<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>

<p dir="ltr">This dissertation reports experiments and simulations of micro-/nanoscale electrical breakdown, connects them to the microscale breakdown theories, relates them to field emission and space-charge-limited conditions, and demonstrates the modification of the approach to microwave fields. It provides the first comprehensive experimental assessment of the transitions between electron emission and gas breakdown mechanisms at microscale and nanoscale and extension of semi-empirical laws for ionization process in DC and microwave. These findings will be valuable in developing theories to predict electron emission and gas breakdown mechanisms, which provides guidance for nanoscale device design.</p>

<p dir="ltr"> Conventional RF switching technologies struggle to simultaneously achieve high-power handling, low loss, high isolation, broadband operation, quick reconfiguration, high linearity, and low cost, which are desirable for many applications, including communications, radar, and sensors. Moreover, they require electrical bias networks, which degrade performance and, in many cases, inhibit wideband applications, including DC operation. On the other hand, plasma (photoconductive) switches use an optical bias to generate free charge carriers. Recently these switches have begun to not only rival conventional technologies in terms of performance metrics such as switching speeds and loss but have exceeded what is possible in terms of power handling. This work details the strides made in placing solid-state plasma technologies at the forefront of advanced, high-power switching applications including a novel high-power tuner and an absorptive/reflective SPnT switch. In various form factors, SSP has achieved analog control of loss as low as 0.09 dB and isolation as high as 54 dB, linearity of 68.8 dBm (IP3), 110 GHz instantaneous bandwidth, including DC, switching speeds as low as 3.5 us, 100+ W power handling, and 30+ W hot switching. In addition, comprehensive physics modeling has been developed to enable seamless design validation before fabrication commences. This thesis discusses the achievements and design considerations for creating optimized plasma switches and proposes a path for future applications.</p>

<p>  </p> <p>Electron emission plays a vital role in many modern technologies, from plasma medicine to heavy ion beams for fusion. An accurate theoretical model based upon the physics involved is critical to efficient operation of devices pushing the boundaries of complexity. The interactions between different electron emission mechanisms can severely alter device performance, especially when operating in extreme conditions. This dissertation studies electron emission from the perspectives of increasing geometric and physical mechanism complexities </p> <p>One half of this dissertation derives new relations for space-charge limited emission (SCLE) in non-planar geometries. SCLE is the maximum stable current that may be produced by electron emission before the electric field of the electrons themselves self-limits further emission. In planar devices, this is modeled by the well-established Child-Langmuir (CL) equation. The Langmuir-Blodgett (LB) equations remain the most commonly accepted theory for SCLE for cylindrical and spherical geometries after nearly a century; however, they suffer from being approximations based on a polynomial series expansion fit to a nonlinear differential equation. I derive exact, fully analytic equations for these geometries by using variational calculus to transform the differential equation into a new form that is fully and exactly solvable. This variational approach may be extended to any geometry and offers a full description of the electric field, velocity, and charge density profiles in the diode. </p> <p>SCLE is also an important mechanism for characterizing the operation of devices with an external magnetic field orthogonal to the electric field. This “crossed-field” problem decreases the limiting current as electrons travel longer, curved paths, effectively storing some charge in the gap (moving parallel to the emitter). At a critical magnetic field called the Hull cutoff, electron paths become so tightly curved that the circuit can no longer be completed, a condition called magnetic insulation. Crossed-field SCLE has been accurately modeled in planar devices by Lau and Christenson. Using the variational approach, I replicate their planar results and extend the calculation to cylindrical geometry, a common choice for magnetron devices. Further, I derive additional equations with simplified assumptions that, for the first time, provide an analytic description of experimental results below the Hull cutoff field. Following this I incorporate a series resistor: device resistance (or impedance) changes non-linearly with current and voltage, so I couple Ohm’s Law (OL) to all the models of crossed-field devices. For devices just below the Hull cutoff, I predict analytically and show in simulation novel bi-modal behavior, oscillating between magnetically insulated and non-insulated modes. With crossed-field device assessment, the variational calculus approach to space-charge may be used for numerous applications, including high power microwave sources, relativistic klystron devices, heavy ion beams, Hall thrusters, and plasma processing. </p> <p>The other half of this dissertation derives analytic theories to solve for emission current with three or more electron emission mechanisms simultaneously. In addition to the CL law, SCLE may also occur in neutral, non-vacuum diodes, modeled by the Mott-Gurney (MG) equation. These are the two limiting mechanisms I study; the other major modality of electron emission is direct electron production, the source of current in the device. Electrons are ejected when impelled by high temperature or electric field at the emission surface. These mechanisms are thermionic (or thermal) emission, modeled by the Richardson-Laue-Dushman (RLD) equation, and field emission, modeled by the Fowler-Nordheim (FN) equation, respectively. Additionally, just as I calculated the impedance of devices operating in a crossed-field configuration, all these models can be similarly coupled to OL. I derive models unifying FN, MG, and CL (with an extension linking OL, mentoring an undergraduate) and RLD, FN, and CL. These models are relevant for modern device design, especially as micro- and nano-scale devices seek to eliminate vacuum requirements and as space and military applications require higher temperature tolerances.</p> <p>While multi-physics models, like the ones described above, are important, the single-physics models (FN, RLD, MG, CL, OL) are still valid (and much easier to use) in their respective asymptotic limits. For example, a circuit behaves purely according to OL for very high resistances, according to MG for very high pressures, and so forth. Importantly, when devices operate in transition regions between these asymptotic limits, <em>none </em>of the asymptotic equations match the predictions of multi-physics models. Yet, intersections between the asymptotic equations are easily found, say for a certain set of voltage, gap distance, and pressure, CL=MG. Since both asymptotic equations give the same prediction, we may conclude that both must be inaccurate for those physical parameters! This gives rise to what I term “nexus theory:” solving two or more asymptotic equations simultaneously to rapidly and accurately predict sets of physical parameters at which multi-physics models (specifically, the physics leading to the “nexus point” parameters, points or curves at which nexus conditions are satisfied) are required for accurate device predictions. In fact, I show that multi-physics models are necessary within roughly one to two orders of magnitude from a nexus. In effect, nexus theory provides a simple, powerful tool to determine how complex a model is necessary for a particular device. Both nexus theory and multi-physics results in this dissertation have been successfully used to design devices to operate in specific transition regimes and identify the resulting device behavior.  </p>

<p dir="ltr">Understanding the acceleration of particles responsible for high-energy non-thermal phenomena in astrophysical jets is a ubiquitous pursuit. A possible culprit for non-thermal particle acceleration is turbulence. Specifically in this thesis, I investigate highly magne- tized or relativistic turbulence, where the magnetic energy to enthalpy ratio of the plasma is much greater than one, as a possible high-energy accelerator inside relativistic jets. I do this through three distinct projects. </p><p dir="ltr">My first project [1] (discussed in Section 3) was built upon a recent study of relativistic turbulence from [2], which found that a non-thermal particle equilibrium can be achieved when a plasma is heated via turbulence but allowed to cool radiatively. I extrapolated these results from PIC (Particle-in-Cell) simulations to larger scales and magnetizations, allowing me to encode key microphysical results of PIC simulations into a Fokker-Planck formalism. Combining these results with a single zone model for a blazar jet, I successfully define the underlying particle distribution with the global parameters of the emission region. To test this model, I fit data from 12 sources and successfully constrain key blazar parameters such as magnetization, bulk Lorentz factor, emission region size, and distance from the central engine. </p><p dir="ltr">My second project covers the development and testing of the open-source toolkit Tleco. This code base was used to evolve the Fokker-Planck equation and solve the resultant emission in my first project. Tleco offers efficient algorithms for evolving particle distributions and solving the resultant emission. It is meant to be user-friendly and easily customizable. </p><p dir="ltr">My third project attempts to enhance our understanding of coherent structures in relativistic turbulence. I employ intermittency analysis to establish a link between statistical fluctuations within the plasma and regions of high-energy dissipation. To achieve this, we used first-principle turbulent PIC simulations across a range of magnetizations and fluctuating magnetic field values. By utilizing the statistical fluctuations to determine the fractal dimension of the structures, I then examine their filling fraction and its dependence on magnetization and the fluctuating magnetic field.</p>

Motivée par deux problématiques liées aux plasmas de tokamak, cette thèse propose deux méthodes d'approximation numérique pour deux problèmes mathématiques s'y rattachant. D'une part, pour l'étude du transport turbulent de particules, une méthode numérique basée sur les schémas de subdivision est présentée pour la simulation de trajectoires de particules dans un champ de vitesse fortement variable. D'autre part, dans le cadre de la modélisation de l'interaction plasma-paroi, une méthode de pénalisation est proposée pour la prise en compte de conditions aux limites de type Neumann ou Robin. Analysées sur des problèmes modèles de complexité croissante, ces méthodes sont enfin appliquées dans des situations plus réalistes d'intérêt pratique dans l'étude du plasma de bord. / Motivated by two issues related to tokamak plasmas, this thesis proposes two numerical approximation methods for two mathematical problems associated with them. On the one hand, in order to study the turbulent transport of particles, a numerical method based on subdivision schemes is presented for the simulation of particle trajectories in a strongly varying velocity field. On the other hand, in the context of modeling the plasma-wall interaction, a penalization method is proposed to take into account Neumann or Robin boundary conditions. Analyzed on model problems of increasing complexity, these methods are finally applied in more realistic situations of practical interest in the study of the edge plasma.

In the aerospace industry, gas discharges have gained importance with the exploration of their performance and capabilities for flow control and combustion. Tunable properties of plasma make gas discharges efficient tools for various purposes. Since the scales of plasma and the available technology limit the knowledge gained from experimental studies, computational studies are essential to understand the results of experimental studies. The temporal and spatial scales of plasma also restrict the numerical studies. It is a necessity to use an idealized model, in which enough physics is captured, while the computational costs are acceptable.<br><br>In this work, numerical simulations of different low-pressure gas discharges are presented with a detailed analysis of the numerical approach. A one moment model is employed for DC glow discharges and nanosecond-pulse discharges. The cheap-est method regarding the modeling and simulation costs is chosen by checking the requirements of the fundamental processes of gas discharges. The verification of one-moment 1-D glow discharges with constant electron temperature variation is achieved by comparing other computational results.<br><br>The one moment model for pulse discharge simulation aims to capture the information from the experimental data for low-pressure argon discharges. Since the constant temperature assumption is crude, the local field approximation is investigated to obtain the data for electron temperature. It was observed that experimental data and computational data do not match because of the stagnant decay of electron number densities and temperatures. At the suggestion of the experimental group, water vapor was added as an impurity to the plasma chemistry. Although there was an improvement with the addition of water vapor, the results were still not in good agreement with experiment.<br><br>The applicability of the local field approximation was investigated, and non-local effects were included in the context of an averaged energy equation. A 0-D electron temperature equation was employed with the collision frequencies obtained from the local field approximation. It was observed that the shape of the decay profiles matched with the experimental data. The number densities; however, are less almost an order of magnitude.<br><br>As a final step, the two-moment model, one-moment model plus thermal electron energy equation, was solved to involve non-local effects. The two-moment model allows capturing of non-local effects and improves agreement with the experimental data. Overall, it was observed that non-local regions dominate low-pressure pulsed discharges. The local field approximation is not adequate to solve these types of discharges.

<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>Multiple industry applications, including combustion, flow control, and medicine, have leveraged nanosecond pulsed plasma (NPP) discharges to create plasma generated reactive species (PGRS). The PGRS are essential to induce plasma-assisted mechanisms, but the rate of generation and permanence of these species remains complex. Many of the mechanisms surrounding plasma discharge have been discovered through experiments, but a consistent challenge of time scales limits the plasma measurements. Thus, a well-constructed model with experimental research will help elucidate complex plasma physics. The motivation of this work is to construct a feasible physical model within the additional numerical times scale limitations and computational resources. This thesis summarizes the development of a one-moment fluid model for NPP discharges, which are applied due to their efficacy in generating ionized and excited species from vacuum to atmospheric pressure. </p><p>From a pulsed power perspective, the influence of pulse parameters, such as electric field intensity, pulse shape and repetition rate, are critical; however, the effects of these parameters on PGRS remain incompletely characterized. Here, we assess the influence of pulse conditions on the electric field and PGRS computationally by coupling a quasi-one-dimensional model for a parallel plate geometry, with a Boltzmann solver (BOLSIG+) used to improve plasma species characterization. We first consider a low-pressure gas discharge (3 Torr) using a five-species model for argon. <a>We then extend to a 23 species model with a reduced set of reactions for air chemistry remaining at low pressure.</a> The foundations of a single NPP is first discussed to build upon the analysis of repeating pulses. Because many applications use multiple electric pulses (EPs) the need to examine EP parameters is necessary to optimize ionization and PGRS formation. </p><p>The major goal of this study is to understand how the delivered EP parameters scale with the generated species in the plasma. Beginning with a similar scaling study done by Paschen we examine the effects of scaling pressure and gap length when the product remains constant for the two models. This then leads to our study on the relationship of pulsed power for different voltages and pulse widths of EPs. By fixing the energy delivered to the gap for a single pulse we determine that the electron and ion number densities both increased with decreasing pulse duration; however, the rate of this increase of number densities appeared to reach a limit for 3 ns. These results suggest the feasibly of achieving comparable outputs using less expensive pulse generators with higher pulse duration and lower peak voltage. Lastly, we study these outcomes when increasing the number of pulses and discuss the effects of pulse repetition and the electron temperature.</p><p>Future work will extend this parametric study to different geometries (i.e. pin-to-plate, and pin-to pin) and ultimately incorporate this model into a high-fidelity computational fluid dynamics (CFD) model that may be compared to spectroscopic results under quiescent and flowing conditions will be discussed.<br></p>

<p> In the past two decades, great interest in weakly ionized plasmas sustained by high voltage nanosecond pulsed plasmas at high repetition rates has emerged. For such plasmas, the electron number density does not significantly decay between pulses, unlike the electron temperature. Such conditions are favorable to reconfigurable plasma antennas where the low electron temperature may enable the reduction of the Johnson–Nyquist thermal noise if an antenna is operated in the plasma afterglow. Moreover, it may be possible to sustain such conditions with RF pulses. Doing so could enable a plasma antenna that transmits the driving frequency when the pulse is applied and receives other frequencies with low thermal noise between pulses.</p> <p>To study nanosecond pulsed plasmas, experiments were performed in a parallel-plate electrode configuration in argon and nitrogen gas at a pressure of several Torr and repetition frequencies of 30-75 kHz. To measure the time-resolved electron number density in the afterglow of each pulse, a custom 58.1 GHz homodyne microwave interferometer was constructed. The voltage and current measurements were made using a back current shunt (BCS). Initial analysis of the measured electron density in both plasmas indicated that the electron thermalization was much faster than the electron decay. In the nitrogen plasma, dissociative recombination with cluster ions was the dominant electron loss mechanism. However, the dissociative recombination rates of the electrons in the argon plasma suggested the presence of molecular impurities, such as water vapor. Therefore, to better understand the recombination mechanisms in argon plasma with trace amounts (0.1% or less by volume) of water vapor under the experimental conditions, a 0-D kinetic model was developed and fit to the experimental data. The influence of trace amounts of water on the electron temperature and density decay was studied by solving electron energy and continuity equations. It was found that in pure argon, Ar⁺ ions dominate while the electrons are very slow to thermalize and recombine. Including trace amounts of water impurities drastically reduces the time for electrons to thermalize and increases their rate of recombination. </p> <p>In addition to large quasi-steady electron number densities and low electron temperature in the plasma afterglow, plasmas sustained by nanosecond pulses use a lower power budget than those sustained by RF or DC supplies. The efficiency of the power budget can be characterized by measuring the ionization cost per electron, defined as the ratio of the energy deposited in a pulse to the total number of electrons created. This was experimentally determined in air and argon plasmas at 2-10 Torr sustained by 1-7 kV nanosecond pulses at repetition frequencies of 0.1-30 kHz. The number of electrons were determined from the measured electron density through microwave interferometry and assuming a plasma volume equivalent to the volume between electrodes. The energy deposited was calculated from voltage and current measurements using both a BCS as well as high frequency resistive voltage divider and fast current transformer (FCT). It was found that the ionization cost in all conditions was within a factor of three of Stoletov’s point (the theoretical minimum ionization cost) and two orders of magnitude less than RF plasma.</p><p> </p><p>Having shown that it is possible to generate high electron density, low electron temperature plasmas with nanosecond pulses, it was necessary to now create a plasma antenna prototype. Initially, commercial fluorescent light bulbs were used and ignited using surface wave excitation at various RF frequencies and powers. The S₁₁ of the antenna response was measured by a VNA through a novel coupling circuit, while the deposited power was measured using a bi-directional coupler. Next, a custom plasma antenna was created in which the pressure and gas composition could be varied. In addition to the S₁₁ and deposited power, the antenna gain, and the electron number density were also measured for a pure argon plasma antenna at pressures of 0.3-1 Torr. Varying the applied power shifts the antenna resonance frequency while increasing the excitation frequency caused an increase in measured electron density for the same deposited power. Initial tests using direct electrode excitation of a twin-tube integrated compact fluorescent light bulb with nanosecond pulses have successfully been achieved. Future efforts include designing the proper circuitry to time-gate out the large pulse voltage to facilitate safe antenna measurements in the plasma afterglow.<br></p>