Steady-state Sheath Research Articles

Loss cone effects and monotonic sheath conditions of a partially magnetized plasma sheath

In this Letter, we propose the conditions for monotonic plasma sheaths adjacent to a floating wall in the presence of an applied, oblique magnetic field. The electron velocity distribution function (VDF) at the sheath edge obtained from a kinetic model exhibits a loss cone shaped truncation. Using an approximation of the truncated VDF, we derive an analytical framework of the sheath edge condition (i.e., Bohm condition), namely, the relation of ion injection velocity and sheath potential drop as a function of the magnetic field angle. The results show that the sheath edge velocity and total potential drop decrease for a steady-state sheath, which eventually collapses when the magnetic field lines become parallel to the wall.

Accurate Evaluation of Steady-State Sheath Voltage and Current in HVDC Cable Using Electromagnetic Transient Simulation

The current and voltage in High Voltage DC (HVDC) line is not pure DC but contain superimposed ripple components. The current ripple in core of HVDC cable magnetically induces a voltage in the sheath, whereas the voltage ripple causes the flow of charging current from core to sheath. The knowledge of sheath voltage is necessary to ensure compliance with the specification of utility companies. In this work, we have reported that the models available in commercial Electromagnetic Transient (EMT) simulation software erroneously introduce a DC bias in steady-state sheath voltage and sheath current. We have also demonstrated that by removing the DC bias accurate steady-state evaluation of sheath voltage and sheath current is possible. Additionally, we have analyzed the sheath voltage and currents in HVDC cable considering different cable lengths and sheath grounding schemes. It has been found that grounding the sheath at the terminal of HVDC cable can limit the sheath voltage to acceptable levels without causing substantial joule loss in the sheath.

Nonuniform plasma diffusion and multi-pulse effect in plasma-based ion implantation

The nonuniform plasma caused by plasma diffusion and the incomplete plasma recovery during short pulse-off time have great influences on the sheath dynamics and the implantation efficiency in plasma-based ion implantation (PBII). In this paper, a magnetized plasma diffusion fluid model is established to describe the plasma diffusion in PBII process. Together with a magnetized sheath fluid model, the full pulse period including sheath dynamics during pulse-on time and plasma recovery during pulse-off time can be described, and the models are verified to be accurate by comparing with experimentally measured electron density profiles. The influence of process parameters on sheath dynamics, the influence of incomplete plasma recovery under multi-pulse bias on implantation efficiency are investigated by solving the presented models with considering plasma diffusion. It is found that the variations of process parameters which accelerate the plasma diffusion reduce the steady-state sheath thickness and increase the ion implantation current, and vice versa. Change the pulse frequency from 1kHz to 100kHz under typical PBII process parameters significantly increases the average ion implantation current density, and the limiting factor which affects the implantation efficiency is converted from duty cycle to plasma diffusion. Increasing the plasma density and decreasing the transverse magnetic field are the effective methods to improve the implantation efficiency as well. The results reported here help to provide a theoretical guidance for the parameters optimization in PBII process.

Effect of atoms evaporated from the anode on the structure of the vacuum arc space charge sheath

The structure of the anode space charge sheath of a vacuum arc is studied with allowance for the dependence of the negative anode fall on the ratio of the directed electron velocity v 0 to the electron thermal velocity v T for different values of the flux density of atoms evaporated from the anode. Poisson’s equation for the sheath potential is solved taking into account the electron space charge, fast cathode ions, and slow ions produced due to the ionization of atoms evaporated from the anode. The kinetic equation for atoms and slow anode ions is solved with allowance for ionization in the collision integral. Analytic solutions for the velocity distribution functions of atoms and slow ions and the density of slow ions are obtained. It is shown that the flux of slow ions substantially affects the spatial distribution of the electric field E(z) in the sheath. As the flux density increases, the nonmonotonic dependence E(z) transforms into a monotonic one and the sheath narrows. For a given flux of evaporated atoms Πa, the increase in the ratio of the directed electron velocity to the electron thermal velocity leads again to a nonmonotonic dependence E(z). As z increases, the electric field first increases, passes through the maximum, decreases, passes through the minimum E min, and then again increases toward the anode. There is a limiting value of the ratio (v 0/v T )* at which E min(z) vanishes. At v 0/v T > (v 0/V T )*, the condition for the existence of a steady-state sheath is violated and the profiles of the field and potential in the sheath become oscillating. The dependence of (v 0/v T )* on the flux density of evaporated atoms Π a is obtained. It is shown that the domain of existence of steady-state solutions in the sheath broadens with increasing Π a .

Simulation of erosion of the tungsten wall by impurities in the divertor plasma

Divertor is a component that directly contacts the plasma in tokamak. To ensure the lifetime of the device, it is necessary to reduce the erosion of the divertor wall by plasma. In this work, a particle-in-cell model is used to study the influences of plasma temperature and impurity concentration on the erosion of tungsten divertor wall by carbon and beryllium ions. The steady-state sheath, particle and energy fluxes to the wall, and the energies and angle of the incident ions can be obtained. Then, these data can be used as the input parameters for the plasma-surface interaction model, to evaluate the erosion rate of the plate based on the empirical formulas for physical sputtering. It is found that the erosion by heating plays a negligible role under the plasma condition of this work. Due to the low physical sputtering threshold energy of tungsten by impurities and the impurity ions accelerated by sheath, the physical sputtering of the tungsten by the impurities plays an dominant role in the total erosion. In addition, the erosion rate increases with the increase of plasma temperature and impurity concentration.

On characteristics of sheath damping near a dielectric wall with secondary electron emission

A preliminary investigation is conducted to study the characteristics of sheath damping near a dielectric wall with secondary electron emission (SEE). Making use of the linear analysis of the sheath stability, we have found two major contributions to the sheath damping, one similar to the Landau damping in uniform plasmas and another determined by local electric field and electron density of the steady-state sheath. It indicates that in a classical sheath regime the damping in the sheath region monotonically increases towards the wall and decreases with the enhancement of SEE effect. In order to verify the theoretical analysis, sheath oscillation processes induced by an initial disturbance are simulated with a time-dependent one-dimensional (1D) sheath model. Numerical results obtained are consistent with the theoretical analysis qualitatively.

The Double Sheath at the Plasma‐Wall Boundary

AbstractThe relation between plasma parameters determining a steady‐state charged sheath at a plasma‐wall boundary was derived and analyzed using a fluid model, which accounted for inertia and partial pressure of both plasma components. The relation generalizes the well‐known Bohm criterion and, in particular, shows that a steadystate sheath may be formed in some cases where the Bohm criterion is not satisfied. Conditions allowing formation of a double sheath structure were formulated and analyzed. It was shown that the structure may be formed only if a) the generalized Bohm criterion is satisfied, and b) the Sagdeev potential has a minimum. The double sheath structure consists of two sub‐sheath of different polarity, which depends on the wall potential polarity and the relation between the plasma component temperatures and inertia. For positive wall potential, the sub‐sheath adjacent to the plasma (the sheath edge) is enriched by negative particles, and the sub‐sheath adjacent to the wall is enriched by positive particles, while the sub‐sheath polarities are reversed for negative wall potential. An analytical theory was formulated and illustrated by numerical solutions of the Poisson equation for special cases. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Disturbance of a Langmuir Probe at the Steady-State Sheath Boundary in a Drifting Plasma

The disturbance resulting from the presence of a positively biased Langmuir probe at an equilibrium steady-state ion sheath boundary in the drifting plasma is simulated by the two-dimensional particle-in-cell (PIC) method in rectangular coordinates. The positive potential of the probe was found to shield electrons on the side of the probe away from the sample surface from the negative potential of the sample surface and thereby cause an increase in the electron current drawn by the probe. The drifting ions on the same side of the probe are slowed by it and thereby raising the ion density in front of the probe. The electron currents of the probe are calculated at different probe positions from the sample surface. It is found that the sheath boundary as defined by the probe current is wider than the unperturbed boundary. The widening of the sheath boundary agrees well with experimental sheath boundary measurements.

Determination of the equilibrium ion sheath in the drifting plasma by numerical simulation

A one-dimensional (1-D) particle-in-cell (PIC) numerical method is developed to determine the equilibrium steady-state sheath width established in a drifting plasma. The simulated and measured steady-state sheath widths are in approximate agreement although the measured width is slightly larger than the simulated. The probe is biased to +90 V and this greatly influences the potential structure within the sheath boundary. The simulation shows that the mean-charge state and mean-atomic-mass approach to dealing with multiple ion species with a range of charge states does not accurately predict the position of the equilibrium sheath when the difference between the charge-to-mass ratios of the ion species is large. A more robust approach is to simulate the steady sheath by a 1-D-PIC method that can handle multiple ion species. In experimental situations where the sample stage is finite in size, the assumption that the equilibrium ion sheath expands from a biased plate of infinite extent may be violated. A two-dimensional PIC numerical method expressed in r-z cylindrical coordinates has been developed to investigate the condition where the 1-D assumption becomes inaccurate. The results confirm that the 1-D-PIC method becomes inaccurate when the steady-state sheath width has dimensions comparable with the sample stage diameter.

Method of matched asymptotic expansions versus intuitive approaches: calculation of space-charge sheaths

A comparison is performed of results predicted by the method of matched asymptotic expansions, by the Child-Langmuir model, and by patching in three problems of the theory of space-charge sheaths: a collisionless steady-state sheath, a matrix sheath, and a collisionless RF sheath. In each problem, results are compared between themselves and with the exact solution. In all the cases, the Child-Langmuir model and patching provide results which are accurate to the first approximation in the sheath voltage but not to the second one, irrespective of details of patching. For a steady-state sheath and a matrix sheath, accuracy of the asymptotic solution is exponential. The asymptotic solution for an RF sheath is accurate to the second approximation and can be further improved, if necessary. Comparison of numerical values shows that the method of matched asymptotic expansions indeed provides a considerably higher accuracy.

Simulation of sheath dynamics and current nonuniformity in plasma-immersion ion implantation of a patterned surface

We study the steady-state structure and the dynamics of a plasma sheath surrounding a patterned surface. A two-dimensional hybrid particle-in-cell code is employed where special attention is given to the plasma side boundaries. The surface consists of a periodic array of wide and narrow trenches. We characterize the conformity of the steady-state sheath and the uniformity of the ion currents to the wall as well as the sheath-expansion dynamics following sudden negative charging of the surface. We study the nonuniform motion of the sheath edge away from the surface, which leads to a planar high-voltage sheath. The inhomogeneous distributions of ion current and impact angle onto the walls are discussed.

Magnetic field effect on the sheath thickness in plasma immersion ion implantation

The sheath thickness in plasma immersion ion implantation has been investigated in the presence of a transverse magnetic field. It has been found that the steady-state sheath thickness increases with increasing magnetic field strength. This result is in line with a simplified model of the sheath in which the steady-state sheath thickness is determined by the plasma density and ion velocity at the sheath edge. These results suggest that a magnetic field may be used to control the high-voltage sheath in plasma immersion ion implantation.

Plasma recovery in plasma immersion ion implantation: dependence on pulse frequency and duty cycle

In the technique of plasma immersion ion implantation, plasmaions are implanted into a target upon pulsed charging to high negativevoltages. Using hybrid particle-in-cell computer simulation, westudy the effects of pulse frequency and duty cycle on the ion flux ontothe target. We find that incomplete plasma recovery between the pulsesaffects the ion current maxima occurring at the beginning of thehigh-voltage pulses; these strongly decrease for small pulse-off times.Also, for large duty cycles, the implantation current becomes dominatedby the saturation current of the steady-state sheath.

Plasma Diffusion in Systems with Particle Losses

In certain experimental fusion devices, plasma is confined by a magnetic field and steadily lost out of the ends of a magnetic bottle. When the scattering rate of particles is high, field diffusion and particle loss can be considered from a macroscopic point of view. The formation of a steady-state sheath into which particles diffuse only to be lost along the lines of force was suggested by Wright and Phillips (1960). The elementary theory is considerably developed here into a state in which the original approximations are evident, though the theory remains essentially phenomenological. It is shown that a steady-state plasma sheath is formed whose thickness depends on the plasma resistivity sound speed and length. The sheath moves into the plasma with a speed which depends on the same parameters.