Collisionless Sheath Research Articles

Numerical investigations of spatiotemporal dynamics of space-charge limited collisional sheaths

Electrostatic particle-in-cell (PIC) and direct simulation Monte Carlo (DSMC) methods are used to compare the plasma dynamics of collisionless with collisional emissive sheaths in partially ionized environments. Space-charge limited emissive sheaths submersed in a plasma with a density of ∼1017 m−3 are examined using a PIC-DSMC solver, CHAOS. Collisionless emissive sheaths with plasma domains sufficiently long (30 and 60 Debye lengths, λD) are subject to strong oscillations due to two-stream electron instability, whereas emissive sheaths in weakly collisional conditions with a short domain (15 λD) exhibit self-spike (sawtooth) oscillations in the plasma field due to the trapped charge-exchange (CEX) ion population within the virtual cathode (VC) region. The two-stream electron instability leads to strong temporal fluctuations in the total emission current, with maximum deviations of 60% and 100% from the time-averaged current for the long plasma domains, whereas CEX collisions cause strong spikes in the emission current if the domain size is short. Our PIC-DSMC simulations show for the first time that the interaction of the two types of instabilities causes the strength of the self-spike to be weakened due to the strong fluctuations caused by the two-stream instability when a sufficiently long computational domain with ion-neutral collisions is employed. By conducting a two-dimensional Fast Fourier Transform (FFT) on the collisional and collisionless sheaths with long domains, we show that the transient evolution of CEX entrapment in the VC increases frequency of sheath oscillations up to two times the ion-acoustic frequencies observed in the collisionless sheath. CEX collisions weaken the VC region and result in a total emission current more than that obtained from the collisionless case for the same domain length. With a more rarefied neutral environment of 1019 m−3 in the plasma sheath, the total emission current increases only 4% in comparison with 14% for one order of magnitude denser environment, within 20 μs. In addition, the spike period is tested with different neutral temperatures and densities. While we do not observe any self-spike in the more rarefied environment, the spike period increased from 5 to 7.5 μs when the neutral temperature is increased from 300 to 2000 K in the denser environment with the simulation time of 20 μs.

Effects of low-frequency voltage on nonlinear standing wave excitation, plasma uniformity, and ion dynamics in dual-frequency asymmetric capacitive discharges

It is known that in very-high-frequency (VHF) capacitively coupled plasmas, the higher harmonics generated by nonlinear sheath motion can enhance the standing wave effect (SWE), which can lead to center-peaked plasma density profiles. In this work, an improved nonlinear electromagnetic model incorporating a transmission line model, an electron momentum balance model, a bulk plasma model, a collisionless nonlinear numerical sheath model, and an ion Monte-Carlo collision (MCC) model is developed to study the effects of low-frequency (LF) voltage VL on the nonlinear standing wave excitation, plasma uniformity, and ion energy and angular distribution functions (IEDFs and IADFs) in dual-frequency (DF) asymmetric capacitive argon discharges at relatively low pressure of 3 Pa. The plasma diffusion in the radial direction and ion dynamics within the LF oscillating sheath are self-consistently considered. The LF voltage VL at 2 MHz varies from 0 to 700 V while the HF voltage VH at 60 MHz is fixed at 100 V. Simulation results indicate that without the addition of an LF source (i.e. VL= 0 V), there are a considerable number of high-order harmonics with short wavelengths, leading to significant SWE and central peak in the radial plasma density profile. Nevertheless, the high-order harmonic excitations tend to be weakened and merely occur around the phase of the full LF sheath collapse due to a shorter characteristic damping time of the surface waves as VL increases. This, combined with increased surface wavelengths of both the driving frequency and the higher harmonics at a higher VL , leads to suppressed standing waves and improved plasma uniformity. Meanwhile, the simulations show that both the low and the high energy peaks of IEDF move towards higher energies, and the energy peak separation width ΔE becomes wider with the increase of VL . The IEDF at the radial center of the powered electrode exhibits a broader ΔE than that at the edge. For the IADF, an increased VL results in more ions incident on the electrode with a smaller deflection angle. Because of a thinner sheath and a higher sheath voltage at the electrode center, the peak value of IADF at the electrode center is greater than that at the edge.

Study of a collisionless magnetized plasma sheath with nonextensively distributed species

A weakly magnetized sheath for a collisionless, electronegative plasma comprising positive ions, electrons, and negative ions is investigated numerically using the fluid approach. The electrons are considered to be non-Maxwellian in nature and are described by Tsalli’s distribution. Such electrons have a substantial effect on the sheath properties. The study also reveals that non-Maxwellian distribution is the most realistic description for negative ions in the presence of an oblique magnetic field. In addition to the negative ion temperature, the sheath potential is also affected by the nonextensive parameters. The present research finds application in the plasma processing and semiconductor industry as well as in space plasmas.

Collisionless magnetized sheath resonance heating induced by a transverse magnetic field in low-pressure capacitive rf discharges

Abstract The mechanism of resonance heating between the gyrating electrons and the oscillating sheath induced by a small transverse magnetic field in low-pressure capacitively coupled plasmas is investigated. The gyrating electrons will coherently collide with the expanding sheath if the electron gyro-frequency coincides with half the driving frequency. These electrons will gain substantial energy from collision-less heating, which strongly enhances the electron power absorption and the plasma density at a constant driving voltage. The electron kinetics is revealed at resonant conditions by particle simulations. Our numerical results demonstrate that the relation between the magnetic field and the driving frequency determines this resonance effect. Besides, it is found that the operating pressure, electrode gap, and driving voltage all strongly affect this electron resonance mechanism. The resonance effect is more pronounced at the conditions of low pressure, large gap, and high voltage.

Isentropic plasma sheath model for improved fidelity

A model is developed for a collisionless plasma sheath assuming isentropic electrons in contrast to the standard isothermal electron assumption. This approach is enabled by the approximation of a Maxwellian electron velocity distribution function across the sheath, which is justified by near wall measurements. The conservation of entropy leads to a modified Boltzmann relation and a modified Bohm criterion. The predicted floating sheath potential is in excellent agreement with experimental data. Takamura's model for a space-charge limited plasma sheath near an emissive surface is also modified for isentropic electrons and with that modification agrees well with numerical results from a full fluid plasma model.

Numerical investigation of a radio frequency sheath with supra-thermal electrons in the presence of the magnetic field

The structure of the magnetized radio-frequency (RF) sheath of plasma containing supra-thermal electrons is studied numerically with a one-dimensional hybrid model consisting of a collisionless sheath model and an equivalent circuit model of the sheath. It is found that effect of the external magnetic field on the sheath structure depends on the direction of the disturbance current source. As the plasma magnetization or angle between the magnetic field and the wall is increased, both the sheath thickness and sheath potential drop increase during the positive disturbance current source at the wall, while they change slightly during the negative disturbance current source at the wall. For the different supra-thermal electron populations parameterized by Kappa velocity distribution, both the sheath thickness and sheath potential drop are always larger in the magnetized sheath than in the unmagnetized sheath. Compared with in the unmagnetized sheath, the width of the ion energy distribution (IED) in the magnetized sheath expands towards the high energy regime, and the low-energy peak of the IED decreases while the high-energy peak of the IED changes a little.

Numerical simulation of anode heat transfer of nitrogen arc utilizing two-temperature chemical non-equilibrium model* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11735004 and 12005010).

A detailed understanding of anode heat transfer is important for the optimization of arc processing technology. In this paper, a two-temperature chemical non-equilibrium model considering the collisionless space charge sheath is developed to investigate the anode heat transfer of nitrogen free-burning arc. The temperature, total heat flux and different heat flux components are analyzed in detail under different arc currents and anode materials. It is found that the arc current can affect the parameter distributions of anode region by changing plasma characteristics in arc column. As the arc current increases from 100 A to 200 A, the total anode heat flux increases, however, the maximum electron condensation heat flux decreases due to the arc expansion. The anode materials have a significant effect on the temperature and heat flux distributions in the anode region. The total heat flux on thoriated tungsten anode is lower than that on copper anode, while the maximum temperature is higher. The power transferred to thoriated tungsten anode, ranked in descending order, is heat flux from heavy-species, electron condensation heat, heat flux from electrons and ion recombination heat. However, the electron condensation heat makes the largest contribution for power transferred to copper anode.

Power deposition behavior of high-density transient hydrogen plasma on tungsten in Magnum-PSI

The lifetime of plasma-facing components (PFCs) will have a strong influence on the efficiency and viability of future fusion power plants. However, the PFCs suffer from thermal stresses and physical sputtering induced by edge-localized modes (ELMs). ELMs in future fusion devices are expected to occur with a high plasma density compared to current day devices such that coupling of recycling neutrals and plasma ions will be strong. Because of the scale hierarchy of future fusion devices compared to the present ones, the influence of this coupling is difficult to predict. Here, we investigate the ELM-like hydrogen plasma induced heat loads on tungsten in the linear device Magnum-PSI, producing ∼1 ms plasma pulses with electron densities up to 3.5 × 1021 m−3. A combination of time-resolved Thomson scattering and coherent Thomson scattering was used to acquire plasma parameters in front of the target. Moreover, a fast infrared camera coupled to finite element thermal analyses allowed to determine the deposited heat loads on the target. We found a significant inconsistency between the plasma power calculated with a conventional collisionless sheath model and the absorbed power by the target. Moreover, plasma stagnation upstream and plasma cooling downstream were observed during the pulses. The observations are explained based on ionization and elastic collisions between the recycling neutrals and plasma ions. The results highlight the impact of plasma-neutral interaction on the power deposition behavior of ELM-like hydrogen plasma on tungsten.

Plasma Sheath Modelling for Computational Aerothermodynamics and Magnetohydrodynamics

ABSTRACT To date, plasma sheath effects have not been incorporated into most CFD simulations of magnetohydrodynamics (MHD) or aerothermodynamics due to the high computational costs involved. The accurate modeling of both collisional and collisionless plasma sheaths for hypersonic applications is here presented. This is followed by the outline of recent numerical methods that enable the sheaths to be integrated with large time steps and to be incorporated within CFD codes. For the first time, the electron energy transport in non-equilibrium is here integrated in coupled form with the plasma sheath within a MHD process. Some example test problems are then presented including MHD in a scramjet combustor and electron transpiration cooling in a hypersonic flow. Such show that the plasma sheath effects can be added within CFD simulations with little loss in computational efficiency. The inclusion of plasma sheath models is seen to be critical in predicting accurately such problems.

Nonlinear harmonic excitations in collisional, asymmetrically-driven capacitive discharges

The standing wave effect, which may lead to center-high density profiles in high frequency capacitive discharges, can be enhanced by nonlinearly excited harmonics. In this work, a nonlinear transmission line model, which solves for the electromagnetic fields in the time domain, is coupled to a two-dimensional bulk plasma fluid model to study nonlinear effects in asymmetric cylindrical capacitive argon discharges. An analytical collisional or collisionless (ion) sheath model is used to determine the stochastic and ohmic sheath heating and the nonlinear dependence of sheath voltage on sheath charge. We first examine a base case of a 20 mTorr argon discharge driven with an electron power P e = 40 W at a frequency f = 60 MHz, using collisionless and collisional sheath models. For the collisionless sheath model, the nonlinearly excited harmonics near the series and spatial resonance frequencies significantly enhance the on-axis power deposition and lead to a sharp peak of electron density at the discharge center. The collisional sheath model gives a smaller sheath width, leading to lower series and spatial resonance frequencies and a smaller source voltage for the fixed electron power. As a result, lower harmonics with broader spatial profiles and decreased magnitude are excited, reducing the center-high plasma nonuniformity. Then, we examine the discharge in a pressure range of 20–100 mTorr at fixed P e = 40 W and f = 60 MHz, using the collisional sheath model. As pressure increases, the harmonics gradually damp out, and the enhancement of on-axis power deposition becomes less significant. At the same time, more power is localized near the powered electrode edge due to decreased skin depth and smaller energy diffusion. As a result, the density peak shifts from the radial center to the powered electrode edge.

Effects of emitting surfaces and trapped ions on the sheath physics and current flow in multidimensional plasma systems

Recent one-dimensional simulations of planar sheaths with strong electron emission have shown that trapping of charge-exchange ions causes transitions from space-charge limited (SCL) to inverse sheaths. However, multidimensional emitting sheath phenomena with collisions remained unexplored, due in part to high computational cost. We developed a novel continuum kinetic code to study the sheath physics, current flow and potential distributions in two-dimensional unmagnetized configurations with emitting surfaces. For small negatively biased thermionic cathodes in a plasma, the cathode sheath can exist in an equilibrium SCL state. The SCL sheath carries an immense density of trapped ions, neutralized by thermoelectrons, within the potential well of the virtual cathode. For further increases of emitted flux, the trapped ion cloud expands in space. The trapped ion space charge causes an increase of thermionic current far beyond the saturation limit predicted by conventional collisionless SCL sheath models without ion trapping. For sufficiently strong emission, the trapped ion cloud consumes the entire 2D plasma domain, forming a mode with globally confined ions and an inverse sheath at the cathode. In situations where the emitted flux is fixed and the bias is swept (e.g. emissive probe), the trapped ions cause a large thermionic current to escape for all biases below the plasma potential. Strong suppression of the thermionic emission, required for the probe to float, only occurs when the probe is above the plasma potential.

The generalized criterion for collisionless plasma sheaths with kappa distributed electrons

In this paper, by analytical estimation and numerical analysis of the ions Mach number domain in an unmagnetized collisionless plasma sheath, the formation condition of a stable sheath has been examined and formulated in the context of kappa distribution formalism. In this study, by theoretical investigation of the normalized variables with respect to well-defined parameters that enclose recent findings of kappa distribution formalism, a generalized Bohm criterion is derived in terms of an extended Mach number, , and a generalized polytropic index, γe, which characterizes different thermodynamic states with a general flexibility from near-thermal equilibrium states to the far-from-thermal equilibrium states. In our formulation, the allowed domains of isothermal and extended Mach number with respect to the connection between the polytropic index and the kappa distribution has been examined. By definition of the extended Mach number, the necessity of supersonic ions at the edge of sheath remains unchanged, but its appearance is different in the terms of the isothermal Mach number. It is shown that the threshold of extended Mach number in far-from-thermal equilibrium states is a sensitive function of the generalized polytropic index and it decreases by closing to the asymptotic limit of anti-equilibrium state (). Moreover, the threshold of isothermal Mach number decreases in the sub-isothermal regions with γe < 1 and so, by closing to the far-from-thermal equilibrium states, the Mach number domain is extended. Specially, it is shown that the thickness of the plasma sheaths reduces for the kappa distributed plasmas with smaller polytropic indices γe. Finally, the possibility for formation of stable plasma sheaths in different thermodynamic sates is discussed, where the validity of the generalized sheath criterion has been confirmed and given by or equivalently .

Deuterium experiment with large-scale negative ion source for large helical device

The deuterium experiment with various source plasma diagnostics was conducted at the research and development negative ion source at National Institute for Fusion Science, which is a cesium-seeded large-scale negative hydrogen-ion source with half discharge volume and the same type of ion source for large helical device. A change of filling gas from hydrogen to deuterium increased line-averaged negative ion density by a factor of 1.3 in beam extraction region as a result of flattened profile with keeping central density. Positive ion and electron densities increased threefold. This was influenced by the increase of plasma density in the plasma generation region. The plasma space potential varied as equivalent of the collisionless sheath theory in electro-positive plasma expected by the hydrogen isotope gas species, although the negative ion could affect sheath formation. Higher atomic cesium density was observed because of higher sputtering yield due to higher momentum of deuterium.

Kinetic sheath in presence of multiple positive ions, negative ions, and particle wall emission

The region between a Maxwellian plasma source and a floating or current-carrying surface is described by a static, one-dimensional collisionless kinetic sheath model. In the plasma source, electrons, negative ions, and several positive ion species with different temperatures can be included. The surface (wall) can emit electrons and/or negative ions. When the flux of surface-emitted negative ions and/or electrons reaches a critical value, the sheath becomes space-charge saturated, which leads to the formation of a virtual cathode in front of the emitting wall and sets the maximum current density that can be transported from the surface to the plasma. The analytical results are benchmarked against a particle-in-cell code.

Stability of different plasma sheaths near a dielectric wall with secondary electron emission

The linear theory stability of different collisionless plasma sheath structures, including the classic sheath, inverse sheath and space-charge limited (SCL) sheath, is investigated as a typical eigenvalue problem. The three background plasma sheaths formed between a Maxwellian plasma source and a dielectric wall with a fully self-consistent secondary electron emission condition are solved by recent developed 1D3V (one-dimensional space and three-dimensional velocities), steady-state, collisionless kinetic sheath model, within a wide range of Maxwellian plasma electron temperature $T_{e}$ . Then, the eigenvalue equations of sheath plasma fluctuations through the three sheaths are numerically solved, and the corresponding damping and growth rates $\unicode[STIX]{x1D6FE}$ are found: (i) under the classic sheath structure (i.e. $T_{e}&lt;T_{ec}$ (the first threshold)), there are three damping solutions (i.e. $\unicode[STIX]{x1D6FE}_{1}$ , $\unicode[STIX]{x1D6FE}_{2}$ and $\unicode[STIX]{x1D6FE}_{3}$ , $0&gt;\unicode[STIX]{x1D6FE}_{1}&gt;\unicode[STIX]{x1D6FE}_{2}&gt;\unicode[STIX]{x1D6FE}_{3}$ ) for most cases, but there is only one growth-rate solution $\unicode[STIX]{x1D6FE}$ when $T_{e}\rightarrow T_{ec}$ due to the inhomogeneity of sheath being very weak; (ii) under the inverse sheath structure, which arises when $T_{e}&gt;T_{ec}$ , there are no background ions in the sheath so that the fluctuations are stable; (iii) under the SCL sheath conditions (i.e. $T_{e}\geqslant T_{e\text{SCL}}$ , the second threshold), the obvious ion streaming through the sheath region again emerges and the three damping solutions are again found.

Analysis of Instability Phenomena at Current Interruption in Vacuum Arc Discharge Compared With Silver or Copper Electrode

Vacuum circuit breakers have an issue of instability arc current and chopping phenomena when the current is cut off from the circuit. To clarify the effect of the instability low-current vacuum arc for circuit breaker development, the instability mechanism of low-current vacuum arc for various parameters was proposed to verify consequent affecting circuit breaker capacity in this paper. This paper used the cathode spot model neglect the transition region between the collisionless sheath and collisional plasma region. The eight equations were used for solving the dependent variable of plasma parameters for both silver and copper electrode materials. The low ionization was supposed in the cathode spot model. The results showed that the changes in the work function and ion current fraction at ambient temperature did not have the main influence on the arc stability limit current both in the silver electrode and copper electrode. On the contrary, the changes in the heat conductivity at ambient temperature affecting the arc stability limit current significantly both in the silver and copper electrodes. Furthermore, the results were found out that the cathode spot can be confirmed unstable current phenomena when the current below 19 A for copper cathode and below 19.4 A for the silver cathode. These phenomena can be explained that the reversed electrons flow from the plasma side to sheath region affecting the electric field at cathode spot become negative zone; thus, the starting noise on the current trace appears before chopping current.

A detailed study on the structures of steady-state collisionless kinetic sheath near a dielectric wall with secondary electron emission. II. Inverse and space-charge limited sheaths

The inverse and space-charge limited (SCL) sheaths between a Maxwellian low-temperature plasma source and a dielectric surface that emits secondary electrons are carefully investigated by a 1D3V, steady-state, kinetic sheath model with fully self-consistent wall secondary electron emission conditions, within a broad range of plasma electron temperature Te. The multistep electron emission (MEE) phenomenon caused by the near-wall potential increment is specially considered, and it is found that the total emitted electron velocity distribution function perpendicular to the dielectric wall approximately obeys two-temperature Maxwellian and half-Maxwellian distribution. Under cold plasma ion assumption, results show that when Te&amp;gt;Tec (the first threshold) the sheath goes into entirely inverse sheath regime: (1) the ion energy E at the sheath edge equals 0, i.e., no presheath is necessary to accelerate ion; (2) the wall total electron emission coefficient Γ=1, i.e., no charge is lost at the wall; (3) the wall potential is just about 0∼0.03(Te/e) for all the cases; (4) the sheath depth is thinner than one Debye length. When Te≥TeSCL (the second threshold), another solution of SCL sheath is also found. It clearly indicates that the two sheaths are possible near plasma-facing dielectric wall with strong electron emission, but they do not always emerge in pairs. Further comparative calculations predict that the MEE phenomenon can significantly decrease the wall potential Φw, virtual cathode potential Φvc, and near-wall potential increment ΔΦ (i.e., Φw-Φvc) for SCL sheath, but slightly increase Φw for the inverse sheath. In addition, the differences between the two sheaths are discussed.

A detailed study on the structures of steady-state collisionless kinetic sheath near a dielectric wall with secondary electron emission. I. Classic sheath and its structure transition

It is well known that plasma sheath presents a classic sheath structure when the dielectric-wall total electron emission coefficient Γ ≤ Γc (&amp;lt;1). However, the structural transition of a classic sheath near a dielectric wall when Γ→Γc is controversial about transiting to a space-charge limited (SCL) sheath or inverse sheath. In this study, the classic sheath between a Maxwellian low-temperature plasma source and a dielectric surface that emits secondary electrons is carefully investigated using a 1D3V, steady-state, kinetic sheath model within a broad range of plasma electron temperatures Te. Using the Monte Carlo method to simulate secondary electron emission (SEE) events that are based on the self-consistent primary electron velocity distribution function at the wall and a detailed SEE model, it is found that the total emitted electron velocity distribution function (EEVDF) perpendicular to the dielectric wall approximately satisfies a three-temperature Maxwellian distribution. Due to the relatively high average energy of this total EEVDF, for cases of Te with cold plasma ion assumption: (1) the critical SCL sheath does not exist; (2) Γc reaches unit; (3) the sheath disappears when Te = Tec (i.e., Γ = 1); and (4) as Te increases, the classic sheath will transit to an inverse sheath structure. Further comparative calculations predict that the magnitude of emitted electrons' average energy may lead to different experimental results between thermionic emitting surfaces which have a “cold” half-Maxwellian EEVDF and SEE surfaces. However, when the finite plasma ion temperature is considered, at the transition point, the cold plasma ion assumption is expected to be invalid, and thus a fully kinetic sheath model should be built to reveal the potential new transition regime.

Comments on the Langmuir probe measurements of radio-frequency capacitive argon–hydrogen mixture discharge at low pressure

In this paper, the current–voltage graphs of discharge in the chamber of capacitive coupled radio frequency (CCRF) at low pressure were presented for Langmuir probe. The Langmuir probe measurements for estimating the electron density and temperature in capacitive coupled discharges at low pressures were presented and the electron temperatures of the Ar–H2 mixture discharge generated at different conditions were reported using the Langmuir probe. The focus of this study is that the CCRF discharge can be determined and explained using the characteristics of plasma by means of Langmuir probe measurements for the different hydrogen rates in Ar–H2 mixture discharge. The measurement results of Langmuir probe gave values around 1015 m−3 for the electron density. The floating potential depended on the electronegative gas amount. It was found that the increase of hydrogen gas amount in the mixture discharge caused the decrease of the floating potential. Also, a decrease in the argon (Ar) metastable with the increase in hydrogen (H2) content was obtained. When the applied radio frequency (RF) power was increased, the thickness and collisionless sheath occurring at lower RF power could transform to thin sheath.

Kinetic features of collisionless sheaths around polarized cylindrical emitters from the orbital motion theory

The kinetic features of the sheath around a cylindrical emitter immersed in collisionless plasma at rest are analysed. After finding self-consistently the electric potential by applying the Orbital Motion Theory to the Vlasov-Poisson system, the local distribution functions are reconstructed and the radial profiles of important macroscopic quantities (plasma densities, currents, and temperatures) are then computed. It is found that there can only be three kinds of holes that are bound by three different boundaries—two related to the constraints from orbital effects and the other due to the electric potential barrier. The results are presented for three regimes: negative probe bias with monotonic and non-monotonic potential and positive probe bias with non-monotonic potential. To understand the variation of macroscopic-quantity radial profiles, three diagrams are presented for kinetic features: the ϵl-diagram for the integration domains of the two orbital invariants, the effective potential, and the local ...