non-Maxwellian Electron Energy Distribution Function Research Articles

A global model and two-dimensional simulation study of a low-pressure inductively coupled CF4 plasma considering non-Maxwellian EEDF

We analyze the discharge characteristics of a low-pressure inductively coupled CF4 plasma using a global model and a two-dimensional (2D) simulation. We first conducted a study comparing the experimental results with the global model, which makes it easier to compare the trend concerning external parameters and less computationally expensive, to validate the chemical reaction data, and then, compared the experimental results with the 2D simulation results. We then analyzed the discharge characteristics by comparing the 2D model results with the global model at various gas pressures and powers. The simulations were performed using COMSOL software, which is based on a fluid model. The electron energy distribution function (EEDF) was solved self-consistently using the Boltzmann equation solver, and then, coupled with the fluid model. The results were more consistent with the experimental results when the EEDF was calculated by solving the Boltzmann equation than for assuming the Maxwellian EEDF. Furthermore, the global model results were similar with the mean values obtained from the 2D model. This indicates that it is efficient to first validate the electron collision cross sections and reaction coefficients using the global model. Our approach is expected to be utilized in the analysis of new gases.

Discussion on the transport processes in electrons with non-Maxwellian energy distribution function in partially-ionized plasmas

In a previous work (Alvarez Laguna et al 2022 Phys. Plasmas 29 083507), we have developed a non-linear moment model for electrons that self-consistently captures non-Maxwellian electron energy distribution function effects. The model does not rely in the local approximation and the transport coefficients are calculated by expanding the distribution function into Hermite polynomials and by taking moments of the Boltzmann equation, including the collision operator for elastic and inelastic collisions with arbitrary cross sections. This model captures the classical Fick’s, Fourier’s, and Ohm’s law as well as Soret, Dufour, and Peltier effects. In addition, novel non-local transport phenomena appear as a result of spatial gradients of the kurtosis of the distribution function. In this paper, we discuss on the transport effects by analyzing two collisional models: constant collision frequency and constant cross section. We estimate the order of magnitude of the transport processes in non-equilibrium electrons by analyzing the Langmuir probe measurements of a low-pressure argon inductively-coupled discharge. The results show that, under these conditions, the transport produced by the spatial gradients in the kurtosis of the distribution function produces a heat-flux contribution that is of the same order of magnitude as the Fourier and Dufour’s effects. These transport effects are beyond the local field or the electron gradient expansions, commonly used in the low-temperature plasma modeling.

Discussion on Electron Temperature of Gas-Discharge Plasma with Non-Maxwellian Electron Energy Distribution Function Based on Entropy and Statistical Physics

Electron temperature is reconsidered for weakly-ionized oxygen and nitrogen plasmas with its discharge pressure of a few hundred Pa, with its electron density of the order of 1017m-3 and in a state of non-equilibrium, based on thermodynamics and statistical physics. The relationship between entropy and electron mean energy is focused on based on the electron energy distribution function (EEDF) calculated with the integro-differential Boltzmann equation for a given reduced electric field E/N. When the Boltzmann equation is solved, chemical kinetic equations are also simultaneously solved to determine essential excited species for the oxygen plasma, while vibrationally excited populations are solved for the nitrogen plasma, since the EEDF should be self-consistently found with the densities of collision counterparts of electrons. Next, the electron mean energy U and entropy S are calculated with the self-consistent EEDF obtained, where the entropy is calculated with the Gibbs's formula. Then, the "statistical" electron temperature Test is calculated as Test=[∂S/∂U]-1. The difference between Test and the electron kinetic temperature Tekin is discussed, which is defined as [2/(3k)] times of the mean electron energy U=⟨ϵ⟩, as well as the temperature given as a slope of the EEDF for each value of E/N from the viewpoint of statistical physics as well as of elementary processes in the oxygen or nitrogen plasma.

Comparison of Langmuir probe and optical emission spectroscopy on a volume-cusp filament ion source using helium.

D-Pace has a Penning ion source that runs with helium for studies of α-particle production. We want to study its plasma properties as a function of varying operational parameters, which results in varied output ion beam characteristics. In order to diagnose the ion source plasma, a collisional-radiative code for helium to be used with optical emission spectroscopy measurements is developed. This has the advantage of being non-invasive, which allows for measurements using the emitted light from the plasma. This collisional-radiative code is shown to compare well with the Yacora on the Web model developed at IPP-Garching, and improves upon it with the addition of radiation trapping. Furthermore, the sensitivity of this model to the inclusion of additional excited state populations and non-Maxwellian electron energy distribution functions is investigated. It is shown that non-Maxwellian distributions can significantly affect spectroscopy measurements. This diagnostic is benchmarked to Langmuir probe measurements on the TRIUMF-licensed volume-cusp ion source to determine whether it can replicate the measured electron density and electron temperature as a function of varied operational parameters. The operational parameters are helium gas flow (10-40 SCCM), arc voltage (100-200V), and arc current (1-5A). The measured plasma properties, while different in absolute value, have similar trends for each operational parameter except when varying arc voltage. It is shown that this mismatch as a function of arc voltage is likely due to high energy non-Maxwellian electrons from the cathode sheath, which are not included within the collisional-radiative model.

Multi-harmonic analysis in a floating harmonic probe method for diagnostics of electron energy and ion density in low-temperature plasmas

A floating harmonic probe (FHP) is used to measure the electron energy and ion density in plasmas. It applies an AC voltage to an electrically floated probe and measures harmonic frequency components in the probe current which contains information about the parameters. In this study, we have quantitatively investigated the effects of stray impedances in an FHP measurement system on the calculated parameters. We also discuss the influence of the electron energy distribution function (EEDF) when it deviates from a Maxwellian shape on the FHP measurement. A new approach of multi-harmonic analysis of FHP data (MHA-FHP) is proposed to analyze the electron energy in plasmas with non-Maxwellian EEDFs. The MHA-FHP method has been compared with the conventional FHP and Langmuir probe methods through the measurement of low-temperature argon plasmas. Experimental results indicated that the MHA-FHP method can provide the shape of the EEDFs, effective electron temperature, and ion density.

A regularized high-order moment model to capture non-Maxwellian electron energy distribution function effects in partially ionized plasmas

A model for electrons in partially ionized plasmas that self-consistently captures non-Maxwellian electron energy distribution function (EEDF) effects is presented. The model is based on the solution of scalar and vectorial velocity moments up to the contracted fourth-order moment. The set of fluid (macroscopic) equations is obtained with Grad's method and exact expressions for the collision production terms are derived, considering the electron–electron, electron–gas, and electron–ion elastic collisions as well as for electron–gas excitation and ionization collisions. A regularization of the equations is proposed in order to avoid spurious discontinuities, existing in the original Grad's moment model, by using a generalized Chapman–Enskog expansion that exploits the disparity of mass between the electrons and the heavy particles (ions and atoms) as well as the disparity of plasma and gas densities, typical of gas discharges. The transport model includes non-local effects due to spatial gradients in the EEDF as well as the impact of the EEDF in the calculation of the elastic and inelastic collision rates. Solutions of the moment model under spatially homogeneous conditions are compared to direct simulation Monte Carlo and a two-term Boltzmann solver under conditions that are representative of high plasma density discharges at low-pressure. The moment model is able to self-consistently capture the evolution of the EEDF, in good quantitative agreement with the kinetic solutions. The calculation of transport coefficients and collision rates of an argon plasma in thermal non-equilibrium under the effect of an electric field is in good agreement with the solutions of a two-term Boltzmann solver, largely improving models with a simplified Bhatnagar–Gross–Krook collisional operator.

Two-dimensional simulation of non-equilibrium silicon nitride plasma discharge for thin film solar cell deposition

In this work, we model a Two-dimensional (2D) radiofrequency (RF) plasma discharge of hydrogenated silicon nitride (SiH4/NH3/H2) in a capacitive coupled plasma reactor (CCP), using non-Maxwellian electron energy distribution function (EEDF). The purpose is to investigate whether there is a real advantage and a significant contribution in 2D simulation using non-Maxwellian EEDF rather than Maxwellian one for determining the fundamental characteristics of radiofrequency plasma discharge. The results show a comparative study of the 2D evolution of the fundamental characteristics of the RF plasma using Maxwellian and non-Maxwellian EEDF. By considering the criterion of the density and the electronic temperature, the analysis of the results shows that the adoption of the non-equilibrium 2D EEDF makes it possible to obtain an approximation closer to the real conditions and thus a better understanding of the plasma discharge.

Influence of the electron kinetics on Ar/NF3 inductively coupled plasma

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.

Characterization of an RF-driven argon plasma at atmospheric pressure using broadband absorption and optical emission spectroscopy

Atmospheric pressure plasmas in argon are of particular interest due to the production of highly excited and reactive species enabling numerous plasma-aided applications. In this contribution, we report on absolute optical emission and absorption spectroscopy of a radio frequency (RF) driven capacitively coupled argon glow discharge operated in a parallel-plate configuration. This enabled the study of all key parameters including electron density and temperature, gas temperature, and absolute densities of atoms in highly electronically excited states. Space and time-averaged electron density and temperature were determined from the measurement of the absolute intensity of the electron-atom bremsstrahlung in the visible range. Considering the non-Maxwellian electron energy distribution function, an electron temperature (Te) of 2.1 eV and an electron density (ne) of 1.1×1019m−3 were obtained. The time-averaged and spatially resolved absolute densities of atoms in the metastable (1s5 and 1s3) and resonant (1s4 and 1s2) states of argon in the pure Ar and Ar/He mixture were obtained by broadband absorption spectroscopy. The 1s5 metastable atoms had the largest density near the sheath region with a maximum value of 8×1017m−3, while all other 1s states had densities of at most 2×1017m−3. The dominant production and loss mechanisms of these atoms were discussed, in particular, the role of radiation trapping. We conclude with comparison of the plasma properties of the argon RF glow discharges with the more common He equivalent and highlight their differences.

Numerical Modeling of Non-equilibrium Plasma Discharge of Hydrogenated Silicon Nitride (SiH4/NH3/H2)

In this work, we model a radiofrequency discharge of hydrogenated silicon nitride in a capacitive coupled plasma reactor using Maxwellian and non-Maxwellian electron energy distribution function. The purpose is to investigate whether there is a real advantage and a significant contribution using non-Maxwellian electron energy distribution function rather than Maxwellian one for determining the fundamental characteristics of a radiofrequency plasma discharge. The results show the evolution of the non-Maxwellian electron energy distribution function, the mobility and the diffusion coefficient required to determine the fundamental characteristics of the radiofrequency plasma discharge of a hydrogenated silicon nitride deposit at low pressure and low temperature, between the two electrodes of the capacitive coupled plasma reactor. By comparing these results using non-Maxwellian electron energy distribution function with those calculated using the Maxwellian one, we conclude that the use of non-Maxwellian electronic energy distribution function is more efficient for describing the evolution of a radiofrequency plasma discharge in a capacitive reactor, which will improve the quality of the deposition of thin films.

Global model of cold atmospheric He + air plasmas: A comparison of Maxwellian and non-Maxwellian EEDFs

The electron energy distribution function (EEDF) of cold atmospheric plasmas (CAPs) is often assumed to be Maxwellian, but it is actually non-Maxwellian, as calculated from the Boltzmann equation. To determine the impact of the Maxwellian assumption, two global models with Maxwellian and non-Maxwellian EEDFs are developed for the He + air CAPs, and the resulting discharge characteristics and reactive species are compared. With the increasing air concentration from 100 to 50 000 ppm in He + air feeding gas, the simulation results of both global models display similar trends, and the orders of the densities of both global models are similar for the reactive species and each air concentration scenario. However, the Maxwellian assumption leads to lower electron temperature, weaker electronegativity, less power dissipation to Ion Joule heating, higher densities, and lagging peak points for most types of reactive species compared to using the non-Maxwellian EEDF. These findings indicate that the Maxwellian assumption is capable of capturing the main physicochemical features of He + air CAPs, but it can lead to large calculation errors at the quantitative level.

A kinetic model for investigating the dielectric properties of rocket exhaust dusty plasmas

The electron kinetic model of rocket exhaust dusty plasma is developed based on the Boltzmann equation. Additional electron-dust elastic and inelastic collision processes are included in the kinetic equation except for the electron-neutral collisions. The dust surface potential and electron density are calculated according to the dust charging balance equation and the quasineutrality condition. The electron energy distribution function (EEDF) is calculated by solving the kinetic equation numerically. It shows that the obtained EEDF results for different dust densities deviate from the Maxwellian distribution obviously. In addition, the dielectric properties of rocket exhausts based on the obtained non-Maxwellian EEDFs are analyzed for different dust and plasma parameters. It is shown that the relative permittivity based on the EEDFs obtained from the kinetic model is larger than that based on Maxwellian distribution, while for the conductivity and attenuation constant, they become smaller. As for the influence of dust particles on the dielectric properties, a high ratio of dust and neutral density (10−9) leads to a smaller absolute value of relative permittivity, electrical conductivity, and attenuation constant for both Maxwellian distribution and EEDF obtained from the kinetic model. When the ratio is low (10−10), the dust particles almost have no influence on the dielectric properties. Meanwhile, it can be seen that the existence of dust particles makes the difference in dielectric properties obtained from the calculated EEDFs and Maxwellian distribution smaller.

Simulation of High-Power Microwave Air Breakdown Modeled by a Coupled Maxwell–Euler System With a Non-Maxwellian EEDF

In this paper, a discontinuous Galerkin time-domain (DGTD) method is developed to simulate high-power microwave air breakdown phenomena, which is modeled by a coupled electromagnetic–plasma system. In the coupled system, the electromagnetic fields are governed by Maxwell’s equations and the plasma is modeled by the five-moment fluid equations (Euler’s equations). The non-Maxwellian electron energy distribution function is used to calculate electron transport coefficients and describe the nonequilibrium collision reactions between electrons and neutral air particles. The coupled Maxwell–Euler equations are solved by the DGTD method with high-order spatial and temporal discretization, which are able to provide a sufficient resolution for the physical quantities in both space and time. Several numerical examples are presented to investigate the physical process and demonstrate the capability of the numerical method.

Comparison of Langmuir probe and multipole resonance probe measurements in argon, hydrogen, nitrogen, and oxygen mixtures in a double ICP discharge

The results of a Multipole Resonance Probe (MRP) are compared to a Langmuir probe in measuring the electron density in Ar, H2, N2, and O2 mixtures. The MRP was designed for measurements in industry processes, i.e., coating or etching. To evaluate a possible influence on the MRP measurement due to molecular gases, different plasmas with increasing molecular gas content in a double inductively coupled plasma at 5 Pa and 10 Pa at 500 W are used. The determined electron densities from the MRP and the Langmuir probe slightly differ in H2 and N2 diluted argon plasmas, but diverge significantly with oxygen. In pure molecular gas plasmas, electron densities measured with the MRP are always higher than those measured with the Langmuir Probe, in particular, in oxygen containing mixtures. The differences can be attributed to etching of the tungsten wire in the Ar:O2 mixtures and rf distortion in the pure molecular discharges. The influence of a non-Maxwellian electron energy distribution function, negative ions or secondary electron emission seems to be of no or only minor importance.

Non-Maxwellian electron energy distribution function in a pulsed plasma modeled with dual effective temperatures

A non-Maxwellian electron energy distribution function (EEDF) has been modeled within a pulsed rf inductively coupled plasma source with the aid of experimental emission spectra and Ar metastable measurements obtained by laser diode absorption. The lower energy portion of the EEDF up to the first excited state energy of 11.5 eV for argon was accurately measured with a Langmuir probe and satisfactorily modeled with a generalized two-parameter expression. Above 11.5 eV, though, inelastic collisions caused the EEDF to deviate from the lower energy generalized expression and soon after, the energy limit of accuracy of the Langmuir probe was approached. In this work, a unique EEDF model was applied for electron energies above 11.5 eV that accounts for spectral effects due to both direct excitation from the Ar ground state and step-wise excitation from the metastable state. Previously tabulated optical cross sections were used with experimental data to simulate the optical emission spectra using a theoretical non-Maxwellian EEDF with dual effective electron temperatures; one for energies below 11.5 eV and one for above. The parameters of the high energy portion of the EEDF were adjusted to produce a least squares fit to up to 10 emission peaks in the 415–428 nm range. The fits provided practical agreement with experimental spectra using the dual effective temperature EEDF. Comparisons were made for the model fitting 10 emission peaks compared to a method of analyzing only the relative intensities of 2 closely spaced emission lines: the 420.1 nm to 419.8 nm line ratio.

Modeling a Langmuir probe in atmospheric pressure plasma at different EEDFs

In this study, we present a computational model of a cylindrical electric probe in atmospheric pressure argon plasma. The plasma properties are varied in terms of density and electron temperature. Furthermore, results for plasmas with Maxwellian and non-Maxwellian electron energy distribution functions are also obtained and compared. The model is based on the fluid description of plasma within the COMSOL software package. The results for the ion saturation current are compared and show good agreement with existing analytical Langmuir probe theories. A strong dependence between the ion saturation current and electron transport properties was observed, and attributed to the effects of ambipolar diffusion.

Numerical modeling of a glow discharge through a supersonic bow shock in air

The interaction between a glow discharge and the bow shock of a Mach 3 air flow around a truncated conical model with a central spike is modeled, and comparison is made with prior experimental results. The KRONOS workflow for plasma modeling in flow fields, which has recently been developed at ONERA, was used for the modeling. Based on the quasi-neutral approximation, it couples hypersonic and reactive flow fields with electron chemistry, including the effect of non-Maxwellian electron energy distribution function. The model used for the discharge involves 12 species and 82 reactions, including ionization, electronic and vibrational excitation, and attachment. The simulations reproduce the main features of the discharge observed experimentally well, in particular, the very recognizable topology of the discharge. It was found from the simulations that behind the bow shock, in the afterglow, the negative ion flow ensures the electrical conduction and the establishment of the glow discharge. The influence of kinetic rates on the voltage-current characteristics is discussed.

Comparative analyses of plasma probe diagnostics techniques

The subject of this paper is a comparative analysis of the plasma parameters inferred from the classical Langmuir probe procedure, from different theories of the ion current to the probe, and from measured electron energy distribution function (EEDF) obtained by double differentiation of the probe characteristic. We concluded that the plasma parameters inferred from the classical Langmuir procedure can be subjected to significant inaccuracy due to the non-Maxwellian EEDF, uncertainty of locating the plasma potential, and the arbitrariness of the ion current approximation. The plasma densities derived from the ion part of the probe characteristics diverge by as much as an order of magnitude from the density calculated according to Langmuir procedure or calculated as corresponding integral of the measured EEDF. The electron temperature extracted from the ion part is always subjected to uncertainty. Such inaccuracy is attributed to modification of the EEDF for fast electrons due to inelastic electron collisions, and to deficiencies in the existing ion current theories; i.e., unrealistic assumptions about Maxwellian EEDFs, underestimation of the ion collisions and the ion ambipolar drift, and discounting deformation of the one-dimensional structure of the region perturbed by the probe. We concluded that EEDF measurement is the single reliable probe diagnostics for the basic research and industrial applications of highly non-equilibrium gas discharge plasmas. Examples of EEDF measurements point up importance of examining the probe current derivatives in real time and reiterate significance of the equipment technical characteristics, such as high energy resolution and wide dynamic range.

Langmuir paradox revisited

Electron energy distribution functions (EEDF) have been measured in the positive column of dc discharges under the condition of the Langmuir paradox (1 and 3 mTorr of argon gas) using a high resolution probe measurement system developed by the authors. An essential deviation of the measured EEDFs from Maxwellian distributions was found for both, slow and for fast electrons. The low energy peaks found in the measured non-Maxwellian EEDFs are attributed to non-local electron kinetics in the axially non-uniform electric field caused by standing striations. Similarly, in rf capacitive and inductive discharges, the formation of the low energy peak is a result of spatially localized electron heating combined with trapping of low energy electrons outside of the heating zone.

Estimating the parameters of a positive column of the halogen-containing glow discharge at moderate pressures

The methodical recommendations for estimating the plasma parameters of an electronegative glow discharge containing halogens at moderate pressures (up to 40 Torr) with the use of simple analytic formulae and without numerical modeling are given. The initial data are easily measureable discharge parameters such as a discharge current, a voltage and a gas mixture pressure and composition as well. It is shown how one can easily consider such important plasma features as non-Maxwellian electron energy distribution function and halogen molecules dissociation by electron impact. As a result, such plasma parameters as the absolute degree of electronegativity, the value of border coordinate between ion–ion and electron–ion plasmas, and the forms of transversal profiles of electron and negative ion concentrations can be evaluated. The comparison of the results with the ones given by a global numerical model shows the suitability of said analytic approach to estimate plasma parameters of real discharges.