On the Kinetics of the Active Zone of the Stationary SF6 Beam Discharge Plasma

The paper deals with the impact of intensive electron attachment on the kinetics of the electrons in the active zone of the stationary band-like beam discharge plasma in SF₆ which is an alternative useful plasma medium for “dry etching”. The energy distribution of the electrons in this plasma was obtained by numerically solving the Boltzmann equation which includes apart from elastic collisions, different exciting collision processes, attachment in electron collisions, direct ionization, the ambipolar loss of electrons, Coulomb interaction between electrons and of electrons with ions and the power input to the electrons by the turbulent electric field. In particular, due to the needed fulfilment of the consistent electron particle balance, for an extended region of the turbulence energy density in this plasma a large impact on the electron kinetics of the intensive electron attachment, which is the prevailing electron loss process, was found enforcing independent of the turbulence energy density always a large power input to the electrons, smooth and only slowly decreasing energy distributions even in the energy region of direct ionization.     

ELectron Collision Rates and Transport Coefficients of a Weakly Ionized dc Plasma in Ar/Sih4-Mixtures

The paper deals with electron kinetics of a Ar/SiH₄ dc plasma, using the stationary and spatially uniform Boltzmann equation. The solution of this kinetic equation has been obtained by applying higher order Legendre polynomial expansion of the electron velocity distribution. For varying mixture composition the energy distribution and relevant macroscopic quantities as mean energy, drift velocity, rate coefficients for excitation, dissociation and ionization and relevant energy transfer rates for these processes have been calculated. In particular, it has been found that a most effective activation of the feed gas occurs for SiH₄ admixtures in the range from 5 to 10 per cents.     

Nonstationary and Quasi-stationary Treatment of Distribution Anisotropy in the Temporal Relaxation of an Energetic Electron Group

A relaxation study of an electron group in collision dominated weakly ionized plasmas has been performed. The study is based on the two-term approximation of the Legendre polynomial expansion of the electron velocity distribution in the nonstationary Boltzmann equation. To overcome the limitation of the conventional quasi-stationary treatment of the distribution anisotropy, a very efficient solution approach of the nonstationary kinetic equation in two-term approximation has been developed which allows for a strict nonstationary treatment of the distribution anisotropy. By using this approach the temporal evolution of the isotropic and anisotropic distribution of the electrons has been investigated for a model plasma, which involves typical features of an inert gas plasma. A comparison of the results with corresponding ones obtained by applying the conventional approach under various parameter conditions clearly indicates a pronounced falsification of the real relaxation course by the latter approach.     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials. 2. Comprehensive Investigation of a Model Plasma

Applying the new technique for finding the converged solution of the Boltzmann equation in a weakly ionized plasma, which was developed in the first part of this paper, a comprehensive study of the electron velocity distribution function for a model plasma with elastic and exciting collisions is performed by solving the Boltzmann equation with increasing order of approximation. The purpose of this investigation is that of calculating the isotropic distribution f₀, the first contribution f₁ to the anisotropy of the velocity distribution, the important macroscopic quantities and, more generally, that of studying the total anisotropy as well as the changes of all these quantities when the approximation degree is enlarged beyond the 2 terms of the conventional Lorentz approximation. By varying some parameters of the model plasma, that is the electric field strength, the magnitude of the excitation cross section and the excitation threshold, the main features of plasmas in inert as well as molecular gases are modelled and the impact of these parameters on the mentioned quantities is analysed. Some of the converged results are compared with results of corresponding Monte Carlo simulations. The approximation degree required to find the converged values of isotropic distribution, main macroscopic quantities and electron distribution in the velocity space (and thus its real anisotropy) is estimated by solving the Boltzmann equation over wide parameter ranges.     

Strict Kinetic Treatment of the Electron Response to Spatial Field Disturbances in Nonequilibrium Plasmas

In generalization of former approaches for the simplified solution of the inhomogeneous electron Boltzmann equation a higher order solution technique has been developed. This technique is based on a multi-term expansion of the electron velocity distribution function and allows a strict study of the electron kinetics in plasmas acted upon by space-dependent electric fields. This solution technique is used to investigate the response of the plasma electrons to spatially limited disturbances of the electric field in weakly ionized plasmas of helium and mercury. By solving the kinetic equation with increasing order of the multi-term expansion the convergent solution of the kinetic problem and thus the strict spatial behaviour of the velocity distribution and of significant macroscopic properties of the electrons has been determined and analysed. Furthermore, the impact of higher order terms of the expansion has been revealed and the falsification of the velocity distribution and of related macroscopic properties has been evaluated when instead of the multi-term solution the simpler two-term solution of the kinetic equation is used.     

A Model for the Bulk Plasma in an RF Chlorne Discharge

A model has been constructed to describe the electrical characteristics of the central bulk plasma region in a 13.56-MHz parallel-plate discharge in chlorine at pressures of about 1 torr. This region is modeled as a volume-controlled plasma with the electron balance dominated by single-step electron-impact ionization and attachment and with the electron energy distribution function in equilibrium with the local instantaneous electric field. Relationships between the ionization frequency, the attachment frequency, the electron drift velocity, and the electric field are provided by solutions of the Boltzmann equation for mixtures of Cl2 and Cl which result from Cl2 dissociation. From a measured current waveform and Cl2/Cl density ratio, the model generates the local electric-field waveform, the time-varying electron density, and the power density in the central portion of the bulk plasma. The calculated time-averaged power input per unit discharge length compares well with experimentally determined values.     

Emission characteristics and parameters of gas-discharge plasma in mixtures of heavy inert gases with chlorine

The ultraviolet (UV) radiation from longitudinal glow-discharge plasma in three- and four-component mixtures of argon, krypton, and xenon with chlorine has been investigated. The total radiation of Ar, Kr, and Xe monochlorides and chlorine molecules in the spectral range 170–310 nm has been optimized with respect to the composition and the pressure of gas mixtures, as well as the discharge current. The mean output power, the electric power of discharge, and the efficiency of a broadband low-pressure exciplex halogen lamp have been determined. Parameters of the glow discharge in Ar-Kr-Cl₂ and Kr-Xe-Cl₂ mixtures have been simulated numerically. The electron energy distribution functions have been determined through the solution of the Boltzmann kinetic equation. These functions have been used to calculate the plasma parameters, namely, electron transfer characteristics, specific losses of discharge power for electronic processes, and ionization and attachment coefficients.     

Self-consistent hybrid model for a stratified positive column of a low pressure glow discharge

The stratification of a positive column of a low-pressure glow discharge in inert gases has been studied with the help of a self-consistent hybrid model. The model is based on the solution of a nonlocal kinetic Boltzmann equation for the electron distribution function, a nonstationary drift-diffusion equation for the ions, and the Poisson equation for the electric field. Spatial electron and ion density distributions and the electric field distribution in the positive column were obtained. The converged solution of the model gives a self-consistent resonant strata length L and the value and the form of the modulated plasma parameters. An unexpected surprising result was obtained: for a given potential drop in the positive column of a low-pressure glow discharge, a self-consistent spatially modulated striation-like electric field does not lead to the resonant increasing of the ionization frequencies in the discharge as compared with a constant electric field with the same potential drop. Usually, it was assumed that, in spatially modulated field distributions, all the parameters in a striated plasma will be more pronounced and have a resonant form. The text was submitted by the authors in English.     

The Impact of Direct Ionization by Beam Electrons on the Electron Kinetics of the Beam Discharge Plasma in Molecular Hydrogen

In a recent paper the stationary beam plasma discharge in partially dissociated hydrogen was investigated where the electron component was described by the Boltzmann equation for a mixture of atomic and molecular hydrogen and the main heavy charged and neutral particles by balance equations. It was assumed that, via the quasilinear beam plasma interaction, the electron beam produces only the turbulent electric field whilst an additional production of plasma electrons due to direct ionization by the beam and thus a direct influence on the balances of charge carriers were neglected. Now the additional production of plasma electrons due to direct ionization by the beam is studied on the basis of a generalized Boltzmann equation but for the simpler model of a purely molecular hydrogen plasma. For experimentally obtainable values of the turbulence energy density, beam energy, beam ionization degree and electron life time the calculation of the electron energy distribution function and of the direct beam contribution to the electron particle balance shows a marked influence of the direct beam ionization with increasing degree of beam ionization.     

Temporal relaxation of plasma electrons acted upon by directcurrent electric and magnetic fields

On the basis of the time-dependent electron Boltzmann equation the temporal relaxation of the electrons in the presence of electric and magnetic fields in weakly ionized, collision dominated plasmas has been studied. The relaxation process is treated by using a strict time-dependent two-term approximation of the velocity distribution function expansion in spherical harmonics. A new technique for solving the time-dependent electron kinetic equation in this two-term approximation for arbitrary angles between the electric and magnetic fields has been developed and the main aspects of the efficient solution method are presented. Using this new approach and starting from steady-state plasmas under the action of time-independent electric fields only, the impact of superimposed DC magnetic fields on the electron relaxation is analyzed with regard to the control of a neon plasma. The investigations reveal an important effect of the magnetic field on the temporal relaxation process. In particular, it has been found that the relaxation time of the electron component with respect to the establishment of steady-state can be enlarged by some orders of magnitude when increasing the magnetic field strength     

Anode plasma structure in a gas discharge with neutral density depleted by ionization

We consider the anode plasma structure in a gas discharge with density of neutral atoms (neutrals) depleted by strong ionization. We obtain analytical solutions of the quasi-neutrality equation for the potential distribution and a condition for the existence of anode plasma in the one-dimensional case for arbitrary potential dependences of the neutral depletion frequency and the electron density. We consider the special cases of a constant neutral depletion frequency, ionization by Maxwellian electrons, and ionization by an intense electron beam under the conditions of collisionless ion motion and Boltzmann thermal electron distribution. The solutions for the first two cases at zero depletion parameter, i.e., at constant gas density, match those obtained in [1] by a power series expansion. In the case of ionization by Maxwellian electrons, the formation of anode plasma at reasonable working-gas flow rates is shown to be possible only at a fairly high electron temperature (if, e.g., xenon is used as the working gas, then T _e ≥ 5 eV). Steady-state solutions of the quasi-neutrality equation under ionization by an intense electron beam exist only if the ratio of the electron beam density to the maximum thermal electron density does not exceed a certain limiting value.     

Theoretical modeling of microwave-pumped high-pressure gas lasers

The electron avalanche and laser excitation processes in high-pressure discharges at microwave frequencies are investigated. In our model, the applied electromagnetic field is treated classically and assumed to be monochromatic. The Boltzmann equation for the electron velocity distribution function under the influence of an alternating electric field is numerically solved for a typical XeCl laser gas mixture. All relevant elastic, inelastic and electron-electron collisions are included in solving the Boltzmann equation. The theoretical modeling of microwwave-pumped high-pressure gas lasers are developed based on the first law of thermodynamics in order to determineE _rms/n (root-mean-square field strength/total number density of gas molecules) which is required by the Boltzmann equation to calculate the electron kinetics rates and microwave-power absorption by the plasma. A sample calculation of the microwave-pumped XeCl laser is presented, and a fair agreement between theory and experiment is seen.Paper partially presented at the 10th Int. Conf. on Lasers and Applications, Lake Tahoe, Nevada, USA (1987)     

Time - Dependent Multi - Term Treatment of Plasma Electrons Acted upon by RF Electric Fields

Using a new method for solving the time-dependent electron Boltzmann equation in higher order accuracy, studies of the temporal behaviour of electrons in weakly ionized, collision-dominated plasmas under the action of rf fields have been performed. The method is based upon a multi-term approximation of the Legendre polynomial expansion of electron velocity distribution function and is applied to investigate the established periodic behaviour of the electron velocity distribution in helium, argon and CO plasmas. The results obtained in various approximation orders are discussed. The analysis has shown that the simplified treatment using only two terms of the velocity distribution expansion can fail in several conditions. In general, the four-term approximation gives already a good representation of the convergent solution of the electron Boltzmann equation at each instant of the rf period. The discrepancies between two-term and convergent results are found to depend sensitively on the specific atomic data, in particular on the magnitude of the various electron collision cross sections involved. Furthermore, the results obtained in the multi-term approximation are compared with corresponding ones obtained by accurate Monte Carlo simulations. Very good agreement between convergent eight-term Boltzmann and Monte Carlo calculations is found.     

Solving the Boltzmann equation in a 2-D-configuration and2-D-velocity space for capacitively coupled RF discharges

A new kinetic scheme, the generalized Monte Carlo flux (GMCF) method, provides the electron particle distribution function in phase space, f(ν, μ, r, z, t) (ν: speed, μ: velocity angle, r: radial position, z: axial position, and t: time), for solving the Boltzmann equation in modeling capacitively coupled RP discharges. For a simulation with spatial- and temporal-varying fields in RF discharges, the GMCF method handles the collision terms of the Boltzmann equation by using one transition matrix to compute the collision transition between velocity space cells. An anti-diffusion flux transport scheme is developed to overcome the numerical diffusion in the velocity and configuration spaces. The major advantages of the GMCF method are the increase in resolution in the tail of distribution functions and the decrease of computation time. The GMCF calculation results in terms of microscopic electron distribution function and macroscopic quantities of density, electric field and ionization rate, are presented for RF discharges and compared with other kinetic and fluid simulation and experimental results. The effects of the induced radial electric field in the sheath close to the radial wall in a cylindrically symmetric parallel-plate geometry are discussed     

Effect of small admixtures of N<Subscript>2</Subscript>, H<Subscript>2</Subscript> or O<Subscript>2</Subscript> on the electron drift velocity in argon: experimental measurements and calculations

The electron drift velocity in argon with admixtures of up to 2% of nitrogen, hydrogen or oxygen is measured in a pulsed Townsend system for reduced electric fields ranging from 0.1 Td to 2.5 Td. The results are compared with those obtained by Monte Carlo simulations and from the solution of the electron Boltzmann equation using two different solution techniques: a multiterm method based on Legendre polynomial expansion of the angular dependence of the velocity distribution function and the S _n method applied to a density gradient expansion representation of the distribution function. An almost perfect agreement between the results of the three numerical methods and, in general, very good agreement between the experimental and the calculated results is obtained. Measurements in Ar-O₂ mixtures were limited by electron attachment to oxygen molecules, which contributes to the measured drift velocity. As a result of this attachment contribution, the bulk drift velocity becomes larger than the flux drift velocity if attachment is more probable for electrons with energy below the mean value and smaller in the opposite case. Attachment also contributes to the negative differential conductivity observed in Ar-O₂ mixtures.     

Geschwindigkeitsverteilungsfunktionen von Atomen und Ionen in verschiedenen Anregungs- und Ionisationsstufen in Niederdruck-Entladungen bei hohem Ionisationsgrad

Supposing free-fall conditions the velocity distribution functions of atoms and ions in various levels in gas discharges at low pressures are calculated. In particular, plasmas at high degrees of ionization are considered. Solving the Boltzmann equation for the motions transverse to the wall of the discharge tube it is shown that the velocity distribution functions can considerably deviate from the Maxwellian and become non-isotropic. Inelastic collisions with electrons and the ionization by electron impacts considerably determine the velocity distribution function of the neutral atoms. The velocity distribution function of the ions is also essentially determined by the electric field within the plasma. For the motions transverse to the wall the half widths of the velocity distribution functions do not only depend on the temperature of the wall, but on the electron density and on the electron temperature as well. At small electron densities the half widths for excited atoms and for ions can be narrower than the one for the ground state atoms. The charge exchange between atoms and ions is shortly taken into consideration.     

Electron transport in semiconductors in the presence of impact ionization

The development of impact ionization in semiconductors has been considered. A transport equation for the electron distribution function in the presence of impact ionization has been derived. It has been found that the collison integral of this equation is nonlinear with respect to the distribution function. The relation between the solutions of this equation and the usual Boltzmann equation have been determined. A Monte Carlo method for numerical calculations in the presence of impact ionization has been developed. The results of numerical calculations carried out using a simplified model of indium antimonide are presented.     

Response of the Electron Kinetics on Spatial Disturbances of the Electric Field in Nonisothermal Plasmas

An efficient method for solving the inhomogeneous electron Boltzmann equation for a weakly ionized collision dominated plasma is represented. As a first application this method is used to investigate in a helium plasma the response of the electron velocity distribution function and of the relevant macroscopic quantities to the impact of spatially limited disturbances in the electric field. In addition to the field action elastic and (conservative) inelastic collisions of electrons with gas atoms are taken into account in the kinetic treatment. In this way the spatial relaxation behaviour of the electrons and their establishment into homogeneous plasma states could be studied on a strict kinetic basis. Unexpectedly large relaxation lengths in electron acceleration direction have been found at medium electric fields.