Electronic processes, such as electronic breakdown and electron emission, in gases and liquids have implications in microplasmas, laser applications, water purification, biomedical applications, geographical mapping, and radiation detection. Electron emission and breakdown mechanisms are heavily researched and characterized in gases. Much of the current research into these mechanisms is focused on unifying breakdown and emission mechanisms. For electron emission, these mechanisms include field emission (FE), space-charge-limited emission (SCLE), thermionic emission (TE), and photoemission (PE), while gas breakdown emission mechanisms include Paschen’s law (PL) and Townsend breakdown (TB)with ion-enhanced FE becoming important at microscale. This research first unified SCLE and FE in vacuum and has been extended to include SCLE with collisions (for a gas at non-vacuum) and TE. This thesis extends this approach in electron emission unification, referred to as “nexus” theory, in two directions. First, we will apply this theory to liquids to examine the transition from FE to SCLE and hypothesize about the implications should there be a phase change. Second, we will incorporate PE, which becomes important with increasing interest in ultrafast laser phenomena at nanoscale and development of solar cells, with SCLE, TE, and FE.<div><br></div><div>Initial nexus theory studies included gas at non-vacuum pressures by including electron mobility in the electron force law. In principle, this behavior should be the same whether the medium is air or liquid. Electron emission and breakdown, which can arise from field emission, are increasingly important in plasma water treatment, pulsed power systems, radiation detection, and even understanding the physics of high electric fields applied to liquid helium for the Spallation Neutron Source. To demonstrate the applicability of nexus theory to liquids, we fit experimental data for electron emission in hydrocarbons to the full theory unifying FE to SCLE with and without collisions. The measured current followed Fowler-Nordheim scaling for FE at lower voltages with space charge beginning to contribute at higher voltages; none of the hydrocarbons study fully transitioned to Mott-Gurney (SCLE with collisions) scaling within the experimentally studied parameter range. Considering a higher mobility representative of a vapor in the theory demonstrates the feasibility of achieving Child-Langmuir (SCLE in vacuum)scaling for the gaps of the size considered experimentally. Thus, this approach may ultimately be applied to model electron emission during both phases changes and transitions between the mechanisms.<br></div><div><br></div><div>We next extended the gas nexus theory to analyze the transitions between PE and the other emission mechanisms. We modified the previous theory that used the generalized thermal-field emission (GTF) theory for electron current to instead use the generalized thermal-field photoemission (GTFP) theory. Using this, we obtained exact solutions for current as a function of applied voltage and demonstrated the asymptotic behavior with regard to the modified Fowler DuBridge (MFD) equation, which models PE. We combined the MFD equation with the other asymptotic solutions to develop state diagrams unifying the various emission mechanisms to provide guidance to the mechanisms and transitions relevant under various conditions of mobility, gap distance, temperature, and laser energy/wavelength/frequency. These diagrams provide guidance on which asymptotic solution or more detailed theory would be necessary to accurately relate current and voltage under various operating conditions.<br></div>

<div>Developments in modern electronics drive device design to smaller scale and higher electric fields and currents. Device size reductions to microscale and smaller have invalidated the assumption of avalanche formation for the traditional Paschen’s law for predicting gas breakdown. Under these conditions, the stronger electric fields induce field emission driven microscale gas breakdown; however, these theories often rely upon semi-empirical models to account for surface effects and the dependence of gas ionization on electric field, making them difficult to use for predicting device behavior a priori.</div><div>This dissertation hypothesizes that one may predict a priori how to tune emission physics and breakdown conditions for various electrode conditions (sharpness and surface roughness), gap size, and pressure. Specifically, it focuses on experiments to demonstrate the implications of surface roughness and emitter shape on gas breakdown for microscale and nanoscale devices at atmospheric pressure and simulations to extend traditional semi-empirical representations of the ionization coefficient to the relevant electric fields for these operating conditions.</div><div>First, this dissertation reports the effect of multiple discharges for 1 μm, 5 μm, and 10 μm gaps at atmospheric pressure. Multiple breakdown events create circular craters to 40 μm deep with crater depth more pronounced for smaller gap sizes and greater cathode surface roughness. Theoretical models of microscale breakdown using this modified effective gap distance agree well with the experimental results.</div><div>We next investigated the implications of gap distance and protrusion sharpness for nanoscale devices made of gold and titanium layered onto silicon wafers electrically isolated with SiO2 for gas breakdown and electron emission at atmospheric pressure. At lower voltages, the emitted current followed the Fowler-Nordheim (FN) law for field emission (FE). For either a 28 nm or 450 nm gap, gas breakdown occurred directly from FE, as observed for microscale gaps. For a 125 nm gap, emission current begins to transition toward the Mott-Gurney law for space-charge limited emission (SCLE) with collisions prior to undergoing breakdown. Thus, depending upon the conditions, gas breakdown may directly transition from either SCLE or FE for submicroscale gaps.</div><div>Applying microscale gas breakdown theories to predict this experimental behavior requires appropriately accounting for all physical parameters in the model. One critical parameter in these theories is the ionization coefficient, which has been determined semi-empirically with fitting parameters tabulated in the literature. Because these models fail at the strong electric fields relevant to the experiments reported above, we performed particle-in-cell simulations to calculate the ionization coefficient for argon and helium at various gap distances, pressures, and applied voltages to derive more comprehensive semi-empirical relationships to incorporate into breakdown theories.</div><div>In summary, this dissertation provides the first comprehensive assessment of the implications of surface roughness on microscale gas breakdown, the transition in gas breakdown and electron emission mechanisms at nanoscale, and the extension of semi-empirical laws for ionization coefficient. These results will be valuable in developing theories to predict electron emission and gas breakdown conditions for guiding nanoscale device design.</div>

<p dir="ltr">Calculating space-charge limited current (SCLC) is a critical problem in plasma physics and intense particle beams. Accurate calculations are important for validation and verification of particle-in-cell (PIC) simulations. The theoretical assessment of SCLC is complicated by the nonlinearity of the Poisson equation when combined with the energy balance and continuity equations. This dissertation provides several theoretical tools to convert the nonlinear Poisson equation into a corresponding linear differential equation, which is then solved for numerous geometries of practical interest.</p><p dir="ltr">The first and second chapters briefly summarize the application of variational calculus (VC) to solve for one-dimensional (1D) SCLC in cylindrical and spherical diode geometries by extremizing the current in the gap. Next, conformal mapping (CM) is presented to convert the concentric cylindrical diode geometry into a planar geometry to obtain the same SCLC solution as VC. In the next chapter, SCLC is determined for several geometries with curvilinear electron flow that cannot be solved using VC because the Poisson equation cannot be written easily. We then map a hyperboloid tip onto a plane to form a non-Euclidean disk (Poincaré disk). These mappings on to Poincaré disk are utilized to solve for SCLC in tip-to-tip and tip-to-plane geometries. Lie symmetries are then introduced to solve for SCLC with nonzero monoenergetic injection velocity, recovering the solutions for concentric cylinders, concentric spheres, tip-to-plane, and tip-to-tip for zero injection velocity. We then extend the SCLC calculations to account for any geometry in multiple dimensions by using VC and vacuum capacitance. First, we derive a relationship between the space-charge limited (SCL) potential and vacuum potential that holds for any geometry. This relationship is utilized to obtain exact closed-form solutions for SCLC in two-dimensional (2D) and three-dimensional (3D) planar geometries considering emission from the full surface of the cathode. PIC simulations using VSim were performed that agreed with the SCLC in 2D diode with a maximum error of 13%. In the final chapters, we extend these multidimensional SCLC calculations to nonzero monoenergetic emission. The SCLC in any orthogonal diode in any number of dimensions is obtained by relating it to the vacuum capacitance. The current in the bifurcation regime is also derived from first-principles from vacuum capacitance. The simulations performed in VSim agreed with the theory with a maximum error of 7%.</p><p dir="ltr">These mathematical techniques form a set of powerful tools that extend prior studies by yielding exact and approximate SCLC in numerous nonplanar and multidimensional diode geometries, thereby not requiring expensive and time-consuming PIC simulations. While more experiments are required to benchmark the validity of these calculations, these results may ultimately prove useful by providing a rapid first-principles approach to determine SCLC for many geometries that can be used to assess the validity of PIC simulations and facilitate multiphysics simulations.</p>