Kinetic Methods in Technological Plasmas

Widespread recent interest in techniques for demonstrating that computer simulation programs are correct ('verification') has been motivated by evidence that traditional development and testing procedures are disturbingly ineffective. Reproducing an exact solution of the relevant model equations is generally accepted as the strongest available verification procedure, but this technique depends on the availability of suitable exact solutions. In this paper we consider verification of a particle-in-cell simulation with Monte Carlo collisions. We know of no exact solutions that simultaneously exercise all of the functions of this code. However, we show here that there can be found in the literature a number of non-trivial exact solutions, each of which exercises a substantial subset of these functions, and which in combination exercise all of the functions of the code. That the code is able to reproduce these solutions is correctness evidence of a stronger kind than has hitherto been elucidated.

One-dimensional particle-in-cell (PIC) simulations of a narrow gap atmospheric pressure He/2%${{\text{H}}_{2}}\text{O}$ radio frequency capacitive discharge showed standing striations in the bulk plasma region while previously conducted PIC simulations of a narrow gap atmospheric pressure He/0.1%${{\text{N}}_{2}}$ discharges [1] showed no such instabilities. We successively modified the base He/${{\text{H}}_{2}}\text{O}$ chemistry to make it more similar to the He/${{\text{N}}_{2}}$ chemistry in order to determine the cause of the striations. Setting the e–${{\text{H}}_{2}}\text{O}$ scattering, attachment, vibrational and rotational excitation rates to zero did not suppress the striations. However, a systematic reduction of the e–ion recombination cross section resulted in a transition to a stable state with no striations. The results are interpreted in terms of a model in which the balance between bulk direct ionization and bulk recombination loss determines the bulk plasma equilibrium. Perturbing the equilibrium, we find that the striations are consistent with an ionization instability induced by non-local electron kinetics that form a spatially-varying high energy tail of the electron energy distribution, causing the ionization rate coefficient to decrease with increasing electron temperature T_e and root-mean-square electric field E in the instability regime.

The influence of negative gas SF₆ admixture on the stability of low-pressure beam-driven discharge is studied by self-consistent 1D particle-in-cell/Monte Carlo collisions model. We find that the plasma parameters as well as its stability are changed drastically when SF₆ is added to argon gas. In an electropositive plasma we obtain the excitation of both fast and slow electrostatic plasma waves which is explained by the propagation of electron beam through the non-homogeneous bounded plasma. These waves lead to the heating of plasma electrons and ions. On the other hand, in electronegative plasma we also obtain a decay of the electron beam. However, since in electronegative beam-generated plasma the electron plasma density is homogeneous the beam is more stable. As a consequence, we obtain only the excitation of two fast electrostatic waves. One wave is excited due to two-stream instability and another wave is excited due to beam decay. The excitation of these fast waves does not influence the plasma homogeneity but influences the heating of plasma electrons.

The control of the vibrational distribution of nitrogen by energy transfer to CO₂ is studied in two closely related experiments. In the first experiment, the time-resolved N₂(v  =  0–3) vibrational level populations and temperature in the afterglow of a diffuse filament nanosecond pulse discharge are measured using broadband coherent anti-Stokes Raman spectroscopy. The rotational–translational temperature in the afterglow is inferred from the partially rotationally resolved structure of the N₂(v  =  0) band. The measurements are performed in nitrogen, dry air, and their mixtures with CO₂. N₂ vibrational excitation in the discharge occurs by electron impact, with subsequent vibration–vibration (V–V) energy transfer within the N₂ vibrational manifold, vibration–translation (V–T) relaxation, and near-resonance V–V' energy transfer from the N₂ to CO₂ asymmetric stretch vibrational mode. The results show that rapid V–V' energy transfer to CO₂, followed by collisional intramolecular energy redistribution to the symmetric stretch and bending modes of CO₂ and their V–T relaxation, accelerate the net rate of energy thermalization and temperature increase in the afterglow. In the second experiment, injection of CO₂ into a supersonic flow of vibrationally excited nitrogen demonstrates the effect of accelerated vibrational relaxation on a supersonic shear layer. The nitrogen flow is vibrationally excited in a repetitive nanosecond pulse/DC sustainer electric discharge in the plenum of a nonequilibrium flow supersonic wind tunnel. A transient pressure increase as well as an upward displacement of the shear layer between the supersonic N₂ flow and the subsonic CO₂ injection flow are detected when the source of N₂ vibrational excitation is turned on. CO₂ injection leads to the reduction of the N₂ vibrational temperature in the shear layer, demonstrating that its displacement is caused by accelerated N₂ vibrational relaxation by CO₂, which produces a static temperature and a pressure increase in the test section. This demonstrates the significant potential of accelerated vibrational relaxation for nonequilibrium flow control, by injection of rapid 'relaxer' species at a desired location, resulting in the rapid thermalization of vibrational energy in nitrogen and air flows, and producing a significant effect on the flow field.

The two particle-in-cell codes EDIPIC and LSP are benchmarked and validated for a parallel-plate glow discharge in helium, in which the axial electric field had been carefully measured, primarily to investigate and improve the fidelity of their collision models. The scattering anisotropy of electron-impact ionization, as well as the value of the secondary-electron emission yield, are not well known in this case. The experimental uncertainty for the emission yield corresponds to a factor of two variation in the cathode current. If the emission yield is tuned to make the cathode current computed by each code match the experiment, the computed electric fields are in excellent agreement with each other, and within about 10% of the experimental value. The non-monotonic variation of the width of the cathode fall with the applied voltage seen in the experiment is reproduced by both codes. The electron temperature in the negative glow is within experimental error bars for both codes, but the density of slow trapped electrons is underestimated. A more detailed code comparison done for several synthetic cases of electron-beam injection into helium gas shows that the codes are in excellent agreement for ionization rate, as well as for elastic and excitation collisions with isotropic scattering pattern. The remaining significant discrepancies between the two codes are due to differences in their electron binary-collision models, and for anisotropic scattering due to elastic and excitation collisions.

Based on accurate representation of the He⁺–He angular differential scattering cross sections consisting of both elastic and charge exchange collisions, we performed detailed numerical simulations of the ion velocity distribution functions (IVDF) by Monte Carlo collision method (MCC). The results of simulations are validated by comparison with the experimental data of the ion mobility and the transverse diffusion. The IVDF simulation study shows that due to significant effect of scattering in elastic collisions IVDF cannot be separated into product of two independent IVDFs in the transverse and parallel to the electric field directions.

Using the Monte Carlo collision method, we have performed simulations of ion velocity distribution functions (IVDF) taking into account both elastic collisions and charge exchange collisions of ions with atoms in uniform electric fields for argon and helium background gases. The simulation results are verified by comparison with the experiment data of the ion mobilities and the ion transverse diffusion coefficients in argon and helium. The recently published experimental data for the first seven coefficients of the Legendre polynomial expansion of the ion energy and angular distribution functions are used to validate simulation results for IVDF. Good agreement between measured and simulated IVDFs shows that the developed simulation model can be used for accurate calculations of IVDFs.

Buffer gas positron traps have been used for over two decades as the prime source of slow positrons enabling a wide range of experiments. While their performance has been well understood through empirical studies, no theoretical attempt has been made to quantitatively describe their operation. In this paper we apply standard models as developed for physics of low temperature collision dominated plasmas, or physics of swarms to model basic performance and principles of operation of gas filled positron traps. The Monte Carlo model is equipped with the best available set of cross sections that were mostly derived experimentally by using the same type of traps that are being studied. Our model represents in realistic geometry and fields the development of the positron ensemble from the initial beam provided by the solid neon moderator through voltage drops between the stages of the trap and through different pressures of the buffer gas. The first two stages employ excitation of N₂ with acceleration of the order of 10 eV so that the trap operates under conditions when excitation of the nitrogen reduces the energy of the initial beam to trap the positrons without giving them a chance to become annihilated following positronium formation. The energy distribution function develops from the assumed distribution leaving the moderator, it is accelerated by the voltage drops and forms beams at several distinct energies. In final stages the low energy loss collisions (vibrational excitation of CF₄ and rotational excitation of N₂) control the approach of the distribution function to a Maxwellian at room temperature but multiple non-Maxwellian groups persist throughout most of the thermalization. Optimization of the efficiency of the trap may be achieved by changing the pressure and voltage drops and also by selecting to operate in a two stage mode. The model allows quantitative comparisons and test of optimization as well as development of other properties.

Negative power deposition in low pressure inductively coupled plasmas (ICPs) is investigated by means of an analytical model which couples Boltzmann's equation and the quasi-stationary Maxwell's equations. Exploiting standard Hilbert space methods an explicit solution for both, the electric field and the distribution function of the electrons for a bounded discharge configuration subject to an unsymmetrical excitation  is  found for the first time. The model is applied to a low pressure ICP discharge. In this context particularly the anomalous skin effect and the effect of phase mixing is discussed. The analytical solution is compared with results from electromagnetic full wave particle in cell simulations. Excellent agreement between the analytical and the numerical results is found.

UV/VUV photon fluxes in plasma materials processing have a variety of effects ranging from producing damage to stimulating synergistic reactions. Although in plasma etching processes, the rate and quality of the feature are typically controlled by the characteristics of the ion flux, to truly optimize these ion and photon driven processes, it is desirable to control the relative fluxes of ions and photons to the wafer. In prior works, it was determined that the ratio of VUV photon to ion fluxes to the substrate in low pressure inductively coupled plasmas (ICPs) sustained in rare gases can be controlled by combinations of pressure and pulse power, while the spectrum of these VUV photons can be tuned by adding additional rare gases to the plasma. In this work, VUV photon and ion fluxes are computationally investigated for Ar/Cl₂ ICPs as used in etching of silicon. We found that while the overall ratio of VUV photon flux to ion flux are controlled by pressure and pulse power, by varying the fraction of Cl₂ in the mixture, both the ratio of VUV to ion fluxes and the spectrum of VUV photons can be tuned. It was also found that the intensity of VUV emission from Cl(3p⁴4s) can be independently tuned by controlling wall surface conditions. With this ability to control ratios of ion to photon fluxes, photon stimulated processes, as observed in halogen etching of Si, can be tuned to optimize the shape of the etched features.

A theoretical model is developed to study the nucleation and catalytic growth of carbon nanofibers (CNFs) in a plasma environment. The model includes the charging of CNFs, the kinetics of the plasma species (neutrals, ions and electrons), plasma pretreatment of the catalyst film, and various processes unique to a plasma-exposed catalyst surface such as adsorption of neutrals, thermal dissociation of neutrals, ion induced dissociation, interaction between neutral species, stress exerted by the growing graphene layers and the growth of CNFs. Numerical calculations are carried out for typical glow discharge plasma parameters. It is found that the growth rate of CNFs decreases with the catalyst nanoparticle size. In addition, the effect of hydrogen on the catalyst nanoparticle size, CNF tip diameter, CNF growth rate, and the tilt angle of the graphene layers to the fiber axis are investigated. Moreover, it is also found that the length of CNFs increases with hydrocarbon number density. Our theoretical findings are in good agreement with experimental observations and can be extended to enhance the field emission characteristics of CNFs.

In this study we have developed a full multi-term space–time solution of Boltzmann's equation for electron transport in gases and liquids. A Green's function formalism is used that enables flexible adaptation to various experimental systems. The spatio-temporal evolution of electrons in liquids in the non-hydrodynamic regime is benchmarked for a model Percus–Yevick (PY) liquid against an independent Monte Carlo simulation, and then applied to liquid argon. The temporal evolution of Franck–Hertz oscillations in configuration and energy space are observed for the model liquid with large differences apparent when compared to the dilute gas case, for both the velocity distribution function components and the transport quantities. The packing density in the PY liquid is shown to influence both the magnitude and wavelength of Franck–Hertz oscillations of the steady-state Townsend (SST) simulation. Transport properties are calculated from the non-hydrodynamic theory in the long time limit under SST conditions which are benchmarked against hydrodynamic transport coefficients. Finally, the spatio-temporal relaxation of low-energy electrons in liquid argon was investigated, with striking differences evident in the spatio-temporal development of the velocity distribution function components between the uncorrelated gas and true liquid approximations, due largely to the presence of a Ramsauer minimum in the former and not in the latter.

The large electron ${\bf{E}}\times {\bf{B}}$ drift (relative to the ions) in the azimuthal direction of Hall-effect thrusters is well known to excite a strong instability. In a recent paper (Lafleur et al 2016 Phys. Plasmas 23 053503) we demonstrated that this instability leads to an enhanced electron–ion friction force that increases the electron cross-field mobility to levels similar to those seen experimentally. Here we extend this work by considering in detail the onset criteria for the formation of this instability (both in xenon, and other propellants of interest), and identify a number of important characteristics that it displays within Hall-effect thrusters (HETs): including the appearance of an additional non-dimensionalized scaling parameter (the instability growth-to-convection ratio), which controls the instability evolution and amplitude. We also investigate the effect that the instability has on electron and ion heating in HETs, and show that it leads to an ion rotation in the azimuthal direction that is in agreement with that seen experimentally.

In this work we study the electron drift instability in Hall-effect thrusters (HETs) using a 2D electrostatic particle-in-cell (PIC) simulation. The simulation is configured with a Cartesian coordinate system modeling the radial-azimuthal ($r\mbox{--}\theta$) plane for large radius thrusters. A magnetic field, ${{\bf{B}}}_{0}$, is aligned along the Oy axis (r direction), a constant applied electric field, ${{\bf{E}}}_{0}$, along the Oz axis (perpendicular to the simulation plane), and the ${{\bf{E}}}_{0}\times {{\bf{B}}}_{0}$ direction is along the Ox axis (θ direction). Although electron transport can be well described by electron–neutral collisions for low plasma densities, at high densities (similar to those in typical HETs), a strong instability is observed that enhances the electron cross-field mobility; even in the absence of electron–neutral collisions. The instability generates high frequency (of the order of MHz) and short wavelength (of the order of mm) fluctuations in both the azimuthal electric field and charged particle densities, and propagates in the ${{\bf{E}}}_{0}\times {{\bf{B}}}_{0}$ direction with a velocity close to the ion sound speed. The correlation between the electric field and density fluctuations (which leads to an enhanced electron–ion friction force) is investigated and shown to be directly responsible for the increased electron transport. Results are compared with a recent kinetic theory, showing good agreement with the instability properties and electron transport.

Low-pressure capacitively coupled radio frequency discharges operated in O₂ and driven by tailored voltage waveforms are investigated experimentally and by means of kinetic simulations. Pulse-type (peaks/valleys) and sawtooth-type voltage waveforms that consist of up to four consecutive harmonics of the fundamental frequency are used to study the amplitude asymmetry effect as well as the slope asymmetry effect at different fundamental frequencies (5, 10, and 15 MHz) and at different pressures (50–700 mTorr). Values of the DC self-bias determined experimentally and spatio-temporal excitation rates derived from phase resolved optical emission spectroscopy measurements are compared with particle-in-cell/Monte Carlo collisions simulations. The spatio-temporal distributions of the excitation rate obtained from experiments are well reproduced by the simulations. Transitions of the discharge electron heating mode from the drift-ambipolar mode to the α-mode are induced by changing the number of consecutive harmonics included in the driving voltage waveform or by changing the gas pressure. Changing the number of harmonics in the waveform has a strong effect on the electronegativity of the discharge, on the generation of the DC self-bias and on the control of ion properties at the electrodes, both for pulse-type, as well as sawtooth-type driving voltage waveforms The effect of the surface quenching rate of oxygen singlet delta metastable molecules on the spatio-temporal excitation patterns is also investigated.

The single particle motion analysis and particle-in-cell merged with Monte Carlo collision (PIC/MCC) simulations are compared to explain substantial breakdown voltage reduction for helium microwave discharge above a critical frequency corresponding to the transition from the drift-dominant to the diffusion-dominant electron loss regime. The single particle analysis suggests that the transition frequency is proportional to the product of ${p}^{-{m}}$ and ${d}^{-({m}+1)}$ where $p$ is the neutral gas pressure, $d$ is the gap distance, and m is a numerical parameter, which is confirmed by the PIC simulation. In the low-frequency or drift-dominant regime, i.e., $\gamma \mbox{-} {\rm{r}}{\rm{e}}{\rm{g}}{\rm{i}}{\rm{m}}{\rm{e}},$ the secondary electron emission induced by ion drift motion is the key parameter for determining the breakdown voltage. The fluid analysis including the secondary emission coefficient, $\gamma ,$ induces the extended Paschen law that implies the breakdown voltage is determined by $pd,$ $\,f/p,$ $\gamma ,$ and$\,d/R$ where f is the frequency of the radio or microwave frequency source, and R is the diameter of electrode. The extended Paschen law reproduces the same scaling law for the transition frequency and is confirmed by the independent PIC and fluid simulations.

Non-equilibrium vibrational distributions (vdf) and non-equilibrium electron energy distribution functions (eedf) in a nitrogen plasma at low pressure (mtorr) have been calculated by using a time-dependent plasma physics model coupled to the Boltzmann equation and heavy particle kinetics. Different case studies have been selected showing the non-equilibrium character of both vdf and eedf under discharge and post-discharge conditions in the presence of large concentrations of electrons. Particular attention is devoted to the electron-molecule resonant vibrational excitation cross sections acting in the whole vibrational ladder. The results in the post-discharge conditions show the interplay of superelastic vibrational and electronic collisions in forming structures in the eedf. The link between the present results in the mtorr afterglow regime with the existing eedf in the torr and atmospheric regimes is discussed.

Comparative studies of capacitively coupled radio-frequency discharges in helium and argon at pressures between 10 and 80 Pa are presented applying two different fluid modeling approaches as well as two independently developed particle-in-cell/Monte Carlo collision (PIC/MCC) codes. The focus is on the analysis of the range of applicability of a recently proposed fluid model including an improved drift-diffusion approximation for the electron component as well as its comparison with fluid modeling results using the classical drift-diffusion approximation and benchmark results obtained by PIC/MCC simulations. Main features of this time- and space-dependent fluid model are given. It is found that the novel approach shows generally quite good agreement with the macroscopic properties derived by the kinetic simulations and is largely able to characterize qualitatively and quantitatively the discharge behavior even at conditions when the classical fluid modeling approach fails. Furthermore, the excellent agreement between the two PIC/MCC simulation codes using the velocity Verlet method for the integration of the equations of motion verifies their accuracy and applicability.

A propagator method (PM), a numerical technique to solve the Boltzmann equation (BE) for the electron velocity or energy distribution function (EVDF/EEDF) of electron swarms in gases, was customized to obtain the equilibrium solution quickly. The PM calculates the number of electrons in cells defined in velocity space using an operator called the propagator or Green's function. The propagator represents the intercellular transfer of electrons corresponding to the electron velocity change due to the acceleration by the electric field and the collisional events with gas molecules. The relaxation of the EVDF to its drift equilibrium solution proceeds with iterative propagator operations for the EVDF. Merits of the PM are that the series expansion of the EVDF as done in the BE analyses is not required and that time evolution of the electron swarm can be observed if necessary. On the other hand, in case only the equilibrium solution of the EVDF is wanted, the relaxation can be accelerated numerically. A demonstration achieved a shortening of the computational time by about three orders of magnitude. Furthermore, this scheme was applied to calculations of a set of electron transport parameters required in fluid-model simulations, i.e. the effective ionization frequency, the centroid drift velocity and the longitudinal diffusion coefficient, using the zeroth-, first- and second-order moment equations derived from the BE. A detailed description on the PM calculation was presented.

Monte Carlo simulation was used to study the translational relaxation of energetic O⁻ ions produced by dissociative electron attachment to O₂ molecules in oxygen plasmas in a strong electric field. Initial O⁻ ions have rather high energies and are more reactive than the ions reaching equilibrium with the electric field. Therefore, there is a noticeable probability that the energetic O⁻ ions participate in endothermic reactions prior to energy relaxation of these ions. The probabilities of charge exchange, electron detachment and ion impact vibrational excitation of O₂ molecules were calculated versus the reduced electric field. It was shown that up to 6% of energetic O⁻ ions produced in oxygen by dissociative electron attachment to O₂ molecules are rapidly transformed to ${{{\rm{O}}}_{2}}^{-}$ ions due to charge exchange collisions. The probability of electron detachment from energetic O⁻ ions and the probability of vibrational excitation were smaller that the probability of charge exchange. Estimates showed that the increase in the effective rates of the ion–molecule reactions due to high reactivity of energetic O⁻ ions can be important in oxygen plasmas for reduced electric fields of 50–100 Td.

This paper is devoted to a presentation of a cross section set for the scattering of He⁺ ions in CF₄, which is assessed by using available experimental data for exothermic charge transfer cross sections that produce ${{{\rm{CF}}}_{3}}^{+}$ and ${{{\rm{CF}}}_{2}}^{+}$ ions and endothermic charge transfer cross sections that produce CF⁺, C⁺ and F⁺ ions. Due to the significant particle losses, experimental transport coefficients have not been measured. Transport properties of He⁺ ions in CF₄ needed for modeling discharges containing mentioned ions are calculated by the Monte Carlo method at a temperature of T = 300 K. Significant differences between flux and bulk transport coefficients are noticed, which is important for fluid models that exploit flux transport coefficients as input data.

Reliable and detailed sets of electron collision cross sections for tetramethylsilane [TMS, Si(CH₃)₄] and tetraethoxysilane [TEOS, Si(OC₂H₅)₄] vapours are proposed. The cross section sets of TMS and TEOS vapours include 16 and 20 kinds of partial ionization cross sections, respectively. Electron transport coefficients, such as electron drift velocity, ionization coefficient, and longitudinal diffusion coefficient, in those vapours are calculated by Monte Carlo simulations using the proposed cross section sets, and the validity of the sets is confirmed by comparing the calculated values of those transport coefficients with measured data. Furthermore, the calculated values of the ionization coefficient in TEOS/O₂ mixtures are compared with measured data to confirm the validity of the proposed cross section set.

The formation process of a microdischarge (MD) in both μm- and nm-sized catalyst pores is simulated by a two-dimensional particle-in-cell/Monte Carlo collision model. A parallel-plate dielectric barrier discharge configuration in filamentary mode is considered in ambient air. The discharge is powered by a high voltage pulse. Our calculations reveal that a streamer can penetrate into the surface features of a porous catalyst and MDs can be formed inside both μm- and nm-sized pores, yielding ionization inside the pore. For the μm-sized pores, the ionization mainly occurs inside the pore, while for the nm-sized pores the ionization is strongest near and inside the pore. Thus, enhanced discharges near and inside the mesoporous catalyst are observed. Indeed, the maximum values of the electric field, ionization rate and electron density occur near and inside the pore. The maximum electric field and electron density inside the pore first increase when the pore size rises from 4 nm to 10 nm, and then they decrease for the 100 nm pore, due to a more pronounced surface discharge for the smaller pores. However, the ionization rate is highest for the 100 nm pore due to the largest effective ionization region.

The characteristics of two-dimensional periodical structures in a magnetized plasma are studied using kinetic simulations. Ridges (i.e. spikes in electron and ion density) are formed and became more pronounced with an increase of magnetic field incidence angle in the plasma volume in the cylindrical chamber. These ridges are shifted relative to each other, which results in the formation of a two-dimensional double-layer structure. Depending on Larmor radius and Debye length up to 19 potential steps appear across the oblique magnetic field. The electrical current gathered into the channels is associated with the electron and ion density ridges.