A. J. Lichtenberg

Stochastic heating in single and dual frequency capacitive discharges

Two electron heating mechanisms in capacitive discharges are ohmic heating due to electron-neutral collisions and stochastic heating at the plasma edge due to momentum transfer from high voltage moving sheaths. In this work, the stochastic heating and its dependence on various parameters are determined, focusing on dual frequency discharges in which the sheath motion is driven by a combination of high and low frequency sources. Particle-in-cell (PIC) simulations are used in order to investigate the electron heating. For a uniform fixed-ion discharge in which the ions are held fixed in a uniform density profile, there is no stochastic heating, as expected. For a two-step fixed-ion discharge in which the ions are held fixed in a two-step density profile with bulk density nb and sheath density nsh<nb, the stochastic heating is nearly proportional to (1−nsh∕nb)2. For a self-consistent discharge with mobile ions, the stochastic heating is well described by a “hard wall model” provided that the bulk oscillation...

Fermi acceleration revisited

Abstract Mappings that have been used to describe the Fermi acceleration mechanism are examined. It is shown that results which appear to be contradictory are due to differences in the mapping equations. For those mappings that can be locally approximated by the standard mapping, the value of the nonlinear parameter of the standard mapping, for which the last isolating KAM surface exists, can be used to predict the loss of KAM stability with action for the more general mappings. Previous results of the variation in the density distribution in the stochastic region of the phase space, averaged over phases, is shown to be consistent with the ergodic hypothesis. Fine scale structure of the mappings is found to be model dependent. The standard mapping is a member of a class of mappings which retains some KAM trajectories at arbitrarily large nonlinearity. This feature is not generic to a wider class of mappings discussed in this paper. The stability of two-iteration fixed points are discussed in detail, including the bifurcation sequence for one type of mapping.

Modelling plasma discharges at high electronegativity

Macroscopic models for the equilibrium of a three-component electronegative gas discharge are developed. Assuming the electrons and the negative ions to be in Boltzmann equilibrium, a positive ion ambipolar diffusion equation is derived. Such a discharge can consist of an electronegative core and may have electropositive edge regions, but the electropositive regions become small for the highly electronegative plasma considered here. In the parameter range for which the negative ions are Boltzmann, the electron density in the core is nearly uniform, allowing the nonlinear diffusion equation to be solved in terms of elliptic integrals. If the loss of positive ions to the walls dominates the recombination loss, a simpler parabolic solution can be obtained. If recombination loss dominates the loss to the walls, the assumption that the negative ions are in Boltzmann equilibrium is not justified, requiring coupled differential equations for positive and negative ions. Three parameter ranges are distinguished corresponding to a range in which a parabolic approximation is appropriate, a range for which the recombination significantly modifies the ion profiles, but the electron profile is essentially flat, and a range where the electron density variation influences the solution. The more complete solution of the coupled ion equations with the electrons in Boltzmann equilibrium, but not at constant density, is numerically obtained and compared with the more approximate solutions. The theoretical considerations are illustrated using a plane parallel discharge with chlorine feedstock gas of p = 30, 300 and 2000 mTorr and , corresponding to the three parameter regimes. A heuristic model is constructed which gives reasonably accurate values of the plasma parameters in regimes for which the parabolic profile is not adequate.

Theory of electron cyclotron resonance heating. II. Long time and stochastic effects

For pt. I see abstr. A4666 of 1973. The theory of single particle electron cyclotron resonance heating in a magnetic mirror is treated analytically and numerically, from the viewpoint of (a) an impulsive heating approximation and (b) a stochastic approximation, using a Fokker-Planck equation. Using (a), numerical calculations of particle heating are performed for 105 half-bounce times tau b. Numerically and analytically from (a), for a given r.f. field strength, are obtained two limiting energies Ws and Wb, with Wb approximately=5Ws. For the transverse particle energy at resonance Wperpendicular to R Wb, invariant curves exist which form a barrier to further particle heating. For a parabolic mirror B(z)=B0 (1+z2/L2) with cyclotron resonance at z=+or-l, Wb=2.88eEL(1+l2/L2)12/( tau b/ tau s)23/, where e is the electronic charge, E the r.f. electric field, and tau s is the period for the cyclotron phase to slip 2 pi with respect to the r.f. field.

Modeling Electronegative Plasma Discharges

A macroscopic analytic model for a three-component electronegative plasma has been developed. Assuming the negative ions to be in Boltzmann equilibrium, a positive ion ambipolar diffusion equation is found. The electron density is nearly uniform, allowing a parabolic approximation to the plasma profile to be employed. The resulting equilibrium equations are solved analytically and matched to an electropositive edge plasma. The solutions are compared to a simulation of a parallel-plane r.f. driven oxygen plasma for two cases: (1) p=50 mTorr, neo = 2.4x10 (15) m-3, and (2) 10 mTorr, new = 1.0x10 (16) m-3. In the simulation, for the low power case (1), the ratio of negative ion to electron density was found to be alpha sub 0 is almost equal to 8, while in the higher power case alpha sub 0 is almost equal to 1.3. Using an electorn energy distribution that approximates the simulation distribution by a two-temperature Maxwellian, the analytic values of alpha sub zero are found to be close to, but somewhat larger, than the simulation values. The average electron temperature found self-consistently in the model is close to that in the simulation. The results indicate the need for determining a two-temperature electron distribution self-consistently within the model.

Instabilities in low-pressure electronegative inductive discharges

Plasma instabilities have been studied in low-pressure inductive processing discharges with SF6 and Ar/SF6 mixtures, i.e. attaching gases. Oscillations are seen in charged particle density, electron temperature and plasma potential using electrostatic probe and optical emission measurements. For SF6, instability onset in pressure and driving power has been explored for gas pressures between 2.5 and 100 mTorr and absorbed powers between 150 and 900 W. For pressures above 20 mTorr, increasing power is required to obtain the instability with increasing pressure, with the frequency of the instability increasing with pressure, mainly lying between 1 and 100 kHz. For Ar/SF6 mixtures, we observe a similar low power transition, with an upper transition to a stable inductive mode. The instability windows become smaller as the argon partial pressure increases. For Ar/SF6 mixtures, we observe a significant effect of the matching network. We improve a previously developed volume-averaged (global) model to describe the instability. We consider a cylindrical discharge containing time varying electrons, positive ions, negative ions, and time invariant excited states. The driving power is applied to the discharge through a conventional L-type capacitive matching network, and we use realistic models for the inductive and capacitive energy deposition. The particle and energy balance equations are integrated, considering quasi-neutrality in the plasma volume and charge balance at the walls, to produce the dynamical behaviour. As pressure or power is varied to cross a threshold, the instability is born at a Hopf bifurcation, with relaxation oscillations between higher and lower density states. The model qualitatively agrees with experimental observations, and also shows a significant influence of the matching network.

Improved volume-averaged model for steady and pulsed-power electronegative discharges

An improved volume-averaged global model is developed for a cylindrical (radius R, length L) electronegative (EN) plasma that is applicable over a wide range of electron densities, electronegativities, and pressures. It is applied to steady and pulsed-power oxygen discharges. The model incorporates effective volume and surface loss factors for positive ions, negative ions, and electrons combining three electronegative discharge regimes: a two-region regime with a parabolic EN core surrounded by an electropositive edge, a one-region parabolic EN plasma, and a one-region flat-topped EN plasma, spanning the plasma parameters and gas pressures of interest for low pressure processing (below a few hundred millitorr). Pressure-dependent effective volume and surface loss factors are also used for the neutral species. A set of reaction rate coefficients, updated from previous model calculations, is developed for oxygen for the species O2, O2(Δg1), O, O2+, O+, and O−, based on the latest published cross-section set...

Macroscopic modeling of radio‐frequency plasma discharges

We describe a self‐consistent model of a symmetric plane parallel rf discharge. The model is built upon basic laws such as conservation of particles and energy, and is capable of predicting rapidly the important discharge parameters from a processing point of view, such as the ion energy and flux to the electrodes. The following physics is incorporated into the model: energy‐dependent electron–neutral ionization, excitation and elastic scattering; nonuniform, self‐consistent collisionless and collisional rf sheaths; electron Ohmic heating by elastic scattering in the sheaths and bulk plasma stochastic heating by the oscillating fields in the sheaths; electron energy losses to neutrals through collisions and to the electrodes; ambipolar ion diffusion; and total rf power balance. A set of equations describing this dynamics has been obtained and used in a code to simulate different discharges. The model has proven to be useful in comparing the effect of varying parameters on the discharge. Comparisons with e...

Self-consistent stochastic electron heating in radio frequency discharges

Fermi acceleration is considered as an underlying mechanism for electron heating in rf discharges, in which the heating arises from the reflection of electrons from moving sheaths. By examining the dynamics of the electron collisions with the sheaths, the map that describes the electron motion is derived. For high‐frequency discharges (ω/2π>50 MHz), the electron motion is shown to be stochastic. By combining these dynamics with collisional effects in the bulk plasma and incorporating self‐consistent physical constraints, a self‐consistent model of the discharge is developed. The model is used to calculate physically interesting quantities, such as the electron temperature and average lifetime, and to predict the minimum pressure necessary to sustain the plasma. The distribution of electron energies is shown to be non‐Maxwellian. These results can be applied to experimentally interesting parallel‐plate rf plasma discharges to predict the operating conditions necessary for stochastic heating to occur.

Theory of electron cyclotron resonance heating. I. Short time and adiabatic effects

The theory of single particle electron cyclotron resonance heating in a magnetic mirror is treated analytically and numerically, using the techniques of (a) integration of the Lorentz force equation and (b) transformation to a Hamiltonian approximation, to study both short time scale and adiabatic effects. The force equation is analytically integrated in the vicinity of the resonance plane to obtain the energy dependence of the effective time spent in resonance per bounce te. For electrons passing through the resonant zone at constant parallel velocity vzR, te varies as vzR-12/. For electrons which turn in or near the resonant zone, te varies as vperpendicular to R, P=2/3, where vperpendicular to R is the transverse velocity at resonance. These results agree with the exact numerical integration of the force equation, for which P approximately=0.5-0.7.