Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

Advanced Analytical Microscopy at the Nanoscale: Applications in Wide Bandgap and Solid Oxide Fuel Cell Materials

The atomic-level structure and chemistry of materials ultimately dictate their observed macroscopic properties and behavior. As such, an intimate understanding of these characteristics allows for better materials engineering and improvements in the resulting devices. In our work, two material systems were investigated using advanced electron and ion microscopy techniques, relating the measured nanoscale traits to overall device performance. First, transmission electron microscopy and electron energy loss spectroscopy (TEM-EELS) were used to analyze interfacial states at the semiconductor/oxide interface in wide bandgap SiC microelectronics. This interface contains defects that significantly diminish SiC device performance, and their fundamental nature remains generally unresolved. The impacts of various microfabrication techniques were explored, examining both current commercial and next-generation processing strategies. In further investigations, machine learning techniques were applied to the EELS data, revealing previously hidden Si, C, and O bonding states at the interface, which help explain the origins of mobility enhancement in SiC devices. Finally, the impacts of SiC bias temperature stressing on the interfacial region were explored. In the second system, focused ion beam/scanning electron microscopy (FIB/SEM) was used to reconstruct 3D models of solid oxide fuel cell (SOFC) cathodes. Since the specific degradation mechanisms of SOFC cathodes are poorly understood, FIB/SEM and TEM were used to analyze and quantify changes in the microstructure during performance degradation. Novel strategies for microstructure calculation from FIB-nanotomography data were developed and applied to LSM-YSZ and LSCF-GDC composite cathodes, aged with environmental contaminants to promote degradation. In LSM-YSZ, migration of both La and Mn cations to the grain boundaries of YSZ was observed using TEM-EELS. Few substantial changes however, were observed in the overall microstructure of the cells, correlating with a lack of performance degradation induced by the H2O. Using similar strategies, a series of LSCF-GDC cathodes were analyzed, aged in H2O, CO2, and Cr-vapor environments. FIB/SEM observation revealed considerable formation of secondary phases within these cathodes, and quantifiable modifications of the microstructure. In particular, Cr-poisoning was observed to cause substantial byproduct formation, which was correlated with drastic reductions in cell performance.

Characterization of Copper Covetic Bulk and Films: Copper with High Carbon Content

Incorporation of carbon nanostructures in metals is desirable to combine the strongly bonded electrons in the metal and the free electrons in carbon nanostructures that give rise to high ampacity and high conductivity, respectively. Carbon in copper has the potential to impact industries such as: building construction, power generation and transmission, and microelectronics. This thesis focuses on the structure and properties of bulk and thin films of a new material, Cu covetic, that contains carbon in concentrations up to 16 at.%. X-ray photoelectron spectroscopy (XPS) shows C 1s peak with both sp2 and sp3 bonded C measuring up to 3.5 wt.% (16 at.%). High resolution transmission electron microscopy and electron diffraction of bulk covetic samples show a modulated structure of ≈ 1.6 nm along several crystallographic directions in regions that have high C content suggesting that the carbon incorporates into the copper lattice forming a network. Electron energy loss spectra (EELS) from covetics reveal that the level of graphitization from the source material, activated carbon, is maintained in the covetic structure. Bulk Cu covetics have a slight increase in the lattice constant, as well as <111> texturing, or possibly a different structure, compared to pure Cu. Density functional theory calculations predict bonding between C and Cu at the edges and defects of graphene sheets. The electrical resistivity of bulk covetics first increases and then decreases with increasing C content. Cu covetic films were deposited using e-beam and pulsed laser deposition (PLD) at different temperatures. No copper oxide or any allotropes of carbon are present in the films. The e-beam films show enhanced electrical and optical properties when compared to pure Cu films of the same thickness even though no carbon was detected by XPS or EELS. They also have slightly higher ampacity than Cu metal films. EELS analysis of the C-K-edge in the PLD films indicate that graphitic carbon is transferred from the bulk into the films with uniform carbon distribution. PLD films exhibit flatter and higher transmittance curves and sheet resistance two orders of magnitude lower than e-beam films leading to a high figure of merit as transparent conductors.