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.     

Impact of the Vibrationally Excited States on the Macroscopic Behaviour of the Band-like Electron Beam Discharge Plasma in H/H2-Mixture

The investigation of the impact of the vibrationally excited molecules in the electronic ground state was performed by simultaneously solving a balance equation system for the main charge carriers, the H atoms, the metastable H atoms, the H₂ molecules in the different vibrational states and for the power transfer of the electrons in the beam discharge mixture plasma. The balance equations for the vibrational states include in particular one-quantum step excitation and deexcitation, electronic excitation, dissociation and ionization from each vibrational level in electron collisions as well as the finite life time of these states because of the gas transfer through the band-like plasma. A main finding is that due to the additional impact of vibrationally excited molecules there is a marked enhancement of the resulting dissociation and ionization degree in the beam discharge plasma at medium power input from the turbulent electric field. For discharge parameters of practical interest the ionization and dissociation budget, the population of the vibrational states, the different energy dissipation processes and the energy pumping into the ladder of the vibrational states were calculated and discussed in detail.     

Modelling of Secondary Electron Impact on the Self-Consistent RF Bulk Plasma Description

The impact of a secondary electron beam, generated at the electrodes and accelerated in the sheaths, on the self-consistent treatment of the electron behaviour in an rf bulk plasma has been investigated by a parametric study. Source of electrons in the plasma are collisional ionization and secondary electron injection. Electrons are lost by ambipolar diffusion to the electrodes of a parallel plate rf discharge configuration. The non-stationary Boltzmann equation is used to determine self-consistently the rf field amplitude necessary for maintaining the steady-state rf bulk plasma as well as the time resolved behaviour of the electron energy distribution function and of all contributions to the electron particle and power balance, at given source rate and energy distribution of secondary electron injection.     

Kinetics of the Electron Component in the Turbulent Electron-Beam Discharge Plasma in Diatomic Gases and Comparison of some Macroscopic Properties between the Plasmas of the Beam and the Glow Discharge

The present paper is devoted to the investigation of the kinetics of the electron component in the stationary beam discharge plasma in molecular nitrogen. Using the Boltzmann equation with the inclusion of the elastic and the main inelastic binary collisions and also the Coulomb interaction between the charged particles we have calculated the energy distribution function and some macroscopic quantities of the electrons within a large range of parameters. Using our earlier results for the beam discharge plasma in hydrogen, also the dependence of the macroscopic quantities on the kind of molecular gas is discussed. Finally, the comparison of some macroscopic properties of the beam and the glow discharge plasma was performed under the condition of equal power input per volume unit in both types of plasmas in nitrogen.     

Kinetic Description of the Stationary Band-Like Weakly Turbulent Beam Discharge Plasma in Hydrogen II. Budget of Ions and Metastables — Analysis of the Balance Equation System

In the first part [1] of our paper a kinetic model for the band-like beam discharge plasma in hydrogen was derived, numerically obtained results on the ionization and dissociation degree were presented and a detailed explanation of the influence of the electron kinetic properties on the macroscopic plasma behaviour was given. Continuing these investigations, we now report and discuss the results on the ion- and metastable atom-budget. On this basis an analysis is made of the resulting balance equation system with a view to finding the most important physical processes, and a simplified equation system is derived which allows to interpret the essential features of the macroscopic plasma properties.     

Die Berücksichtigung der Ionisation durch Elektronenstoß in der Elektronenkinetik des schwachionisierten anisothermen Plasmas. I. Analyse des Ionisierungsstoßintegrals und Ableitung von geeigneten Darstellungen für eine numerische Behandlung

The influence of the Ionization by Electron Collisions on the Electron Kinetics of the Low Ionized Anisothermal Plasma. I. Analysis of the Ionization-Collision-Integral and the Derivation of Representations Appropriate ot Numerical Calculations Starting form the balance of the electrons in the velocity space we determined the contribution of the ionization by electron collisions to the Boltzmann equation for the electrons. After integrating stepwise the derived collision-integral for ionization we obtained the development of the ionization-collision-integral in spherical harmonics, using the impulse and energy balance. From the latter the contributions to the concentration and the energy balance of the electrons were determined. Using special asumptions concerning the differential scattering cross section for the ionization we derived representations of the collision-integral suitable for numerical calculations. Especially the important contribution of this collision-integral to the equation of isotropic distribution function depends on the total cross section for ionization only.     

On the Question of Increasing Maxwellization of the Electron Energy Distribution Function due to Electron — Electron Interaction with Growing Ionization Degree in the Beam Discharge Plasma

The question of the establishment of an almost complete Maxwellian energy distribution of the electrons in a given discharge plasma is of importance, especially under the aspect of a more convenient determination of transport and rate coefficients. In [5] an energy space averaged criterion for the establishment of such a structure of the electron energy distribution function in the stationary beam discharge plasma of molecular gases was formulated from a heuristic point of view. This paper investigates this question in detail starting from the adequate Boltzmann equation of such plasmas. Energy resolved conditions for the establishment of an almost complete Maxwellian distribution are derived from the kinetic equation. Using one of these conditions, the formal derivation of the mentioned averaged condition is performed and its limitations are shown. The fulfilment of the energy resolved and of the averaged conditions are illustrated and discussed using solutions of the kinetic equation in purely molecular hydrogen for two parameter sets yielding larger and smaller deviations of the real distribution function from the Maxwellian form.     

Glow discharge in rare-gas and metal vapour mixture

Applying the Boltzmann equation to a He-Cd mixture discharge the electron energy distribution functions, kinetic coefficients and collision frequencies are numerically calculated. Calculations are made for a homogeneous and stationary discharge plasma subjected to an externally applied electric field. The collision processes which have been taken into account are elastic and inelastic collisions of electrons with He and Cd atoms as well as mutual encounters of electrons. In this case the electron energy distribution and all the quantities calculated from it are dependent on the reduced electric field, the ionization degree and the relative cadmium concentration.     

Electron Kinetics with Attachment and Ionization from Higher Order Solutions of Boltzmann's Equation

An appropriate approach is presented for solving the Boltzmann equation for electron swarms and nonstationary weakly ionized plasmas in the hydrodynamic stage, including ionization and attachment processes. Using a Legendre-polynomial expansion of the electron velocity distribution function the resulting eigenvalue problem has been solved at any even truncation-order. The technique has been used to study velocity distribution, mean collision frequencies, energy transfer rates, nonstationary behaviour and power balance in hydrodynamic stage, of electrons in a model plasma and a plasma of pure SF₆. The calculations have been performed for increasing approximation-orders, up to the converged solution of the problem. In particular, the transition from dominant attachment to prevailing ionization when increasing the field strength has been studied. Finally the establishment of the hydrodynamic stage for a selected case in the model plasma has been investigated by solving the nonstationary, spatially homogeneous Boltzmann equation in twoterm approximation.     

Probe Measurements in a Beam Plasma Discharge

The influence of the beam energy and beam current density on the parameters of the plasma is investigated. The efficiency of carrier generation by collective interactions (Langmuir-oscillations) is compared with that by the single collision mechanism. The additional negative charging of the static probe caused by the beam electrons is used for the determination of the beam electron density.     

Kinetic Description of the Stationary Band-Like Weakly Turbulent Beam Discharge Plasma in Hydrogen. I. Basic Equations — Numerical Approach — Ionization and Dissociation Budget

On the basis of the electron Boltzmann equation and of the balances for the charge carriers (e, H⁺, H₂⁺, H₃⁺), the H-atoms and the metastable H(2s)-atoms for the H/H₂-mixture the behaviour of the weakly turbulent, band-like electron-beam discharge plasma in hydrogen has been calculated taking into account the main collision and transport processes. In dependence of the normalized discharge length, the life time of the neutral particles in the band-like plasma and the electron beam generated turbulence energy density the ionization and dissociation budget and in particular the most important electron-heavy particle collision rates, which appear in the balance equation system, are investigated in the present part of our paper. In a second part the results related to the budget of the ions and the metastable atomic state will be reported and an analysis of the main processes in the balances will be made.     

Modeling of plasma-etch processes using well stirred reactorapproximations and including complex gas-phase and surfacereactions

A 0-D or well stirred reactor model determines spatially and time-averaged species composition in plasma-etch reactors, through solution of species, mass, and electron-energy balance equations. The use of well stirred reactor approximations reduces the computational expense of detailed kinetics calculations and allows investigation of the dependence of plasma chemistry on etch-process parameters. The reactor is characterized by a chamber volume, surface area, net mass flow or residence time, pressure, energy loss to surroundings, power deposition, and inlet-gas composition. The electron-energy equation includes a detailed power balance with losses to ions and electrons through the sheath, as well as inelastic and elastic collision losses. The model employs reaction-rate coefficients for electron-impact reactions, which require an assumption of the electron energy distribution function (EEDF). We compare model results using Maxwellian EEDF's, as well as reaction-rate coefficients determined as a function of average electron energy through solution of the Boltzmann equation, for chlorine chemistry. The Boltzmann rates are determined by time-lagging the equilibration of electrons with applied electric fields. The Maxwellian reaction rates give higher ionization fractions than the Boltzmann rates, affecting the predicted electronegativity and positive ion composition for chlorine plasmas. The model also shows a strong sensitivity of the plasma composition to the assumed surface-recombination probability of atomic chlorine     

Die Berücksichtigung der Ionisation durch Elektronenstoß in der Elektronenkinetik des schwachionisierten anisothermen Plasmas III. Das instationäre Verhalten bei großer impulsförmiger Feldstörung

The Influence of Ionization by Electron Collisions on the Electron Kinetics of the Low Ionized Anisothermal Plasmas The time behaviour of the electron component was calculated during the additional application of a single pulse to the electric field in the plasma. The investigations were performed for the weakly ionized Ne-plasma as a typical example taking into account supplementarily the direct ionization due to electron collisions and an electron loss term with a constant life time besides elastic and exciting collisions. Using the instationary Boltzmann equation we determined the time behaviour of the essential macroscopic quantities. Besides the caluclation of the marked temporal development of such quantities as the electron concentration, the electron collision frequencies for excitation and ionization and the different energy transfer rates especially the relaxation of the electron component was analysed after switch on and switch off the additional rectangular pulse.     

The Influence of Admixtures of Molecular Gases on the Efficiency of Radiation Production by Ar?Hg Mixture Plasmas used in Fluorescent Lamps

Starting from former investigations of pure Ar? Hg mixture plasmas in parameter ranges typical of fluorescent lamps we studied the influence of additional admixtures of molecular gases (N₂, H₂) on the energy transfer from the electrons heated by an electric field to the lowest excited states of Hg atoms which are the energy source for the resonance radiation production. By calculation of the different power loss rates via solving the appropriate Boltzmann equation for three component mixture plasmas it was found that already a threshold level of molecular impurities of about 10^?4 Torr leads to a marked energy dissipation by the impurities and thus to a pronounced reduction of the efficiency of the resonance radiation production. This is caused by the great effectivity of vibrational excitation of molecules in electron collisions due to the great cross sections for such collisions and their low thresholds.     

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.     

Eine quantitative Deutung der Elektronengruppen im elektronenstrahlerzeugten He-Niederdruckplasma mit Hilfe der Boltzmannschen Stoßgleichung

Neglecting electron-electron collisions the velocity distribution of plasma electrons in a beam generated He-low pressure plasma is evaluated with help of Boltzmann equation. The energy distribution of secondary electrons generated by the electron beam is considered applying the atom collision theory ofGryzinski. The resulting velocity distribution of plasma electrons shows group character. The “temperatures” of the ultimate and secondary electrons and their density ratio are in satisfying agreement with experiment.     

Field Free Relaxation of the Electron Component in the Temporally Decaying Molecular Gas Plasmas Hydrogen and Nitrogen

This paper deals with the temporal relaxation of the electron component of weakly ionized, anisothermal and collision dominated plasma in the molecular gases hydrogen and nitrogen after jump-like switching off of the electric field starting from stationary states. The investigation is based upon a solution of the non-stationary Boltzmann equation using a finite difference approach of the resulting partial differential equation. Besides the temporal development of the energy distribution and of some macroscopic quantities of the electrons especially the characteristic relaxation times and their physical nature are discussed and compared with former results on the relaxation in an inert gas plasma.     

Tonks-Langmuir problem for a bi-Maxwellian plasma

An analytical solution of the Tonks-Langmuir (TL) problem with a bi-Maxwellian electron energy distribution function (EEDF) is obtained for a plasma slab. The solution shows that the ambipolar potential, the plasma density distribution, and the ion flux to the wall are mainly governed by the cold electrons, while the ionization rate and voltage drop across the wall sheath are governed by the hot electrons. The ionization rate by direct electron impact is found to be spatially rather uniform, contrary to the T-L solution where it is proportional to the plasma density distribution. The temperature of hot electrons defined by the ionization balance is found to be close to that of the T-L solution for a mono-Maxwellian EEDF, and is in reasonable agreement with experiments carried out in a low pressure capacitance RF discharge. The energy balance for cold electrons in this discharge shows that their heating by hot electrons via Coulomb interaction is equalized by the cold electrons' escape to the RF electrodes during collapse of the RF sheath     

电子束泵浦氩中高能电子分布的理论计算

 改进了文献中报导的Boltzmann基本方程。与Boltzmann基本方程相比，改进后的Boltzmann方程更全面地描述了电子与基态氩原子碰撞的物理过程，并能计算出整个能量区间的电子分布。利用Boltzmann基本方程和改进的Boltzmann方程，对电子束泵浦氩中能量大于氩原子第一激发态能量(11.56eV)的高能电子分布函数进行了理论计算。计算中，选取了电子碰撞氩的微分电离截面和激发截面的解析表达式。对计算所得的稳态电子分布函数以及达到稳态分布所需的特征时间进行了分析和讨论。