-Hubbard Model
S. Johri and R. Bhatt developed a real-space renormalization group approach aimed at extracting the localized single-particle eigenstates of the Anderson model from a large system by identifying clusters of resonant site potentials. E. Campbell generalized this real-space renormalization group approach using standard perturbation theory. Both approaches were intended to approximate the single-particle density of states of the Anderson model. In this thesis, we aimed to test the potential of applying a similar real-space renormalization group approach to calculate the density of states of the interacting Anderson-Hubbard model. Our interest in the density of states of this model is due to a V-shaped zero-bias anomaly in two-dimensional systems. A real-space renormalization group approach is best applied to a one-dimensional system. We found that the zero-bias anomaly is not V-shaped in one-dimension. To test the potential of a real-space renormalization group approach, we used the cluster approach which is the same as the non-interacting renormalization group approach but without the perturbation theory and found that for strong disorder this technique could accurately calculate the density of states over a wide range of energies but deviated from exact results at the band edge, at $\omega=\pm U$ and near $\omega=0$. The first two inaccuracies will be reduced with a proper real-space renormalization group approach. We suspect that the last inaccuracy is associated with long range physics and may be difficult to recover. We also developed a technique that adjusts the identification of clusters in the cluster approach to improve the computation time of the density of states with minimal loss of accuracy in a tunable range around the Fermi level. We found that this technique significantly reduced the computation time and was able to preserve the density of states near the Fermi level, except at the smallest energies near $\omega=0$. Author Keywords: Anderson-Hubbard model, renormalization group, Strong electron correlations, Zero-bias anomaly
Role of Dielectric Screening in SrTiO3-Based Interfaces
We build a theoretical model for exploring the electronic properties of the two-dimensional (2D) electron gas that forms at the interface between insulating SrTiO3 (STO) and a number of perovskite materials including LaTiO3, LaAlO3, and GdTiO3. The model treats conduction electrons within a tight-binding approximation, and the dielectric polarization via a Landau-Devonshire free energy that incorporates STO's strongly nonlinear, nonlocal, field-, and temperature-dependent dielectric response. We consider three models for the dielectric polarization at the interface: an ideal-interface model in which the interface has the same permittivity as the bulk, a dielectric dead-layer model in which the interface has permittivity lower that the bulk, and an interfacial-strain model in which the strain effects are included. The ideal-interface model band structure comprises a mix of quantum 2D states that are tightly bound to the interface, and quasi-three-dimensional (3D) states that extend hundreds of unit cells into the STO substrate. We find that there is a substantial shift of electrons away from the interface into the 3D tails as temperature is lowered from 300 K to 10 K. We speculate that the quasi-3D tails form the low- density high-mobility component of the interfacial electron gas that is widely inferred from magnetoresistance measurements. Multiple experiments have observed a sharp Lifshitz transition in the band structure of STO interfaces as a function of applied gate voltage. To understand this transition, we first propose a dielectric dead-layer model. It successfully predicts the Lifshitz transition at a critical charge density close to the measured one, but does not give a complete description for the transition. Second, we use an interfacial-strain model in which we consider the electrostrictive and flexoelectric coupling between the strain and polarization. This coupling generates a thin polarized layer whose direction reverses at a critical density. The transition occurs concomitantly with the polarization reversal. In addition, we find that the model captures the two main features of the transition: the transition from one occupied band to multiple occupied bands, and the abrupt change in the slope of lowest energy band with doping. Author Keywords:
Real-space renormalization group approach to the Anderson model
Many of the most interesting electronic behaviours currently being studied are associated with strong correlations. In addition, many of these materials are disordered either intrinsically or due to doping. Solving interacting systems exactly is extremely computationally expensive, and approximate techniques developed for strongly correlated systems are not easily adapted to include disorder. As a non-interacting disordered model, it makes sense to consider the Anderson model as a first step in developing an approximate method of solution to the interacting and disordered Anderson-Hubbard model. Our renormalization group (RG) approach is modeled on that proposed by Johri and Bhatt [23]. We found an error in their work which we have corrected in our procedure. After testing the execution of the RG, we benchmarked the density of states and inverse participation ratio results against exact diagonalization. Our approach is significantly faster than exact diagonalization and is most accurate in the limit of strong disorder. Author Keywords: disorder, localization, real-space renormalization, strong correlations
Correlating density of states features with localization strength in disordered interacting systems
Johri and Bhatt found singular behavior near the band edge in the density of states as well as in the inverse participation ratio of the Anderson model. These singularities mark a transition to an energy range dominated by resonant states. We study the interacting case using an ensemble of two-site Anderson-Hubbard systems. We find the ensemble-averaged density of states and generalized inverse participation ratio have more structure than in the non-interacting case because there are more transitions and in particular the transitions depend on the ground state. Nonetheless, there are regions of sharp decline in the generalized inverse participation ratio associated with specific density of state features. Moreover these features move closer to the Fermi level with the addition of interactions making them more experimentally accessible. Unfortunately resonances unique to interacting systems cannot be specifically identified. Author Keywords: Correlated electrons, Disorder, Localization
TWO-DIMENSIONAL CONDUCTIVITY AT LaAlO3/SrTiO3 INTERFACES
Experiments have observed a two-dimensional electron gas at the interface of two insulating oxides: strontium titanate (SrTiO3) and lanthanum aluminate (LaAlO3). These interfaces exhibit metallic, superconducting, and magnetic behaviours, which are strongly affected by impurities. Motivated by experiments, we introduce a simple model in which impurities lie at the interface. We treat the LaAlO3 as an insulator and model the SrTiO3 film. By solving a set of self-consistent Hartree equations for the charge density, we obtain the band structure of the SrTiO3 film. We then study the relative contributions made by the occupied bands to the two-dimensional conductivity of the LaAlO3/SrTiO3 interface. We find that the fractional conductivity of each band depends on several parameters: the mass anisotropy, the filling, and the impurity potential. Author Keywords: conductivity, impurities, insulating oxides, Two-dimensional electron gases

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