Plasma Sheath Edge Research Articles

Characterization of a radially emitting spray jet in a cylindrical inertial electrostatic confinement plasma source based on particle in cell simulation

This paper investigates the outflow mechanism of a cylindrical inertial electrostatic confinement plasma source using Particle in Cell plasma simulations. Various phenomena such as ion-wall interactions, impact ionization, excitation, and secondary electron generation are considered. Together with a fine time and cell resolution to reveal relevant phenomena. The results indicate a high electron and ion density in front of the cathode at the widest grid opening, which suggests the formation of a cascade-like charge carrier emission, commonly referred to as the “spray jet”. Initially, a virtual anode is formed within the cathode grid and is shifted toward the grid opening as time elapsed. Ions are accelerated in the plasma sheath around the cathode and collide with the cathode grids or oscillate through the grid opening. Electrons are accelerated and ionize the background gas at the plasma sheath edge. While the electron flow is disordered within the cathode it forms a cascade-like quasi-neutral plasma at the outlet of the plasma source. Through the formation of a virtual anode the discharge becomes self-sustained. With this simulation results the working principle of the inertial electrostatic confinement plasma source can be derived.

Dust particle charging process in a radio frequency sheath of plasma containing kappa distribution electrons

Based on self-consistent modelling of the radio-frequency sheath parameters, such as the ion and electron densities and the ion velocity, the dust particle charging process in an RF sheath is investigated by employing the kappa$(\kappa )$distribution for the electrons. It is shown that the charge number of the dust particle decreases near the sheath–wall interface while it shows the opposite tendency near the plasma-sheath edge, as the$\kappa$value is decreased. The fluctuation of the dust particle charge modified by the$\kappa$value depends on the dust particle radius. With an increases in the$\kappa$value, the fluctuation of the dust particle charge has a slight increase for a small dust particle, and it shows a significant increase for a large dust particle. In addition, as the$\kappa$value is decreased, the charge number of the dust particle obtained from the time-averaged plasma parameters deviates from the results obtained from the instantaneous plasma parameters. Moreover, a smaller deviation can be found for a large dust particle under the same$\kappa$value conditions.

Experimental investigation on optimal plasma generation in inductively coupled plasma

Total energy loss per ion–electron pair lost (εT) is investigated to optimize the plasma generation at various RF powers and gas pressures in an argon inductively coupled plasma (ICP). The ion densities and electron temperatures are measured to obtain εT at the plasma–sheath edge. At a fixed RF power, the obtained εT has a minimum at a certain electron temperature, and at this condition, an optimal plasma generation is achieved according to a global model. Since the electron temperature is a function of the gas pressure, at that certain gas pressure the energy loss in the plasma is minimized and plasma is generated most efficiently. Interestingly, the electron temperature at which εT becomes the minimum decreases as the RF power increases. This is explained by multistep ionization and the electron density dependence of the density of the excited states. Measured εT is compared with the calculated result from the global model that includes multistep ionization, and these are consistent with each other.

The effect of energetic electrons on the collisionless fusion plasma sheath in ion cyclotron range of frequencies heating

A one-dimensional model, consisting of a hydrodynamic radio-frequency (RF) sheath model for the ion cyclotron range of frequencies heating and an equivalent circuit model, is used to study the structure of the collisionless RF sheath of a fusion plasma containing a beam of energetic electrons. For various energetic electron concentrations and velocities at the plasma-sheath edge, a set of equations describing the model are solved numerically to obtain the potential drop across the RF sheath and the sheath thickness, as well as the spatiotemporal variations of the potential, the ion density, and the background electron and energetic electron densities inside the sheath. Under the current EAST ion cyclotron range of heating conditions, it is observed that even at small beam fluxes, the potential drop across the sheath is enhanced at any time in an RF cycle for different bulk plasma densities, ion temperatures, and frequencies and amplitudes of the disturbance current. When the energetic electron component is included, the physical sputtering yields of the RF sheath wall materials such as titanium and iron become significant as a result of the enhancement of the drop in the sheath potential.

Introduction to the theory and application of a unified Bohm criterion for arbitrary-ion-temperature collision-free plasmas with finite Debye lengths

At present, identifying and characterizing the common plasma–sheath edge (PSE) in the conventional fluid approach leads to intrinsic oversimplifications, while the kinetic one results in unusable over-generalizations. In addition, none of these approaches can be justified in realistic plasmas, i.e., those which are characterized by non-negligible Debye lengths and a well-defined non-negligible ion temperature. In an attempt to resolve this problem, we propose a new formulation of the Bohm criterion [D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields (McGraw-Hill, New York, 1949)], which is here expressed in terms of fluid, kinetic, and electrostatic-pressure contributions. This “unified” Bohm criterion consists of a set of two equations for calculating the ion directional energy (i.e., the mean directional velocity) and the plasma potential at the common PSE, and is valid for arbitrary ion-to-electron temperature ratios. It turns out to be exact at any point of the quasi-neutral plasma provided that the ion differential polytropic coefficient function (DPCF) of Kuhn et al. [Phys. Plasmas 13, 013503 (2006)] is employed, with the advantage that the DPCF is an easily measurable fluid quantity. Moreover, our unified Bohm criterion holds in plasmas with finite Debye lengths, for which the famous kinetic criterion formulated by Harrison and Thompson [Proc. Phys. Soc. 74, 145 (1959)] fails. Unlike the kinetic criterion in the case of negligible Debye length, the kinetic contribution to the unified Bohm criterion, arising due to the presence of negative and zero velocities in the ion velocity distribution function, can be calculated separately from the fluid term. This kinetic contribution disappears identically at the PSE, yielding strict equality of the ion directional velocity there and the ion sound speed, provided that the latter is formulated in terms of the present definition of DPCFs. The numerical values of these velocities are found for the Tonks–Langmuir collision-free, plane-parallel discharge model [Phys. Rev. 34, 876 (1929)], however, with the ion-source temperature extended here from the original (zero) value to arbitrary high ones. In addition, it turns out, that the charge-density derivative (in the potential “space”) with respect to the potential exhibits two characteristic points, i.e., potentials, namely the points of inflection and maximum of that derivative (in the potential space), which stay “fixed” at their respective potentials independent of the Debye length until it is kept fairly small. Plasma quasi-neutrality appears well satisfied up to the first characteristic point/potential, so we identify that one as the plasma edge (PE). Adopting the convention that the sheath is a region characterized by considerable electrostatic pressure (energy density), we identify the second characteristic point/potential as the sheath edge (SE). Between these points, the charge density increases from zero to a finite value. Thus, the interval between the PE and SE, with the “fixed” width (in the potential “space”) of about one third of the electron temperature, will be named the plasma–sheath transition (PST). Outside the PST, the electrostatic-pressure term and its derivatives turn out to be nearly identical with each other, independent of the particular values of the ion temperature and Debye length. In contrast, an increase in Debye lengths from zero to finite values causes the location of the sonic point/potential (laying inside the PST) to shift from the PE (for vanishing Debye length) towards the SE, while at the same time, the absolute value of the corresponding ion-sound velocity slightly decreases. These shifts turn out to be manageable with employing the mathematical concept of the plasma-to-sheath transition (different from, but related to our natural PST concept), resulting in approximate, but sufficiently reliable semi-analytic expressions, which are functions of the ion temperature and Debye length.

Magnetized electronegative plasma presheath-sheath containing two positive ion species

The structure of the presheath-sheath for a magnetized electronegative discharge in the presence of two species of positive ions is investigated. In the absence of collision, for different magnetic field strength, the densities of the charges and the profiles of the positive ions velocities and electric potentials are obtained. It is shown that a slightly magnetized uniform structure discharge converts to a stratified multiple layer oscillatory structure for an intermediate magnetic field. But, in contrast to a single positive ion species up to high values of magnetic field, oscillatory behavior continues to exist. We also observe oscillations in the profile of electron densities which are hidden in the single positive ion species case. A competition between the two positive ions species exists which depends on their different masses. The density profiles of the species follow each other along the sheath and the heavier component always occupies higher density values. The influence of the elastic collisions between positive ions and neutrals on the structure of a magnetized electronegative discharge is also investigated. Finally, the positive ion flux at the plasma-sheath edge for each species is obtained as a function of magnetic field strength, for different values of the collision frequency.

Plasma polymerization at different positions in an asymmetric ethylene discharge

The characteristics of plasma polymerization are investigated in an asymmetric, capacitively coupled plasma discharge. Here, the deposition in different plasma zones, i.e. on the driven electrode, within the plasma bulk and the plasma sheath as well as approximately at the plasma-sheath edge, is investigated. Principal expectations are perfectly met, though new interesting dependences of the obtained a-C : H coatings with respect to film properties and deposition rates are also found. That is, the deposition rates as measured on thin, small glass slides placed directly on the electrode are considerably higher than everywhere else in the plasma, yet only single-sided. In contrast, the deposition rates on the samples within the plasma are lowered depending on the exact placement, while a double-sided coating is obtained. Furthermore, film properties, such as the film density, are highly dependent on the sample placement in the plasma, which can even be higher under floating conditions. With simple physical arguments we are able to show the relations between the deposition rate and the energy input into the plasma as well as between the energy density during film growth and the film density itself.

Potential profile near singularity point in kinetic Tonks-Langmuir discharges as a function of the ion sources temperature

A plasma–sheath transition analysis requires a reliable mathematical expression for the plasma potential profile Φ(x) near the sheath edge xs in the limit ɛ≡λD/ℓ=0 (where λD is the Debye length and ℓ is a proper characteristic length of the discharge). Such expressions have been explicitly calculated for the fluid model and the singular (cold ion source) kinetic model, where exact analytic solutions for plasma equation (ɛ=0) are known, but not for the regular (warm ion source) kinetic model, where no analytic solution of the plasma equation has ever been obtained. For the latter case, Riemann [J. Phys. D: Appl. Phys. 24, 493 (1991)] only predicted a general formula assuming relatively high ion-source temperatures, i.e., much higher than the plasma-sheath potential drop. Riemann’s formula, however, according to him, never was confirmed in explicit solutions of particular models (e.g., that of Bissell and Johnson [Phys. Fluids 30, 779 (1987)] and Scheuer and Emmert [Phys. Fluids 31, 3645 (1988)]) since “the accuracy of the classical solutions is not sufficient to analyze the sheath vicinity” [Riemann, in Proceedings of the 62nd Annual Gaseous Electronic Conference, APS Meeting Abstracts, Vol. 54 (APS, 2009)]. Therefore, for many years, there has been a need for explicit calculation that might confirm the Riemann’s general formula regarding the potential profile at the sheath edge in the cases of regular very warm ion sources. Fortunately, now we are able to achieve a very high accuracy of results [see, e.g., Kos et al., Phys. Plasmas 16, 093503 (2009)]. We perform this task by using both the analytic and the numerical method with explicit Maxwellian and “water-bag” ion source velocity distributions. We find the potential profile near the plasma–sheath edge in the whole range of ion source temperatures of general interest to plasma physics, from zero to “practical infinity.” While within limits of “very low” and “relatively high” ion source temperatures, the potential is proportional to the space coordinate powered by rational numbers α=1/2 and α=2/3, with medium ion source temperatures. We found α between these values being a non-rational number strongly dependent on the ion source temperature. The range of the non-rational power-law turns out to be a very narrow one, at the expense of the extension of α=2/3 region towards unexpectedly low ion source temperatures.

Existence of subsonic plasma sheaths

The location of the plasma sheath edge, where quasineutrality is broken, is rigorously derived by using a kinetic description of the plasma. It is shown that sheaths can exist with arbitrarily small ion velocity at the sheath edge, thus violating the Bohm criterion, V(i)=c(s) at the sheath edge. Bohm's criterion is recovered in the case of large enough ion current through the wall, and it is found to be a reasonable approximation in floating potential conditions. However, in the case of a predominant electron current through the wall, Bohm's criterion is not able to describe the sheath-edge transition. The analytical results are supported by numerical simulations performed with a fully kinetic particle-in-cell code modeling a source-driven, weakly collisional plasma, bound between two absorbing walls.

Collisional effects in magnetized plasma sheath with two species of positive ions

The effects of ion–neutral collision on the characteristics of a magnetized plasma sheath which consists of two species of positive ions are investigated. It is assumed that the ions have different masses. In contrast to our previous work, the effects of ion–neutral collision frequency are added to the magnetized plasma sheath. Using a three-fluid hydrodynamic model and some dimensionless variables, the dimensionless equations are obtained and solved numerically. By taking into account the ion–neutral collision effects on a magnetized three-component plasma sheath, it is shown that apart from the presence of the second ion species, by increasing the collision frequency of two ions with neutrals, the amplitude of fluctuations of ion species density distributions increases and the position of these fluctuations is shifted towards the plasma sheath edge. Also, by increasing the ion–neutral collision frequency these fluctuations turn off faster than those in a collisionless case. It is shown that in the collisional magnetized plasma sheath, the effects of the presence of the heavier ion species on the lighter ion density turn off much faster in comparison with what happens in a collisionless magnetized plasma sheath. Furthermore, it is found that in a collisionless plasma sheath by increasing the density of the heavier ion species, the normalized electrostatic potential decreases while in a collisional plasma sheath the presence of the heavier ion species does not have any considerable effect on the normalized electrostatic potential. In addition, it is shown that when the distance of each ion species from the plasma sheath boundary becomes larger than five times the electron Debye length (x > 5λDe) the fluctuations of the ion species velocities disappear by increasing the ion–neutral collision frequency. Also, it is found that the electron density distribution decreases by increasing the ion–neutral collision frequency.

Edge-to-center plasma density ratio in high density plasma sources

The flux of positive ions leaving a classical low-temperature plasma discharge is proportional to the plasma density at the plasma–sheath edge, and the edge-to-center plasma density ratio, the so-called hl factor, normally depends only on the discharge size and the neutral gas pressure. The ion flux leaving the discharge is therefore linearly proportional to the central plasma density. The hl factor has been previously derived by solving the plasma transport equations over a large pressure range, with the assumption of constant neutral gas density within the discharge. Tonks and Langmuir derived the low pressure (collisionless) solution of this problem in 1929. More recent works have shown that the neutral gas density is no longer constant when the plasma pressure becomes comparable to the neutral gas pressure. In this paper, we solve the plasma transport equations in this new situation and we propose a new expression for the hl factor. It is shown that hl becomes a function of the central plasma density which implies that the ion flux leaving the discharge is no longer proportional to this density. This effect has to be included in particle and energy balance equations used in global models of high density plasma sources.

Probe measurements in a nonstationary plasma

Single and double probes are simple and common tools for plasma measurements. In the case of nonstationary plasmas, the values of the plasma density obtained with these tools may differ significantly from the correct values measured, e.g., by microwave methods. The reason for such discrepancy could be the Bohm criterion failure during the plasma transition to the steady state. Indeed, the Bohm criterion, which is commonly used as a boundary condition at the plasma-sheath edge, directly determines the ion saturation current to the probe surface. The transition-time duration is studied and explained quantitatively for various plasmas produced by a version of a ferroinductor-coupled plasma source, which has its magnetic core fully immersed in the plasma. Corresponding conversion factors for probe measurements have been evaluated. Also, the influence of a certain amount of “hot” non-Maxwellian electrons on probe characteristics has been investigated.

Effect of rise-time patterns on dynamics of sheath expansion during plasma immersion ion implantation

Plasma immersion ion implantation (PIII) has been developed as a low_cost and efficient surface modification technique of irregularly-shaped objects. The effect of six pulse waves with different rise_time patterns on the spatio-temporal evolution of plasma sheath, energy and dose of ion implantation has been simulated by particle-in-cell modeling. Statistical results may be obtained through assuming the Boltzmann distribution of electrons, and solving Poisson and Newton equations for tracing each ion in the plasma sheath. The results show that rise_time pattern has a critical influence on the evolution of plasma sheath. There exists maximum thickness difference of plasma sheath for different waveforms. The acceleration of ions is non_uniform due to the non-uniformity of electrical field strength. The maximum gradient of electrical field appears near the edge of plasma sheath. The results also show that optimization of dose and energy of incident ions may be achieved through modification of rise_time pattern. The numerical simulation of sheath expansion can be effectively used to provide a scientific basis for optimizing the PIII process.

Three-dimensional effects for two-dimensional samples in plasma immersion ion implantation

Plasma immersion ion implantation is a fast method for ion implantation into complex-shaped three-dimensional objects as the ions are accelerated from the receding plasma sheath edge towards the substrate during the negative high-voltage pulses. Except for the region near the edges, homogeneous implantation can be obtained for the surface of flat samples. Here, it is shown that a much larger absolute dose and a much larger dose variation is observed on the back side of rectangular stainless steel samples—resulting in highly localized corrosion—due to the interference of the supporting rod connected to the high voltage-feed through.

Global Existence of Plasma Ion-Sheaths and Their Dynamics

When a negative high voltage pulse is applied to a target material immersed in a plasma, a boundary layer (sheath) forms around the target and the motion of cold ions in the plasma is governed by the Euler-Poisson system. In this paper, by simplifying the Euler-Poisson system on suitable physical regimes, we present a theory for the existence and dynamics of time-dependent sheaths with planar, cylindrical and spherical symmetry. For the construction of ion-sheaths, we employ the method of characteristics and study the dynamic behavior of a plasma-sheath edge based on ODEs which are formally derived from the Euler-Poisson system.

Bohm criterion failure in nonstationary plasmas

The Bohm criterion, generally used as a boundary condition at the plasma-sheath edge, is not compatible with the time evolution of the voltage fall across the collisionless cathode sheath in a pulsed hollow-cathode discharge. Similarly this criterion is not valid for the interpretation of probe measurements of the plasma in this discharge. The discrepancy disappears only when the discharge reaches its steady state. The duration of this invalidity is much longer than the time needed to fill the device with plasma and than the time needed for the formation of the plasma sheath. The sheath evolution in nonstationary plasmas is described quantitatively by assuming at the sheath edge a temporary growth of the ion speed from an initial low value up to the ion sound velocity, i.e., up to the Bohm criterion requirement.

The multi-aperture hollow target: a method for determining ion incidence angles in plasma immersion ion implantation

As in plasma immersion ion implantation ions are accelerated from a spherical plasma sheath edge which surrounds the target, a distribution of angles of ion incidence can be expected on a non-spherical target. The angle of ion incidence is of importance, as it determines the projected range of ions in the solid, their depth distribution and also the saturation dose, which is given by sputter limitations. In the present contribution, a simple but very effective method for determining the angles of ion incidence is described. A hollow cylindrical target with an upper cover plate with small drill holes is used. Ions from the plasma enter the target through these apertures and impinge onto the inner bottom where a cellulose sheet is placed which is blackened. By optically determining the positions of ion incidence on the bottom with respect to the positions of the aperture, the angles under which the ions enter the aperture, i.e. the angle of ion incidence on the plate, can be calculated. The obtained data show that in the center of the cylindrical target the ions impinge at right angles whereas in the outer parts smaller angles of incidence are found. The distribution of angles is given as a function of the process parameters such as pulse voltage and pulse duration.

Stability analysis of a model equilibrium for a gravito-electrostatic sheath in a colloidal plasma under external gravity effect

The present contribution tries to find a scientific answer to the question of stability of an equilibrium plasma sheath in a colloidal plasma system under external gravity effect. A model equilibrium of hydrodynamical character has been discussed on the basis of quasi-hydrostatic approximation of levitational condition. It is found that such an equilibrium is highly unstable to a modified-ion acoustic wave with a conditional likelihood of linear driving of the so-called acoustic mode too. Thus, it is reported (within fluid treatment) that a plasma-sheath edge in a colloidal plasma under external gravity effect could be highly sensitive to the acoustic turbulence. Its consequential role on possible physical mechanism of Coulomb phase transition has been conjectured. However, more rigorous calculations as future course of work are required to corroborate our phenomenological suggestions.

A study of methods for moving particles in RF processing plasmas

Methods for moving charged particles in RF processing plasmas are investigated. These methods include varying RF power, varying chamber pressure, attraction and repulsion by an electrostatic probe, and movement with magnetic fields. Varying RF power changes the depth of the potential wells where particles are trapped. The RF power affects shape and location of the traps and the bulk plasma potential. Increasing the chamber pressure moves the sheath edge closer to the wafer being processed. Since particle traps are found at the plasma sheath edge increasing the chamber pressure will move the particle traps (and any trapped particles) closer to the wafer being processed. The Langmuir probe can repel particles when under negative bias and attract them when positively biased. This probe can also distort the sheath edge when the tip resides within the sheath. Applying a magnetic field can change the characteristics of the particle traps and produce a force on the charged dust particles. >

The Bohm criterion for radio-frequency discharges: A numerical verification based on Poisson’s equation

Recently it was shown that, by using the analysis of electrostatic waves entering the plasma–sheath edge, the direct-current (dc) Bohm criterion also holds for discharges under radio-frequency (rf) conditions. In this paper, the influence of Bohm’s criterion on the sheath characteristics for generator frequencies much higher than the ion plasma frequency has been examined by means of an analysis which is based directly on the numerical solution of the Poisson equation. The present calculations indicate that for high rf voltages the time-dependent potential profile does not always increase monotonically, even if Bohm’s criterion is fulfilled. However, as this nonmonotonic behavior does not lead to a more stringent Bohm criterion, the statement that the original dc Bohm criterion also holds in the high-frequency regime is confirmed. The calculations show further that the time-dependent sheath potential is almost completely modulated in the major part of the sheath and that the modulation drops abruptly to zero close to the plasma–sheath edge.