Sheath Edge Research Articles

Parametric analysis of a magnetized cylindrical plasma

The relevant macroscopic model, the spatial structure, and the parametric regimes of a low-pressure plasma confined by a cylinder and an axial magnetic field is discussed for the small-Debye length limit, making use of asymptotic techniques. The plasma response is fully characterized by three-dimensionless parameters, related to the electron gyroradius, and the electron and ion collision mean-free-paths. There are the unmagnetized regime, the main magnetized regime, and, for a low electron-collisionality plasma, an intermediate-magnetization regime. In the magnetized regimes, electron azimuthal inertia is shown to be a dominant phenomenon in part of the quasineutral plasma region and to set up before ion radial inertia. In the main magnetized regime, the plasma structure consists of a bulk diffusive region, a thin layer governed by electron inertia, a thinner sublayer controlled by ion inertia, and the non-neutral Debye sheath. The solution of the main inertial layer yields that the electron azimuthal energy near the wall is larger than the electron thermal energy, making electron resistivity effects non-negligible. The electron Boltzmann relation is satisfied only in the very vicinity of the Debye sheath edge. Ion collisionality effects are irrelevant in the magnetized regime. Simple scaling laws for plasma production and particle and energy fluxes to the wall are derived.

Effect of the electron energy distribution function in plasma on the Bohm criterion and on the drop voltage through the sheath: Case of microwave expanding plasma

This work is devoted to the study of the sheath expansion above a charged species collector immersed into the plasma in the general case of the electron energy distribution function (EEDF) (not necessarily a Boltzmann distributed function). In the first part, considering the general form of the electron kinetic energy distribution function (EEDF) in the plasma and assuming a one dimensional system, we propose an equation describing the limit value of the ion velocity at the sheath edge. Then, we extend this equation to the three dimensional problem. These results are discussed assuming Boltzmann distribution for the one dimensional system and Maxwell–Boltzmann distribution in the case of the three dimensional system. Both one and two temperature distributions are considered. Then, the method is applied to experimental results obtained in expanding microwave discharge sustained in argon. In the second part, we study the drop voltage through the sheath that is produced around a cylindrical collector biased at the floating potential. A theoretical model is given, and results are compared with experimental values measured by means of a cylindrical Langmuir probe immersed in microwave expanding plasma. The model agrees with experiments and points out the role of the EEDF profile on the ion velocity at the sheath edge and, consequently, on the drop voltage through the sheath.

Plasma parameters of pulsed-dc discharges in methane used to deposit diamondlike carbon films

Here we approximate the plasma kinetics responsible for diamondlike carbon (DLC) depositions that result from pulsed-dc discharges. The DLC films were deposited at room temperature by plasma-enhanced chemical vapor deposition (PECVD) in a methane (CH4) atmosphere at 10 Pa. We compared the plasma characteristics of asymmetric bipolar pulsed-dc discharges at 100 kHz to those produced by a radio frequency (rf) source. The electrical discharges were monitored by a computer-controlled Langmuir probe operating in time-resolved mode. The acquisition system provided the intensity-voltage (I-V) characteristics with a time resolution of 1 μs. This facilitated the discussion of the variation in plasma parameters within a pulse cycle as a function of the pulse waveform and the peak voltage. The electron distribution was clearly divided into high- and low-energy Maxwellian populations of electrons (a bi-Maxwellian population) at the beginning of the negative voltage region of the pulse. We ascribe this to intense stochastic heating due to the rapid advancing of the sheath edge. The hot population had an electron temperature Tehot of over 10 eV and an initial low density nehot which decreased to zero. Cold electrons of temperature Tecold∼1 eV represented the majority of each discharge. The density of cold electrons necold showed a monotonic increase over time within the negative pulse, peaking at almost 7×1010 cm−3, corresponding to the cooling of the hot electrons. The plasma potential Vp of ∼30 V underwent a smooth increase during the pulse and fell at the end of the negative region. Different rates of CH4 conversion were calculated from the DLC deposition rate. These were explained in terms of the specific activation energy Ea and the conversion factor xdep associated with the plasma processes. The work deepens our understanding of the advantages of using pulsed power supplies for the PECVD of hard metallic and protective coatings for industrial applications (optics, biomedicine, and electronics).

Electron trajectories and betatron oscillation in the wake bubble in laser-plasma interaction

The trajectories of electrons originating from different initial locations in the unperturbed plasma during the interaction of an ultraintense laser with underdense plasma in the bubble regime are followed by particle-in-cell simulation. It is found that plasma electrons initially aligned with the rim of the laser focal spot contribute most to the bow wave in front of the bubble and those aligned with the lateral bubble sheath edge contribute most to the self-injection at the back of the bubble. A scaling law between the transverse electric and magnetic fields for the case where there are many electrons in the bubble is given and discussed in terms of betatron oscillations of the injected electrons.

Modeling positive ion current to a planar probe in low-pressure electronegative discharges

A fluid model is utilized to describe the plasma-sheath boundary for a negatively biased planar probe immersed in electronegative plasmas. The model equations are solved on the scale of the electron Debye length and calculate the spatial distributions of electric potential, velocity, and density of positive ions in front of the probe. The position of sheath edge, the positive ion velocity at sheath edge (the Bohm velocity), and the positive ion flux collected by the probe are determined and compared with analytic (or scaling) formulas. Effects of control parameters on the Bohm velocity, the sheath thickness, and on the positive ion flux are investigated. A larger thermal motion of negative ions causes the Bohm velocity to increase, the sheath to increase, and the positive ion flux collected by the probe to increase. An increase in collision causes the Bohm velocity to decrease and the sheath to decrease resulting in a decrease in the positive ion flux. An increase in electronegativity causes both the Bohm velocity and the sheath thickness to decrease, resulting in an increase in the positive ion flux. As the value of the non-neutrality parameter q increases, the Bohm velocity and the sheath thickness are found to decrease, and the positive ion flux collected by the probe increases. The behavior of the positive ion flux entering the sheath is discussed as functions of control parameters. A careful comparison of theoretical positive ion flux with the experimental flux can allow us to obtain the electronegativity, the plasma ionization rate (q), and the collision parameter (δ).

Sheath and boundary conditions for plasma simulations of a Hall thruster discharge with magnetic lenses

The effect of magnetic lens configuration on sheath in a Hall thruster discharge channels is considered. A model of the plasma-wall transition is developed in the case of large magnetic field incidence angle with respect to the wall. Peculiarity of such case consists in that the potential drop across the magnetic field is set externally. In this case, standard boundary conditions at the sheath edge for plasma simulations fail and a new formulation for those boundary conditions is proposed. The results obtained demonstrate importance of the effect of the magnetic field incidence angle on the sheath boundary conditions for plasma simulations as well as on the energy balance in the Hall thruster discharge.

Large-area and low-damage processes for hybrid flexible device fabrications with reactive high-density plasmas driven by multiple low-inductance antenna modules

Plasma generation and control technologies for meters-scale ultra-large-area plasma sources have been developed with multiple low-inductance antenna (LIA) modules, as a promising candidate of ultra-large-area and high-density plasma sources for next-generation processing of hybrid flexible devices. Properties of argon-oxygen mixture plasmas sustained with multiple LIA units have been investigated and surface modifications of polymer substrates using the plasmas have been performed. Ion energy distribution at the sheath edge of the argon-oxygen mixture plasmas showed considerable suppression of ion energies as small as or less than 10 eV. We have examined effect of plasma exposure on surface modification and/or degradation of polymer. Surface analysis of polytetrafluoroethylene (PTFE) exhibited nano-surface modification of the polymer surface without suffering degradation of molecular structures beneath the nano- surface layer.

Sheath structure in front of an electron emitting electrode immersed in a two-electron temperature plasma: a fluid model and numerical solutions of the Poisson equation

A one-dimensional fluid model of sheath formation in front of a large, planar, floating, electron emitting electrode (collector) immersed in a two-electron temperature plasma is presented. For certain values of the parameters the Bohm criterion is triple valued. Three methods to select the correct Bohm velocity are compared. The simplest is to find the parameters, where the floating potentials that correspond to the high and to the low Bohm velocity are equal. But sometimes—when the electron emission is small—such parameters cannot be found. The second method is the ‘maximum positive ion flux criterion’, proposed recently by Fernández Palop et al (2002 J. Appl. Phys. 91 2587), for electronegative plasmas. In this work it is modified for a two-electron temperature plasma with emitted electrons and applied successfully. This method is sometimes—when the electron emission from the collector is increased—not applicable because the positive ion flux as a function of potential has only 1 maximum. The third method is a numerical solution of the Poisson equation. The transition from the low to the high Bohm velocity occurs at the parameters where the numerically found space charge density profile indicates the formation of a double layer in the sheath. These parameters are always identical to the parameters found with the maximum positive ion flux method, when the latter can be applied. The ‘shooting method’ for the correct determination of electric field at the collector is presented. This method reveals the existence of regular and irregular numerical solutions of the Poisson equation. The irregular solutions of the Poisson equation fulfill only one of the two sheath edge conditions and they correspond to the solutions of the fluid model, that are not physically acceptable. The parameters, where the numerical solutions of the Poisson equation become irregular, are identical to those found with the ‘positive square of electric field’ test, which is also presented.

The effect of a magnetic field on a cylindrical object in a plasma

A two fluid model is applied to the presheath of a cylindrical object immersed in plasma under the influence of an axial magnetic field. The impact of the magnetic field on presheath behavior is studied and quantified by calculating the sheath edge potential and floating potential of the object. The latter is of great interest since it determines the cylinder’s charge. It is found that the density distribution in the presheath deviates significantly from the Boltzmann relation for electrons due to the presence of the magnetic field. This occurs because B opposes the radial flow of charge. However, the field not only inhibits transport of charge to the cylinder, but also deflects both fluids, setting up currents in the system, which increase the externally applied field. This effect can alter the field topology around the object significantly and makes it necessary to solve for the magnetic field self-consistently in some situations. The relevant parameter range is identified and a sample calculation with nonuniform B is shown. Interestingly, the magnetic field contribution due to the cylinder adds to the applied field, such that the object behaves as if it were ferromagnetic.

Effect of surface produced negative ions on near wall sheath

The role of surface produced negative ions in a negative ion source on the sheath structure, which takes place in the surroundings of a metallic plane immersed in a plasma, is analysed using a simple model. It is well known that the Bohm criterion which determines the electric potential at the sheath edge can be multi-valued in electronegative plasmas. It is observed in this study that a multi-valued sheath potential equally holds well in the case of a negative ion source where surface production mechanism of negative ions also exists. Due to the presence of surface produced negative ions, the positive ion current flux towards the metallic plane increases and the sheath potential with respect to the metallic plane increases.

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.

Electron heating in radio-frequency capacitively coupled atmospheric-pressure plasmas

In atmospheric-pressure plasmas the main electron heating mechanism is Ohmic heating, which has distinct spatial and temporal evolutions in the α and γ modes. In γ discharges, ionizing avalanches in the sheaths are initiated not only by secondary electrons but also by metastable pooling reactions. In α discharges, heating takes place at the sheath edges and in contrast with low-pressure plasmas, close to 50% of the power absorbed by the electrons is absorbed at the edge of the retreating sheaths. This heating is due to a field enhancement caused by the large collisionality in atmospheric-pressure discharges.

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.

Analytical expression for sheath edge around corner cathodes

A simple analytical expression for the position of the sheath edge around a two-dimensional corner cathode with included angle θc has been discovered. This expression is valid for weakly collisional sheaths in the Child–Langmuir regime ϕc ≫ kTe/e, where −ϕc < 0 is the cathode bias and Te is the electron temperature. In polar coordinates (r, θ), the sheath edge is given by (r/s0)sin[πθ/(2π − θc)] = [π/(2π − θc)] where s0 is the planar sheath width far from the vertex of the corner. This result is verified by comparison with previous numerical solutions (Watterson P A 1989 J. Phys. D: Appl. Phys. 22 1300) for the knife edge (θc = 0) and convex square corner (θc = π/2). The observed agreement suggests that this expression gives the sheath edge for all corner angles, both concave and convex. The utility of this result is demonstrated by computing the full sheath solution for a knife-edge cathode with ϕc = 100kTe/e.

Numerical investigation of the magnetized plasma sheath characteristics in the presence of negative ions

In this work, a planar and low-pressure discharge in a mixture of a noble gas (argon) and an electronegative gas (oxygen) is considered. It is assumed that the produced plasma consists of electrons, two species of positive ions, and one species of negative ion. The behavior of the density distribution and kinetic energy of these charged particles in the sheath region are studied. Also, it is assumed that a weak external magnetic field which is nearly perpendicular to the wall is exerted to the sheath region. The positive ion species are considered as a cold, collisionless fluid while both electron and negative ion densities obey the Boltzmann distribution. By using a hydrodynamic approach and ignoring ionization and recombination, it is shown numerically that by increasing the density of the negative ions in the plasma the density distribution of both positive ion species and the kinetic energy of these ion species decreases and increases, respectively. Also, it is shown that in the presence of the negative ions the normalized electrostatic potential of the sheath region changes and by increasing the negative ion densities the normalized electrostatic potential in the sheath increases. In addition, the effect of the density ratio of both positive ion species, density and temperature of the negative ions, and the magnitude of the external magnetic field are studied on the net density distribution of the charged particles in the sheath region. The obtained numerical results show that by decreasing the temperature of the negative ions the amplitude of the fluctuations of the net density distribution of the charged particles in the sheath region increases and the position of these fluctuations shifts toward the sheath edge. Finally, it is seen that in the absence of negative ions the net density distribution of charged particles in the sheath region is monotonic while in the presence of negative ions it is nonmonotonic.

Analytical approximations for the sheath potential profile

The space charge sheath in front of a plane negative wall is frequently described with rather poor approximations since Poisson's equation cannot be solved analytically. Accounting for the correct boundary conditions at the sheath edge, convenient analytical approximations are constructed both in the fluid and in the kinetic approach. The fluid solution is universal and does not depend on the special plasma properties. In the kinetic analysis, in contrast, different plasma models must be distinguished because the space charge formation depends on the ion velocity distribution at the sheath edge. The differences are significant in particular at low sheath potentials. Explicit formulae are given for the Tonks–Langmuir model and for the charge exchange model. All approximations represent the sheath potential profiles with surprising accuracy.

Plasma and sheath

In the analysis of a finite plasma system in contact with an absorbing wall separate model zones of a quasi-neutral plasma body and a thin space charge sheath may be distinguished. This subdivision goes back to Langmuir and is based on a small ratio of the Debye length λD and the plasma dimension L. It is strictly valid only in the asymptotic limit λD/L → 0. In this limit the plasma–sheath interface, the ‘sheath edge’, is characterized by a formal singularity of the plasma solution and closely related to the Bohm criterion. For finite λD, in contrast, there are difficulties in the interpretation of the plasma–sheath transition, in the application of the Bohm criterion, and in a reasonable definition of the plasma–sheath boundary. These difficulties are discussed both in the framework of a fluid and of a kinetic analysis on the basis of an ‘intermediate scale’ analysis. Special emphasis is given to the matching problem and its relation to the eigenvalue problem caused by the plasma balance.

The plasma–sheath boundary: its history and Langmuir's definition of the sheath edge

The introduction of the terms sheath in 1923 and plasma in 1928 by Langmuir is described, followed by their use in the Tonks and Langmuir theory of the positive column at low pressures in 1929. Attention is drawn to the development of Langmuir's ideas during the period from 1923 to 1929. The well-known Bohm criterion for sheath formation, published in 1949, is shown to be closely related to the earlier work of Tonks and Langmuir. The much-used version of the Bohm criterion with the equality sign is obtained by employing the two-scale theory of the plasma and sheath, for the case where λD/L → 0.A generalized Bohm criterion is obtained by introducing the ion velocity distribution; the resulting expression can be understood by considering the propagation of ion-acoustic waves. The plasma–sheath boundary is found to be a sonic surface. Other generalizations of the Bohm criterion are given, including a mixture of positive ions, the presence of negative ions and a non-Maxwellian electron velocity distribution.

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.

Measurement of potential distributions in sheath and presheath near a mesh and metal plate

The potential distributions in the boundary regions of a mesh and solid metal plate were measured using a fine structured emissive probe and compared with theoretical models. It is shown that the sheath region near the mesh is thinner than that near the plate, owing to the fact that the mesh is partially transparent to the ions. The measured electric field at the Bohm sheath edge is in agreement with the asymptotic result in the transition region, but the field at the Child–Langmuir (CL) sheath edge is larger than the values of several models found in the literature. The potential distribution in the CL sheath region agrees with the theoretical curve in the case of the plate, but deviates from the theoretical curve in the case of the mesh.