Free algebras in Von Neumann-Bernays-Gӧdel set theory and positive elementary inductions in reasonable structures

This thesis consists of two independent chapters. The first chapter deals with universal algebra. It is shown, in von Neumann-Bernays-Gӧdel set theory, that free images of partial algebras exist in arbitrary varieties. It follows from this, as set-complete Boolean algebras form a variety, that there exist free set-complete Boolean algebras on any class of generators. This appears to contradict a well-known result of A. Hales and H. Gaifman, stating that there is no complete Boolean algebra on any infinite set of generators. However, it does not, as the algebras constructed in this chapter are allowed to be proper classes. The second chapter deals with positive elementary inductions. It is shown that, in any reasonable structure ᶆ, the inductive closure ordinal of ᶆ is admissible, by showing it is equal to an ordinal measuring the saturation of ᶆ. This is also used to show that non-recursively saturated models of the theories ACF, RCF, and DCF have inductive closure ordinals greater than _ω.

Numerical solution of parabolic equations by the box scheme

The box scheme proposed by H. B. Keller is a numerical method for solving parabolic partial differential equations. We give a convergence proof of this scheme for the heat equation, for a linear parabolic system, and for a class of nonlinear parabolic equations. Von Neumann stability is shown to hold for the box scheme combined with the method of fractional steps to solve the two-dimensional heat equation. Computations were performed on Burgers' equation with three different initial conditions, and Richardson extrapolation is shown to be effective.

Growth and characterization of novel, III-V semiconductor heterostructures

The material presented in this thesis concerns the growth and characterization of III-V semiconductor heterostructures. Studies of the interactions between bound states in coupled quantum wells and between well and barrier bound states in AlAs/GaAs heterostructures are presented. We also demonstrate the broad array of novel tunnel structures realizable in the InAs/GaSb/AlSb material system. Because of the unique broken-gap band alignment of InAs/GaSb these structures involve transport between the conduction- and valence-bands of adjacent layers. These devices possess a wide range of electrical properties and are fundamentally different from conventional AlAs/GaAs tunnel devices. We report on the fabrication of a novel tunnel transistor with the largest reported room temperature current gains. We also present time-resolved studies of the growth fronts of InAs/GainSb strained layer superlattices and investigations of surface anion exchange reactions.

Chapter 2 covers tunneling studies of conventional AlAs/GaAs RTD's. The results of two studies are presented: (i) A test of coherent vs. sequential tunneling in triple barrier heterostructures, (ii) An optical measurement of the effect of barrier X-point states on Γ-point well states. In the first it was found if two quantum wells are separated by a sufficiently thin barrier, then the eigenstates of the system extend coherently across both wells and the central barriers. For thicker barriers between the wells, the electrons become localized in the individual wells and transport is best described by the electrons hopping between the wells. In the second, it was found that Γ-point well states and X-point barrier states interact strongly. The barrier X-point states modify the energies of the well states and increase the escape rate for carriers in the quantum well.

The results of several experimental studies of a novel class of tunnel devices realized in the InAs/GaSb/AlSb material system are presented in Chapter 3. These interband tunnel structures involve transport between conduction- and valence-band states in adjacent material layers. These devices are compared and contrasted with the conventional AlAs/GaAs structures discussed in Chapter 2 and experimental results are presented for both resonant and nonresonant devices. These results are compared with theoretical simulations and necessary extensions to the theoretical models are discussed.

In chapter 4 experimental results from a novel tunnel transistor are reported. The measured current gains in this transistor exceed 100 at room temperature. This is the highest reported gain at room temperature for any tunnel transistor. The device is analyzed and the current conduction and gain mechanisms are discussed.

Chapters 5 and 6 are studies of the growth of structures involving layers with different anions. Chapter 5 covers the growth of InAs/GainSb superlattices for far infrared detectors and time resolved, in-situ studies of their growth fronts. It was found that the bandgap of superlattices with identical layer thicknesses and compositions varied by as much as 40 meV depending on how their internal interfaces are formed. The absorption lengths in superlattices with identical bandgaps but whose interfaces were formed in different ways varied by as much as a factor of two. First the superlattice is discussed including an explanation of the device and the complications involved in its growth. The experimental technique of reflection high energy electron diffraction (RHEED) is reviewed, and the results of RHEED studies of the growth of these complicated structures are presented. The development of a time resolved, in-situ characterization of the internal interfaces of these superlattices is described. Chapter 6 describes the result of a detailed study of some of the phenomena described in chapter 5. X-ray photoelectron spectroscopy (XPS) studies of anion exchange reactions on the growth fronts of these superlattices are reported. Concurrent RHEED studies of the same physical systems studied with XPS are presented. Using the RHEED and XPS results, a real-time, indirect measurement of surface exchange reactions was developed.

Investigation of hypervelocity impact phenomena using real-time concurrent diagnostics

Hypervelocity impact of meteoroids and orbital debris poses a serious and growing threat to spacecraft. To study hypervelocity impact phenomena, a comprehensive ensemble of real-time concurrently operated diagnostics has been developed and implemented in the Small Particle Hypervelocity Impact Range (SPHIR) facility. This suite of simultaneously operated instrumentation provides multiple complementary measurements that facilitate the characterization of many impact phenomena in a single experiment. The investigation of hypervelocity impact phenomena described in this work focuses on normal impacts of 1.8 mm nylon 6/6 cylinder projectiles and variable thickness aluminum targets. The SPHIR facility two-stage light-gas gun is capable of routinely launching 5.5 mg nylon impactors to speeds of 5 to 7 km/s. Refinement of legacy SPHIR operation procedures and the investigation of first-stage pressure have improved the velocity performance of the facility, resulting in an increase in average impact velocity of at least 0.57 km/s. Results for the perforation area indicate the considered range of target thicknesses represent multiple regimes describing the non-monotonic scaling of target perforation with decreasing target thickness. The laser side-lighting (LSL) system has been developed to provide ultra-high-speed shadowgraph images of the impact event. This novel optical technique is demonstrated to characterize the propagation velocity and two-dimensional optical density of impact-generated debris clouds. Additionally, a debris capture system is located behind the target during every experiment to provide complementary information regarding the trajectory distribution and penetration depth of individual debris particles. The utilization of a coherent, collimated illumination source in the LSL system facilitates the simultaneous measurement of impact phenomena with near-IR and UV-vis spectrograph systems. Comparison of LSL images to concurrent IR results indicates two distinctly different phenomena. A high-speed, pressure-dependent IR-emitting cloud is observed in experiments to expand at velocities much higher than the debris and ejecta phenomena observed using the LSL system. In double-plate target configurations, this phenomena is observed to interact with the rear-wall several micro-seconds before the subsequent arrival of the debris cloud. Additionally, dimensional analysis presented by Whitham for blast waves is shown to describe the pressure-dependent radial expansion of the observed IR-emitting phenomena. Although this work focuses on a single hypervelocity impact configuration, the diagnostic capabilities and techniques described can be used with a wide variety of impactors, materials, and geometries to investigate any number of engineering and scientific problems.