Rf Plasma Sheath Research Articles

Buckling of two-dimensional plasma crystals with nonreciprocal interactions.

Laboratory realizations of two-dimensional (2D) plasma crystals typically involve monodisperse microparticles confined into horizontal monolayers in radio-frequency (rf) plasma sheaths. This gives rise to the so-called plasma wakes beneath the microparticles. The presence of wakes renders the interactions in such systems nonreciprocal, a fact that can lead to a quite different behavior from the one expected for their reciprocal counterparts. Here we examine the buckling of a hexagonal 2D plasma crystal, occurring as the confinement strength is decreased, taking explicitly into account the nonreciprocity of the system via a well-established point-wake model. We observe that for a finite wake charge, the monolayer hexagonal crystal undergoes a transition first to a bilayer hexagonal structure, unrealizable in harmonically confined reciprocal Yukawa systems, and subsequently to a bilayer square structure. Our theoretical results are confirmed by molecular dynamics simulations for experimentally relevant parameters, indicating the potential of their observation in state-of-the-art experiments with 2D complex plasmas.

On the role of secondary electron emission in capacitively coupled radio‐frequency plasma sheath: A theoretical ground

AbstractWe propose a theoretical ground for emissive capacitively coupled radio‐frequency (rf) plasma sheath under low pressure. The rf sheath is assumed to be collisionless and oscillates with external source. A known sinusoidal voltage instead of current is taken as prerequisite to derive sheath dynamics. Kinetic studies are performed to determine mean wall potential as a function of secondary emission coefficient and applied voltage amplitude, with which the complete mean direct current sheath is resolved. Analytical analyses under homogeneous model and numerical analyses under inhomogeneous model are conducted to deduce real‐time sheath properties including space potential, sheath capacitance, and stochastic heating. Obtained results are validated by a continuum kinetic simulation without ionization. The influences of collisionality and ionization induced by secondary electrons are elucidated with a particle‐in‐cell simulation, which further formalizes proposed theories and inspires future works.

Influence of dust particles on the bulk electron density in radio frequency plasmas measured by microwave interferometry

The influence of dust particles, inserted in the rf plasma sheath of a capacitively coupled argon plasma, on the bulk electron density is investigated. The line integrated electron density has been measured using 160 GHz Gaussian beam microwave interferometry. A significant electron density increase compared to the dust free plasma was observed for high number densities of larger dust particles (d=12.3 μm). Furthermore, the rising electron density is combined with increasing optical plasma emission. For smaller dust particles (d=3.6 μm), no clear effect, but a tendency to a weak electron density reduction, was found. The results are compared to previous simulations of the impact ionization and excitation in dusty plasmas.

Wave mode coupling due to plasma wakes in two-dimensional plasma crystals: In-depth view

Experiments with two-dimensional (2D) plasma crystals are usually carried out in rf plasma sheaths, where the interparticle interactions are modified due to the presence of plasma wakes. The wake-mediated interactions result in the coupling between wave modes in 2D crystals, which can trigger the mode-coupling instability and cause melting. The theory predicts a number of distinct fingerprints to be observed upon the instability onset, such as the emergence of a new hybrid mode, a critical angular dependence, a mixed polarization, and distinct thresholds. In this paper we summarize these key features and provide their detailed discussion, analyze the critical dependence on experimental parameters, and highlight the outstanding issues.

Microparticles in a Collisional Rf Plasma Sheath under Hypergravity Conditions as Probes for the Electric Field Strength and the Particle Charge

We used microparticles under hypergravity conditions, induced by a centrifuge, in order to measure nonintrusively and spatially resolved the electric field strength as well as the particle charge in the collisional rf plasma sheath. The measured electric field strengths demonstrate good agreement with the literature, while the particle charge shows decreasing values towards the electrode. We demonstrate that it is indeed possible to measure these important quantities without changing or disturbing the plasma.

Energy Distribution of Ions Incident on Radio-Frequency Biased Electrodes in External Magnetic Field

A radio-frequency (rf) plasma sheath model in an oblique magnetic field is established and the energy distribution of ions (IED) incident on the rf sheath biased electrodes is numerically investigated. The simulation results reveal that the external magnetic field can have a decisive impact on the ion flux and energy distribution of the sheath. The ion energy can be transferred between the perpendicular and parallel components under the action of a magnetic field.

Effect of radio frequency discharge power on dusty plasma parameters

The parameters of a two-dimensional dusty plasma consisting of six, 9 μm diameter particles trapped inside a radio frequency (rf) plasma sheath have been measured as a function of rf power in a 13.5 mtorr (1.8 Pa) argon discharge. The center-of-mass and breathing frequencies are found by projecting the cluster’s Brownian motion onto the associated normal mode. The center-of-mass frequency (i.e., radial confinement) is insensitive to rf power. The Debye shielding parameter κ, as found from the breathing frequency, increases from ≈0.5 to 2 as the square root of rf power. The Debye length decreases from ≈2.7 to 0.7 mm as the inverse of the square root of rf power. The average particle charge q≈−17 000e is effectively independent of rf power. These results are consistent with an electron temperature that is independent of rf power and an ion density that is directly proportional to rf power, where the Debye length is determined by the ion density in combination with the electron temperature.

Dust particles in collisionless plasma sheath with arbitrary electron energy distribution function

Dust particles often appear in industrial plasmas as undesirable product of the plasma-wall interactions. Large particles of several micrometers in diameter are concentrated in a thin layer (the sheath) above the lower electrode of the rf driven parallel plate device, where the electric force is strong enough to compensate particle’s gravity. Experimental and theoretical uncertainties are significantly increased in the plasma sheath. Common models of dust charging in the plasma sheath suppose the Maxwellian electron energy distribution function (EEDF) in conjunction with a flux of cold ions satisfying classical Bohm criterion at the sheath edge. In this paper we generalize this model to arbitrary EEDF with adapted Bohm criterion. We limit our considerations to collisionless or slightly collisional plasma, where the EEDF inside the sheath is expressed through the EEDF in the plasma bulk. Derived theoretical formulas are incorporated into numerical model, describing collisionless radio frequency (rf) plasma sheath together with the electrical charge, various kinds of forces, balancing radius and oscillation frequency of particles.

Electrical characterization of a capacitive rf plasma sheath

The authors report on an experimental system designed to investigate and characterize capacitive radio frequency (rf) sheaths. An electrode mounted in an inductive plasma reactor and driven with separate rf and direct current (dc) power sources is used. The advantage of this design is that the electrode sheath is decoupled from the plasma parameters. This allows detailed investigation of the sheath with different bias conditions without perturbing the bulk plasma parameters. Power coupled to ions and electrons through the sheath, at low pressure, is investigated and a method to determine the electron conduction current to the electrode, using the external dc bias, is presented.

A new diagnostic to characterize a plasma crystal

We describe experiments aimed at characterizing complex plasma crystals and their interaction with the space charge surrounding an RF discharge. By analysing the complex plasma radiofrequency impedance, we were able to measure the density of the free electrons of the crystal. The density of the ions was derived by Langmuir probe measurements in the bulk plasma combined with a model of the RF plasma sheath. Surprisingly the full characterization of the crystal reveals that the 3D ensemble is quasi-neutral, i.e. a three component plasma and the particles levitate in their self-built electric field, much lower than the sheath field. This result is inherent to the 3D geometry.

RF plasma sheath in an oblique magnetic field

A sheath model in an oblique magnetic field is build, which has one-dimension coordinate space and three-dimension speed space. The effects of magnetic field on the structure of RF sheath and parameter characteristics are discussed. The numerical simulation result shows that magnetic field has great effects on the sheath structure, especially in the region near the edge of sheath. Also, the magnetic field affects ion energy in the direction vertically to the board and incidence departure angle.

Two-dimensional radio-frequency sheath dynamics over a nonflat electrode with perpendicular magnetic field

A self-consistent fluid model is developed to simulate the radio-frequency (rf) plasma sheath dynamics over a nonflat electrode, with a magnetic field applied perpendicular to it. The model consists of the two-dimensional (2D) time-dependent fluid equations, coupled with the Poisson equation, and it uses an equivalent-circuit model to self-consistently determine the relationship between the instantaneous voltage at the rf-biased electrode and the sheath thickness. In addition to the usual plasma molding effects, different properties of the azimuthal ion flow associated with the E×B drift are observed and studied under varying discharge pressures and the magnetic field intensities. It is found that the azimuthal flow exhibits rather nonuniform distribution with a peak around the edge of a hole in the electrode. In addition, when the discharge pressure increases, the velocity of the azimuthal ion flow, as well as the velocities of ion flow in all other directions, are found to decrease, whereas the sheath edge is found to move closer to the electrode. While the variation of the magnetic field is found to have no significant effect on the sheath structure, the azimuthal ion flow velocity is found to increase in proportion to the magnetic field.

Charge of a macroscopic particle in a plasma sheath

Charging of a macroscopic body levitating in a rf plasma sheath is studied experimentally and theoretically. The nonlinear charge vs size dependence is obtained. The observed nonlinearity is explained on the basis of an approach taking into account different plasma conditions for the levitation positions of different particles. The importance of suprathermal electrons' contribution to the charging process is demonstrated.

Radiofrequency sustained double layer in a plasma reactor

This article presents experimental evidence of a coupling of power between the rf supply and a collisionless plasma in a capacitively coupled plasma reactor. The reactor is driven in plasma sheath resonance to let high rf currents circulate in the system. These currents allow the formation of strong rf fields inside the nonuniform plasma leading to a double layer with a mechanism similar to the rf plasma sheath at the electrodes. Electrons accelerated by the double layer are very effective in ionizing, because of the increased cross section, and create a dense plasma at low pressures. The power is consequently transferred by the ions accelerated in the electrode sheath to the substrate to be processed with high improvement in both performance and process rate. Diagnostics include a retarding field analyzer on the grounded electrode to measure the energy spectrum of ions hitting the substrate and a capacitive probe to obtain the rf potential variations in the plasma.

Wake effects on vertical alignment of two dust particles in a rf plasma sheath

A theoretical method is proposed to study the motion of two dust particles in a radio-frequency plasma sheath with flowing ions. It is shown that, owing to the asymmetric wake force, the particles move in a coordinated way, remaining aligned in a direction perpendicular to an electrode, when a modulating laser beam hits the upper particle. These results imply that the wake effects may be responsible for formation of the vertically aligned Coulomb crystal structure.

Coaxial mode excitation and dissipation in ion Bernstein wave experiments

In recent ion Bernstein wave (IBW) heating experiments on the Tokamak Fusion Test Reactor (TFTR) [J. R. Wilson. R. E. Bell, S. Bernabei, K. Hill et al., Phys. Plasmas 5, 1721 (1998)] a velocity shear layer in the plasma core was obtained. The magnitude of velocity shear was believed to be too small to create an internal transport barrier, because of parasitic edge processes which reduced the power coupled to the core. In this paper we investigate these rf (radio frequency) edge processes by employing a model which includes both coaxial modes and their dissipation in rf plasma sheaths. The coaxial mode (here, an electron plasma wave trapped in the halo plasma between the lower hybrid layer and the vessel wall) can propagate at low poloidal wave numbers. This feature is shown to relate to the observed poloidal phasing dependence of the antenna loading. Results of analytical models and a three-dimensional antenna code are presented. The experimentally observed loading is also nonlinear, being larger at very low powers. This feature is explored using an rf sheath dissipation model. Loading into the coaxial mode is expected to maximize when the density gradient at the lower hybrid layer is steep, preventing efficient mode transformation to the IBW. The role of ponderomotive force in modifying the density profile is also discussed.

Dynamics of collisionless rf plasma sheaths

The behavior of rf plasma sheaths has been the subject of much scientific study and also is technologically important for plasma etching and deposition in the manufacture of integrated circuits. This paper presents a semianalytic model of rf sheaths and describes an experiment that tested the model. An approximation to the first integral of the Poisson equation allows solving for the response of plasma sheaths to an imposed rf bias voltage. This approximation enables the plasma sheaths to be included within an electrical model of the plasma and external rf circuit components, and affords a prediction of the ion energy distributions impacting the electrodes, which are in contact with the plasma. The model is a significant advance beyond previous sheath models because it has no restriction on the ratio of the rf period to the ion transit time across the sheath. The model is applicable to those high-density, low-pressure plasmas in which the Debye length is a small fraction of the ion mean-free path, which itself is a small fraction of the plasma dimension. The experimental test of the model was conducted by comparing the predicted and measured rf potential, current, and power at the sheath adjacent to a capacitively coupled, rf-biased electrode in a plasma reactor with argon discharges sustained by an inductively coupled plasma source. The comparisons included both linear and nonlinear components of the rf electrical parameters. Results of the experiment were in substantial agreement with model predictions.

Subharmonics and rf-plasma sheaths.

Period doubling and concomitant subharmonic generation have been reported in rf-excited plasmas. We show that this phenomenon is caused by the nonlinear interaction of the plasma-sheath capacitance with the external rf power circuits. As a result, changes in the external circuits can be used to control period doubling. Sheath capacitance can be derived from electrical measurements of period doubling.

Particle simulation of the magnetized rf plasma sheath

In the sheath regions strong electric fields can develop that accelerate ions to sufficient energy to cause sputtering and increase the plasma impurities to an intolerable level. Simulations of these sheaths are performed with a one-and-two-half-dimensional (y,vx,vy,vz) electrostatic particle model with a homogeneous magnetic field tilted in relation to y; guiding-center electrons and full dynamic ions are used. The plasma is taken to be bounded by metallic walls in which a time-dependent potential difference V(t) is applied and the average energies and flux of ions hitting the walls are computed. The results show that the magnetic fields inhibits the sputtering caused by D atoms. Comparisons of the present model with previous work, which used a Vlasov model and fluid approach, are also made with good agreement in the appropriate regime.

Analysis of a kinematic model for ion transport in rf plasma sheaths.

An idealized model for ion transport across an oscillating plasma sheath is analyzed to obtain insight into qualitative features of the ion energy distributions observed in low-pressure rf discharges. The sheath is characterized by a constant electric field over an extent that varies sinusoidally with time, and ions incident on it correspond to a monoenergetic flux independent of phase \ensuremath{\varphi} in the rf cycle. The dimensionless parameters \ensuremath{\alpha}=${\mathit{qV}}_{\mathit{s}}$/m${\mathrm{\ensuremath{\omega}}}^{2}$${\mathit{d}}^{2}$ and \ensuremath{\beta}=${\mathit{v}}_{0}$/\ensuremath{\omega}d (where d and ${\mathit{V}}_{\mathit{s}}$ are the mean sheath thickness and potential drop, \ensuremath{\omega} is the excitation frequency, and ${\mathit{v}}_{0}$ and q/m are the incoming ion speed and charge-to-mass ratio) govern the ion trajectories, which are found to divide into groups, delimited by two ``critical'' values of \ensuremath{\varphi}, that undergo N and N+1 encounters with the field. The first critical phase depends only weakly on \ensuremath{\beta}, whereas the second is sensitive to both \ensuremath{\alpha} and \ensuremath{\beta} and cycles continuously as these parameters diminish. Correspondingly, within the ``transition regime'' where \ensuremath{\alpha} and \ensuremath{\beta} are neither very small nor greater than (or comparable to) unity, the precise form of the incident-ion energy spectrum exhibits rapid variations, superposed on a systematic narrowing, as the frequency \ensuremath{\omega} is increased.