The Calculations of the Concentrations,of the Radial Distributions,and of the Radial Particle Currents of Excited and of Doubly Charged Ions in Low Pressure Discharges

A system of differential equations describing the radial profiles of the number densities and of the radial drift velocities of the ions and of the electrons in positive columns at low pressure containing several species of ions is derived. Excited ions and doubly charged ions, generated in two-step processes by electron impacts, the inertia of the ions and space charge effects are taken into account. For the excited ions de-excitation processes by electron collisions and by spontaneous emission are regarded. A set of nonlinear equations to determine the population densities and the initial values of the differential equations and corresponding boundary conditions are put up. Numerical solutions are given for discharges in argon under free-fall conditions similiar to argon ion lasers. One notices that without stepwise processes via excited ion levels the concentration of double charged ions remains small. In some cases the radial drift of the ions considerably reduces the population of the metastable ion levels. The radial density profiles of the double charged ions and of long-living excited ions considerably deviate from the squared radial profile of the electron density. In addition, for low degrees of ionization the theory of the free-fall column given by Tonks and Langmuir is extended to plasmas containing two species of ions.     

Ladungsträgerdichte,Neutralgasdichte, elektrisches Potential und Elektronentemperatur in der zylindrischen Diffusionssäule unter Berücksichtigung des Elektronendruckes

The profiles of charged particle density, neutral gas density, charged particle generation and the electron temperature itself are calculated in the cylindrical positive column under diffusion conditions. The electron pressure is taken into account. The nonlinear differential equations for the charged particle density and the neutral gas density are solved by means of relatively well konverging power series. Inside the column the neutral gas partial pressure decreases with increasing electron partial pressure. Thereby, the profiles for charged particle density and electrical potential are flattened near the axis and fall more rapidly near the wall, and the electron temperature increases. The electron pressure increases with rising electric power input into the column. The necessary condition for the existence of a steady-state low pressure discharge with high current densities is investigated shortly.     

The Theory of the Low Pressure Discharge in the Strong Ionization Regime

A system of integro-differential equations describing the velocity distribution functions of the neutral atoms and of the ions and the radial profiles of the electric potential and the particle densities in the positive column at low pressures and high degrees of ionization is derived. This system is solved for a simplified, but self-consistent, plane model. Both the initial velocity of the ions and the neutral gas depletion caused by the ionization processes are taken into account. For a cylindrical model a solution is obtained for the region near the axis. With rising neutral gas temperature the radial ion flux density is slightly increasing. The pressure tensor, the heat flux tensor, and several components of the corresponding tensor of the fourth order for the ion gas are calculated. It is shown that on the axis the radial and the azimuthal ion temperatures are smaller than the neutral gas temperature because the ions generated out of neutral atoms are retarded during the flight to the axis.     

Längsfeldstärke,Elektronentemperatur, mittlere Neutralgasdichte,Ionenstromdichte auf die Wand und mittlere Entladungsstromdichte in der Freifallsäule bei hohem Ionisationsgrad

On the basis of a foregoing paper new theoretical results for the positive column at low pressure and strong ionization, especially for discharges in noble gas ion lasers, are given. The mean velocity v_n0 of the neutral atoms reemitted from the wall is taken into account. The electric conductivity is calculated for an argon plasma. The formulas connecting the electron temperature, the mean neutral gas density, and the electric field strength are derived. The electron temperature, the axial electric field intensity, the degree of ionization, the axial electron drift velocity, the ion flux to the wall, and the force density causing the main part of gas pumping along the column are calculated as functions of the product of the mean current density and the tube radius, and of v_n0 for argon. The axial drift velocity of the electrons is still smaller than the mean thermal electron velocity for high discharge currents, except at very low gas pressures. In general, the ion flux to the wall is not directly proportional to the discharge current. The factor for the determination of the charged particle density by means of probe measurements at the wall is discussed. The self-magnetic field affects the discharge only at high electron temperature, high degree of ionization, and relatively large tube radius, i.e. at high current density and low gas pressure in not too narrow discharge channels.     

Geschwindigkeitsverteilung,Teilchendichte, transversale Driftgeschwindigkeit und Energiedichte des Neutralgases in der Freifallsäule unter Berücksichtigung der „Neutralgasaufzehrung”︁ durch die Ionisationsvorgänge

A neutral gas rarefaction caused by ionization processes occurs in the plasma of the low pressure gas discharges. The velocity distributions, the particle density, the transversal drift velocity and the energy density of the neutral gas are calculated both for the plane and for the cylindrical positive column under free fall conditions. The neutral gas rarefaction is taken into account. It is shown, that the velocity distributions is non-Maxwellian and anisotropic. The pressure tensor is anisotropic, too. Particle density and energy density of neutral gas decrease with increasing electron density and electron temperature relatively homogeneously over the cross section of the column. Only, if the degree of ionization is high, these densities are much smaller near the axis than at the wall. Decreasing neutral gas temperature causes a similar change in the particle density profile as increasing electron density and electron temperature do. The transverse neutral gas pressure decreases from the axis to the wall in all cases. In the steady-state column an upper limit exists for the transverse particle current density of the neutral gas and of the ion gas. This limit depends on the gas temperature, the filling pressure and the atomic mass of the filling gas. In the appendix the Boltzmann equation is given in a form, which is suitable to investigate cylindrical problems not only for simple examples.     

Plasma Properties and Neutral Gas Densities in Hg-Rare Gas Discharges

The density of the neutral gases in Hg-rare gas discharges and their spatial distribution is controlled by the discharge parameters as well as by the externally adjustable partial pressures. Essential quantities in this context are the gas temperature produced by elastic collisions, the ion transport by the discharge current, the processes governing the wall temperature, and ambipolar diffusion. Despite equal partial pressures different densities may occur, which, in turn, will influence the parameters of the discharge. This has effects on the assessment of the methods of Hgvapour pressure adjustment and on the evaluation of the measured values. These effects are demonstrated, especially by field strength measurements, for a wide parameter range.     

Electron temperature in a decaying krypton plasma in the presence of a low electric field

The influence of low electric fields on the average electron energy in an afterglow krypton plasma is studied by means of probe diagnostics and theoretical analysis. It is shown that, when the average electron energy is lower than the energy corresponding to the minimum scattering transport cross section, the degree of plasma ionization substantially affects the shape of the electron energy distribution function (EEDF). The nonequlibrium character of the EEDF results in the density dependence of the coefficient of ambipolar diffusion, which leads to a change in the radial profile of the charged particle density, an increase in the drop in the ambipolar potential across the plasma, and an increase in the rate of diffusive plasma decay. These effects substantially enhance the diffusive cooling of electrons, which is probably a decisive factor influencing the electron energy balance in high-Z noble gases.     

Theoretical Description of the Radial Distributions of the Ions and of the Electrons in the Subnormal Positive Column

A relatively simple set of differential equations describing the radial profiles of the number densities and of the radial drift velocities of the ions and the electrons and of the radial electric field intensity in the subnormal positive column is derived. The inertia of the ions is taken into consideration, but the ion temperature is supposed to be zero. Corresponding boundary conditions are used. For discharges in argon under diffusion and under free-fall conditions numerical solutions are given.     

Gas Density Distributions and Reduced Field Strenghts of the Positive Column of Low Pressure XeCl* Glow Discharges

Interferometric measurements of radial gas density distributions have been performed on the cylindrical positive column of DC low pressure glow discharges (LPGD) in pure Xe and Xe/Cl₂ gas mixtures. Absolute gas temperatures have been measured by thermocouples. In the mixtures, the gas temperature is several hundred Kelvin above the temperature in pure Xe. Additionally, the radial distribution of the gas density in the mixtures cannot be described by Bessel profiles, which would result from Schottky's diffusion theory. Combined with field strength measurements, radial profiles of E/N (electric field strength/neutral density) have been determined. Results of this work will be useful for model developments of LPGD in rare-gas/Cl₂ mixtures but also for the general understanding of the positive column in attaching gases.     

Effect of Parallel-Plate Geometry on Mode Transition Behavior in Argon Microplasmas: Two-Dimensional Simulation

A two-dimensional self-consistent fluid model is employed to investigate radio-frequency process parameters on the plasma properties in Ar microdischarges. The neutral gas density and temperature balance equations are taken into account. We mainly investigate the effect of the electrode gap on the spatial distribution of the electron density and electron temperature profiles, due to a mode transition from the regime(secondary electrons emission is responsible for the significant ionization) to the regime(sheath oscillations and bulk electrons are responsible for sustaining discharge) induced by a sudden decrease of electron density and electron temperature.The pressure, radio-frequency sources frequency and voltage effects on the electron density are also elaborately investigated.     

A theoretical study of the influence of ambipolar diffusion on the population distribution of excited states in a plasma

The effect of charged particle diffusion on excitation temperature was investigated theoretically, assuming that the diffusion processes are ambipolar and that the charged particles return to the plasma in the form of ground state atoms, after recombination at the plasma boundary. An approximate expression was obtained by solving a set of balance equations for the excited state atom densities. It shows that, with increase of the plasma pressure, the excitation temperature rises and approaches the electron temperature. It was found also that, as the electron density increases, the diffusion effect on the excitation temperature diminishes. A criterion was obtained which should be satisfied so that the excitation temperature may be in agreement with the electron temperature. The theoretical results of this paper are in good agreement with experimental results.     

Simplified calculations of a thermionic converter in the ignited mode

The charged particle transport in the plasma of a thermionic converter in the ignited mode is treated to be due to the charged particle density gradient and the electric field. The corresponding coefficients as well as the numerical treatment of the ionization and recombination processes are taken from the literature, the latter one in a suitable approximation. The electron temperature is assumed to be uniform within the gap. Taking account of the boundary conditions for the electron and ion currents and for the flux of the kinetic electron energy analytical solutions are found whose numerical evaluations can easily be performed. To make allowance for the Schottky effect and a double sheath at the emitter surface is shown to be necessary and possible with moderate effort for the calculation of I-V curves. The validity limit of the model is discussed.     

A study of density in electron-cyclotron-resonance plasma

A theory is developed for the density profile of low temperature plasmas confined by applied magnetic field and an experiment of the electron-cyclotron-resonance (ECR) plasma is conducted to compare the theoretical prediction and experimental measurements. Due to a large electron mobility along the magnetic field, electrons move quickly out of the system, leaving ions behind and building a space charge potential, which leads to the ambipolar diffusion of ions. In a steady-state condition, the plasma generation by ionization of neutral molecules is in balance with plasma loss due to the diffusion, leading to the electron temperature equation, which is expressed in terms of the plasma size, chamber pressure, and the ionization energy and cross section of neutrals. The power balance condition leads to the plasma density equation, which is also expressed in terms of the electron temperature, the input microwave power and the chamber pressure. It is shown that the plasma density increases, reaches its peak and decreases, as the chamber pressure increases from a small value (0.1 mTorr). These simple expressions of electron temperature and density provide a scaling law of ECR plasma in terms of system parameters. After carrying out an experimental observation, it is concluded that the theoretical predictions of the electron temperature and plasma density agree remarkably well with experimental data     

Untersuchungen der kontrahierten Edelgassäule bei mittleren Drücken. I Verfahren zur Ermittlung der Plasmaparameter und ihrer Radialverteilungen

Presented is a method to determine iteratively electron density, electron temperature, and gas temperature in medium pressure rare gas discharges. This method bases on the linear relationship between emission coefficient of continuum radiation and electron density according to electron-atom-bremsstrahlung. The radial profile of gas temperature is determined by solving the energy balance equation. The electron temperature is calculated from the reduced electric field strength and the degree of ionization. The received profiles and the values at axis permit to examine present theoretical conceptions and to estimate the influence of various elementary processes on discharge mechanism.     

Two-dimensional simulation of polysilicon etching with chlorine ina high density plasma reactor

A two-dimensional fluid simulation of polysilicon etching with chlorine in an inductively-coupled high density plasma source is presented. A modular approach was used to couple in a self-consistent manner the disparate time scales of plasma and neutral species transport. This way, complex plasma chemical reactions (involving electrons, ions and neutrals) as well as surface chemistry can be included in the simulation, The power deposited into the plasma was calculated by an electromagnetics module which solves Maxwell's equations. The power deposition was used in the electron energy module to find the electron temperature and the rate coefficients of electron-impact reactions. These were in turn used as source terms in separate neutral and charged species transport modules. By iterating among the modules, a self-consistent solution was obtained. Quantities of interest, such as power deposition, species density and flux, and etch rate and uniformity were thus calculated, As power deposition was increased, the electron density increased linearly, the plasma became less electronegative, the degree of gas dissociation increased, and the plasma potential remained constant. The radial uniformity of the Cl atom flux was better than that of the ion flux. The reactivity of the wafer as compared to that of the surrounding electrode surface significantly affected the etch uniformity, despite the low pressure of 10 mtorr     

不同电离度下大气等离子体粒子行为的数值模拟

利用一个空间零维大气等离子体模型对其中的主要粒子在不同电离度情况下的变化规律进行了研究.得到放电后不同初始电子密度下的电子寿命,同时给出了主要带电粒子和中性粒子密度随时间的演化.结果表明,电子密度随时间快速衰减,电子寿命随电离度的增大而减小.对一些重要的中性粒子(如O,N,O₃和NO)随电离度增大的行为进行了分析. 关键词： 电离度 大气等离子体 数值模拟     

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.