Modern industry heavily relies on inductively coupled plasmas for the generation of energetic ions. In electric propulsion, these ions generate thrust. In the semiconductor industry, they are used to etch high aspect ratio features. In materials science, they are crucial in the deposition and modification of thin films, which are essential for developing advanced materials with unique properties. There are a lot of studies of ion distribution functions at the biased electrodes for the low- and moderate ion energies, which are of interest for electric propulsion and film deposition. However, there are limited numbers of studies examining the regimes relevant to semiconductor applications, such as high aspect ratio etching. These regimes use high-density plasmas and high-power RF biasing. The accessibility of these regimes is difficult due to requirements on hardware. Their self-consistent computational studies are also difficult due to the high computational cost. This paper uses a one-dimensional collisionless sheath model coupled with a basic circuit equation to analyze how radio frequency bias affects the peak energies of ions extracted from the inductively coupled plasma. It is observed that, to extract high-energy ions from this plasma, the electron and ion conduction currents from the plasma must be such that they cannot compensate for the radio frequency current. Then, the capacitive high-voltage sheath builds up in front of the biased electrode, enabling ion acceleration to kiloelectron volt energies. It is found that at low radio frequencies, the ion energy at the biased electrode increases linearly with radio frequency power.

This paper reviews the state of the art of our understanding of the mechanisms of runaway electron generation in pressurized gases from the numerical modeling perspective. Since the energy relaxation length of these electrons is comparable to the interelectrode spacing, these electrons can be captured only using the kinetic approach. Therefore, only the results from kinetic models are discussed here. Special attention is given to pulsed discharges, which play an important role in modern industry. It is concluded that the mechanisms of runaway electron generation are defined by the gap overvoltage and the discharge gap geometry. For small and moderate overvoltages, runaway electrons are primarily generated at the heads of fast ionization waves or streamers. Due to their long energy relaxation length, these electrons can pre-ionize the discharge gap far from their origin, accelerating ionization and starting new avalanches. At high overvoltages, cathode surface irregularities enhance the local electric field, leading to electron emission into the interelectrode space. These electrons, injected into the strong electric field, gain high energy and reach discharge walls with extremely high energies measuring tens and hundreds of electron volts. These electrons not only pre-ionize the gas but also stimulate the emission of high-energy photons, which can further contribute to the pre-ionization of the discharge gap.

The main objective of the present paper is the detailed analysis of the electron kinetics near the Paschen's curve minimum of pulsed breakdown of nitrogen gas. Three main questions are to be answered. First, whether the breakdown curve minimum corresponds to the threshold electric field necessary for the runaway electron generation. Second, what is the role of these electrons in the vicinity of the Paschen's curve minimum. Third, what is the ionization cost near the minimum. To answer these questions, the one-dimensional model is used in which electrons are modeled using the Particle-in-Cell approach, while ions are modeled in the drift-diffusion approximation.

Data-driven techniques developed in recent years for the discovery of equations describing complex physical phenomena open unique opportunities for plasma physics. These methods allow getting insights into the processes difficult for analytical description. Since gas discharges can be represented as complex electrical circuits consisting of impedances and capacitances, it looks natural to use the data-driven techniques to study their complex dynamics. In the present paper, the sparse identification of nonlinear dynamics (SINDy) method is applied to analyze the self-oscillations of direct current discharge in argon. It is obtained that the third order polynomials describe best the oscillations of the discharge voltage and current. They allow an accurate capturing of the oscillations amplitudes as well as the harmonics of these oscillations. To understand the physical meaning of each term, an analytical model is presented which describes the discharge self-oscillations.

In recent years, interest in striation phenomena in radio frequency (rf) discharges has risen due to the availability of new experimental data and the implementation of new computational models. Depending on the conditions, different mechanisms of discharge striations are realized. These are the ionization instability, the instability due to the electron attachment to electronegative gases, or the instability due to thermoelectric transport. Although the first two mechanisms were modeled quite extensively in recent years, the understanding of the influence of the Dufour effect originating from plasma density gradients on the stability of radio frequency discharges in long tubes remains poor. In this paper, the influence of this mechanism on the longitudinal striations of radio frequency discharge is presented using a one-dimensional model of argon discharge driven with rf excitation under intermediate pressure conditions of 0.5 Torr. It is found that striation formation is sensitive to the value of the thermoelectric heat transport coefficient in the low electron temperature range. The critical value of this coefficient necessary for the instability onset is derived using the linear stability analysis.

In recent years, interest in synthetic C4F7N as a gas insulator has been growing due to its unique insulation properties and low global warming potential. In spite of this, very few studies are devoted to the analysis of the electric breakdown of C4F7N and its mixtures with other dilutants such as CO2 and N2. In the present paper, we use a two-dimensional fluid model to analyze the electrical breakdown of the atmospheric pressure C4F7N/N2 mixture. We establish the influence of the C4F7N fraction in the mixture and the cathode voltage rise rate on the breakdown voltage. We find that the ratio between the electron attachment frequency and the voltage rise rate defines the streamer parameters. Namely, if the time scale of electron attachment to C4F7N is much faster than the voltage rise rate, the plasma electronegativity of the streamer body and of its head is extremely high, and it is difficult to define exactly the streamer head location. In the opposite case of the fast rise rates, the conventional streamers with sharp heads were obtained.

The objective of this paper is the validation of a surface reaction mechanism for silicon etching in low-pressure chlorine plasmas. We employ a quasi-one-dimensional fluid model to model the experimental conditions of Khater and Overzet [Plasma Sources Sci. Technol. 13, 466 (2004)]. This model couples self-consistently the plasma fluid equations with the surface reaction mechanism derived from the available literature. Based on the comparison between the experiments and modeling results, the best set of etch yield parameters is proposed for the conditions typical for industrial plasma processing. The influence of these etch yield parameters on the gas-phase plasma is also discussed.

The main aim of the present paper is to clarify the influence of the SF6 fraction in the SF6/N2 mixture on the breakdown voltage. For this, the two-dimensional axisymmetric fluid model coupled with the comprehensive mechanism of plasma chemical reactions is used. In addition, the influence of various parameters such as the voltage rise time and the SF6 fraction in the mixture is analyzed. It is observed that depending on the voltage rise time an admixture of only 1% of SF6 to N2 results in an increase in the breakdown voltage by 7%–43%. The sensitivity of breakdown voltage decreases with decreasing voltage rise time and is caused by the electron attachment time scale becoming comparable to the breakdown time. The results of simulations confirm that the increase in the SF6 fraction in the mixture leads to an increase in the breakdown voltage. This is explained by the influence of the SF6 fraction on the electron attachment rate coefficients rather than on the ionization reactions.

In nanosecond gas breakdown, runaway electrons are responsible for the interelectrode space pre-ionization, which supports the propagation of fast ionization waves or streamers. However, a detailed understanding of mechanisms of how this happens is still elusive because of the wide range of energies of runaway electrons, from the hundreds of eV to the hundreds of keV. While the low-energy part of the runaway electron population can only ionize the background gas, the high-energy part can produce x-rays that can also ionize the gas. The objective of this paper is to understand the influence of different electron groups on the fast ionization wave dynamics for the conditions typical for runaway electrons generation. We find that there are two main groups, one having energies below 100 eV and another having energies above 1 keV. The first group defines the establishment of the fast ionization waves and their plasma parameters, while the second group defines the speed of the ionization front propagation.

Using a self-consistent plasma model coupled with Maxwell's equations, the limitations of independent control of ion fluxes and their energy distribution functions extracted from the high-density inductively coupled chlorine plasma are studied. Two extreme cases of discharge power are considered: 100 W and 1 kW. We find that in the low-power case, plasma is mainly generated by electromagnetic waves while the radio-frequency biased electrode primarily enables plasma ion extraction. Therefore, the ion fluxes and distribution functions are controlled independently. For the high-power case of 1 kW, the bias electrode significantly contributes to plasma generation but has only a small effect on sheath voltage. As a consequence, independent control of ion fluxes and distribution functions becomes impossible. Namely, the increase in the power driving the radio-frequency electrode leads to the increase in the ion fluxes but has little effect on their energy and angular distributions.

The mechanism of gas heating in low-pressure inductively coupled chlorine plasma is analyzed using a self-consistent two-dimensional axisymmetric fluid plasma model that is coupled with the compressible Navier–Stokes equations. For gas pressures of 10 and 20 mTorr and the discharge power in the range 0.1–1.3 kW, the main reactions contributing to gas heating were the ion–ion recombination reactions and the quenching of electronically excited chlorine atoms. At the same time, the energy released by the electron impact dissociation reaction of molecular chlorine is negligible due to its high degree of dissociation within the plasma bulk. The comparison between the results of our simulations and the fitting equation proposed in the literature show qualitative agreement, although there is significant quantitative discrepancy.

A computational model for the interaction of a high-pressure microplasma with an electromagnetic wave is presented. A one-dimensional particle-in-cell Monte Carlo collision model is used to investigate the plasma non-ideality effects in a second-stage laser-heated xenon plasma with a comprehensive chemistry mechanism, including excited species. A xenon microplasma at a temperature of 300 K and pressure of 10 bar becomes non-ideal after the laser heating stage with lower ionization and electronic excitation barriers, which makes the presence of excited species important to the plasma generation process. For these conditions, two-step ionization via excited species becomes a major contributor to plasma ionization. We find that full ionization of the plasma is obtained in ∼2 ps when excited species are included in the chemistry model, as opposed to ∼2.5 ps for a chemistry mechanism that does not consider their effect. With excited species, the ionization mode transitions from direct ionization via electron-neutral collisions to direct and stepwise ionization as the plasma generation progresses.

The influence of electron field emission on the sub-normal mode of a magnetized direct current high-pressure helium discharge is analyzed using the two-dimensional axisymmetric fluid model. It is observed that in spite of accounting for a more intense electron emission mechanism, discharge still operates in the sub-normal mode. However, the field emission driven discharge is characterized by a smaller discharge voltage and a larger discharge current. For large values of the electric field enhancement factor, the discharge voltage can be as low as ∼40 V, and the discharge current is a few amperes. It is also seen that for large values of the field enhancement factor and small values of the ballast resistor, rather dense plasma (density ∼ 1017 m−3) can be generated on the nanosecond time scale.

A chemical reaction mechanism of chlorine plasma under low-pressure conditions that is widely used in the literature is validated against the experimental data of Y. Wang and J. K. Olthoff [J. Appl. Phys. 85, 6358 (1999)] for an inductively coupled plasma reactor. The model used in the present study is a self-consistent two-dimensional fluid plasma model coupled with Maxwell’s equations. The quantities of interest in the plasma are the fluxes and energy distribution functions of Cl2+ and Cl+ ions. We find that the charge exchange reaction between Cl+ and Cl2 that is typically included in chlorine plasma reaction mechanisms results in poor predictability of the model compared to experiments. Neglecting this reaction allows for a correct prediction of the dominant ion species in the low-pressure chlorine plasma and dependence of their fluxes on the gas pressure. Additionally, neglecting the charge exchange reaction allows for a rather accurate prediction of ion energy distribution functions at the grounded electrode. Overall, we conclude that the rate coefficient of the charge exchange reaction between Cl+ and Cl2 reported in the literature significantly exceeds what may in fact explain the role of this process in a low-pressure plasma discharge.

We report on transient generation of highly ionized (ionization degree ∼10%) argon microplasma using a self-consistent fluid plasma model coupled with the compressible Navier–Stokes equations. The plasma is generated within a micrometer size cathode spot immediately after the onset of intense secondary electron emission from the cathode and exists over a relatively short duration of ∼10 ns. We observe the electron pressure within this microplasma exceeding the background gas pressure by a few times and discuss the mechanisms of the energy transfer from this plasma to the heavy species. The localized gas heating generates a compression wave that propagates from the cathode to the anode.

We present a multi-fidelity process simulator ‘VizGlow-MPS’ that combines well-established high-fidelity models with a reduced-order model surrogate. The high-fidelity model provides experimentally validated results for equipment operation that informs the reduced-order model that predicts results in a few minutes of wall-clock time. The approach constitutes a ‘digital twin’ for process reactors with multiple levels of fidelity that a process engineering can choose from. This approach is demonstrated on c-C4F8 inductively coupled plasma and pulsed CF4/H2 capacitively coupled plasma widely used in etching applications.

The dynamics of a subnormal mode of direct-current micro-discharge in pressurized nitrogen is studied using a self-consistent one-dimensional plasma fluid model coupled with the compressible Navier–Stokes equations. To clarify different time scales, we use a hierarchy of plasma reaction mechanisms. The simplest mechanism considers the two-species plasma without any excited species. The excitation of electronic and vibrational levels of nitrogen is only considered in the electron energy balance. The second mechanism considers multiple ions and electronically excited species. The most comprehensive third mechanism expands the second mechanism by tracking the vibrationally excited species as well as the reactions of vibrational-translational and vibrational-vibrational relaxation. By using this third mechanism, we analyze the dynamics associated with the relaxation of the vibrational distribution function of nitrogen.

In this work, we improve the plasma kinetics of perfluorocyclobutane (c-C4F8), one of the most important gases in plasma etching applications. We use the self-consistent plasma fluid simulation model coupled with a comprehensive finite-rate chemical reaction mechanism. First, we discuss the deficiencies of the existing mechanisms of plasma chemical reactions found in the literature and the approach to improve these mechanisms. Second, we compare the results of our self-consistent simulations of inductively coupled plasmas in pure c-C4F8 with the experimental results obtained using the Gas Electronics Conference reference cell plasma reactor. Finally, we analyze the influence of various model parameters such as the surface reactions mechanism, gas pressure, discharge power, and electron stochastic heating length scale on the plasma parameters. We discuss how these parameters influence the kinetics of the dominant plasma species.

A self-consistent two-dimensional axisymmetric fluid plasma model coupled with Maxwell's equations is used to analyze the plasma kinetics of inductively coupled Ar/NF3 low-pressure plasma. We have developed a plasma chemical reaction mechanism for Ar/NF3 plasma and validated the same against the experimental data found in the literature. We analyze two sets of the electron-neutral reaction rate coefficients obtained for assumed Maxwellian and non-Maxwellian electron energy distribution functions. We find that the electron density in the discharge is sensitive to the choice of the electron energy distribution, while the fluorine atoms density is less sensitive. Since the density of atoms is controlled by a balance between production by dissociation reactions and diffusion to the walls, their density is very sensitive to the wall sticking probability.

The aim of the present paper is to examine the influence of assumption on the electron energy distribution function on the relation between the plasma potential and the electron temperature for both electropositive (argon) and electronegative (chlorine) plasmas. A one-dimensional fluid model is used for simplicity although similar results were obtained using a self-consistent two-dimensional fluid model coupled with the Maxwell's equations for inductively coupled plasmas. We find that for electropositive plasma only a bi-Maxwellian electron energy distribution function provides reasonable results compared to measurements in low-pressure inductively coupled plasmas, namely, the increasing plasma potential for increasing electron temperature. For electronegative plasma, the plasma potential is an increasing function of the electron temperature for all electron distributions considered in the model. However, the scaling factors do not agree with the conventional plasma theory. We explain these results by the deviation of electrons from a Boltzmann distribution, which is due to non-equilibrium and non-local nature of plasma at the low-pressure conditions.