non-Maxwellian Electron Energy Distribution Function Research Articles

Characteristics of a non-Maxwellian electron energy distribution in a low-pressure argon plasma

The generality of the non-Maxwellian electron energy distribution function (EEDF) is demonstrated by using optical emission spectroscopy (OES) and Langmuir probe measurements in inductively- and capacitively-coupled low-pressure argon plasmas to analyze the shape factor of the EEDF. To measure the shape factor of the EEDF, we propose a corona — equilibrium (CE) — based analysis model operating at low density, which uses the line intensity ratio of the Ar I to the Ar II emission lines. The Ar I line is chosen to represent the relatively low-energy state, and the Ar II line is chosen to represent the high-energy state. Thus, an analysis of the shape factor is equivalent to monitoring the variation in the high-energy electron fraction represented in the tail of the EEDF. Results show a depleted tail for the Maxwellian distribution in most of the low-density argon plasmas. The analysis reveals that the generation and the stepwise ionization of metastable argon atoms by inelastic collisions with high-energy (∼ 10 eV) electrons are dominant processes of argon plasma generation and cause serious high-energy electron loss in a low-density system compared to the loss in an ideal Maxwellian plasma. The existence of argon metastable states is inevitable; thus, the general shape of the electron energy distribution in low-pressure argon plasmas is non-Maxwellian.

Effect of non-Maxwellian electron energy distribution functions on the simulation of an inductively coupled Ar–Hg discharge

In this paper, we simulate an inductively coupled Ar–Hg plasma with a hybrid model combining Maxwellian equations, plasma fluid equations and the Boltzmann equation of electrons. The calculated space distribution of Hg 6-3P1 atoms is compared with measurements. It turns out that in the situation of a non-Maxwellian electron energy distribution function (EEDF), the calculation agrees quite well with the measurement. This is because the calculated non-Maxwellian EEDFs have more electrons with intermediate kinetic energy. This suppresses the coupling reactions between excited Hg atoms, while increasing direct excitation, stepwise ionization and so on. Further analyses indicate that chemi-ionization of Hg 6-P atoms is not important in the ionization balance due to the effect of non-Maxwellian EEDFs, but it has an observable influence on the space distribution of Hg 6-3P1 atoms.

Possibilities of determining non-Maxwellian EEDFs from the OES line-ratios in low-pressure capacitive and inductive plasmas containing argon and krypton

Optical emission spectroscopy (OES) is classically used to determine the electron temperature in low-temperature plasmas by assuming a Maxwellian electron energy distribution function (EEDF). However, non-Maxwellian EEDFs are often found in these plasmas. In this work, we propose an OES line-ratio method that is aimed at obtaining non-Maxwellian EEDFs, with a collisional–radiative model of argon and krypton. When applied for both a capacitively coupled plasma and an inductively coupled plasma, this method provides satisfactory results compared with those measured using a Langmuir probe.

Determination of metastable level densities in a low-pressure inductively coupled argon plasma by the line-ratio method of optical emission spectroscopy

The line-ratio method of optical emission spectroscopy (OES) is used for the diagnosis of plasma parameters. In this work, electrostatic probe-assisted OES is employed to measure metastable level densities from spectral lines and electron energy distribution functions (EEDFs) in a low-pressure inductively coupled argon plasma. Emission spectroscopy is based on plasma modelling through a simple collisional–radiative model. The line intensities of Ar(3p54p → 3p54s) are modified due to the plasma reabsorption at relatively high pressures where the plasma becomes optically thick. To consider this effect, a pressure dependence factor αij(P) is first derived from both the measured intensity and pressure-dependent cross-section for electron excitation. It is found that the obtained metastable densities range from 1.3 × 109 to 1.2 × 1010 cm−3 and their ratios are nearly constant by a factor of about 3–5 in the investigated pressure range (3–50 mTorr). The effect of non-Maxwellian EEDF on the metastable densities is also discussed. The results measured by the line-ratio method are consistent with that of the OES-branching fraction method taking into account the photon escape factor to treat the radiation trapping.

Determination of Non-Maxwellian Electron Energy Distributions in Low-Pressure Plasmas by Using the Optical Emission Spectroscopy and a Collisional-Radiative Model

A Maxwellian electron energy distribution function (EEDF) is often assumed when using the optical emission line-ratio method to determine the electron temperature in low-temperature plasmas. However, in many cases, non-Maxwellian EEDFs can be formed due to the non-local electron heating or the inelastic-collisional energy loss processes. In this work, with a collisional-radiative model, we propose an approach to obtain the non-Maxwellian EEDF with a ‘two-temperature structure’ from the emission line-ratios of Paschen 2p levels of argon and krypton atoms. For applications of this approach in reactive gas (CF4, O2, etc) discharges that contain argon and krypton, recommendations of some specific emission line-ratios are provided, according to their sensitivities to the EEDF variation. The kinetic processes of the relevant excited atoms are also discussed in detail.

A simple analysis on the abnormal behavior of the argon metastable density in an inductively coupled Ar plasma

The abnormal behavior of the argon metastable density during the E–H mode transition in argon ICP discharge was investigated. Lots of investigations including global models expected that during and after the mode transition of ICP discharge the argon metastable density increases with applied rf power (i.e. electron density). However, recent measurement of metastable density [A.M. Daltrini, S.A. Moshkalev, T.J. Morgan, R.B. Piejak, W.G. Graham, Appl. Phys. Lett. 92 (2008) 061504] revealed that the argon metastable density decreases with the applied power during and after the mode transition. This result may not be explained by the previous global model which is based on the assumption of the Maxwellian electron energy distribution function (EEDF). In this paper, to explain this abnormal behavior with simple manners, a simple global model taking account of the effect of the non-Maxwellian EEDFs is proposed. The calculated result from the proposed global model showed that the argon metastable density can exhibit an abnormal behavior with electron density which is in good agreement with the measurement results, indicating the close coupling of electron kinetics and the behavior of metastable density. The proposed simple analysis is expected to provide qualitative kinetic insight to understand the behavior of the metastable density in various plasma discharges which typically exhibit non-Maxwellian distribution.

Effects of Microfilaments on DC Plasma Torches and Afterglows

In this paper, we investigate the effects of microfilaments on the operation of direct-current (dc) atmospheric-pressure nitrogen plasma torches. We measure the roto-vibrational temperatures of the first negative and second positive systems of the N2 and N2 + species, as well as the electron energy distribution function (EEDF) in the plasma plume and at several distances away from the source. The EEDF is measured using both planar and double-point Langmuir probes. We found that the operation of a dc nitrogen plasma torch in a mixed glow/microfilament regime results in the ejection of high-energy electron bullets from the torch nozzle. The frequency of these electron bullets outside the plasma coincides with the microfilament frequency. Furthermore, the microfilament frequency is shown to have a direct correlation with the N2 + roto-vibrational temperature. The electron bullets possess a highly non-Maxwellian EEDF and maintain their nonequilibrium energies while contained within the N2 substrate gas.

Electrical characteristics of (N2-H2) gas mixture DC glow discharge

The electrical characteristics of DC Nitrogen-Hydrogen mixture glow discharge were investigated. The discharge current-voltage (I-V) characteristic curves of the discharge were measured at different gas pressures and gas mixture percentage. The axial distribution of both potential and electric field were studied at different H2percentage. The existence of the reversal electric field in the negative glow region was confirmed. Using the axial electric field distribution, the cathode fall thickness (Xc) was calculated. Second derivative of the single Langmuir probe electron current method was used to measure the electron energy distribution function (EEDF) in (N2-H2) gas mixture DC glow discharge at different H2 percentage. In the positive column region, a Maxwellian EEDF was found at different conditions, while in both cathode fall and negative glow regions, a non-Maxwellian EEDF was observed. Two groups of electrons were detected in these two regions. Key words: Glow discharge, axial potential and electric field distributions, Paschen's curves, Langmuir probes.

Reconstruction of an electron energy distribution function using integrated data analysis

The non-Maxwellian electron energy distribution function (EEDF) in the positive column of a neon dc-discharge was reconstructed from the visible emission spectrum obtained with an overview spectrometer. The analysis is based on Bayesian probability theory (integrated data analysis), which allows for the use of the full information in the spectral data by incorporating all important underlying physical mechanisms. The data are described by a collisional–radiative model and a statistical description of the spectroscopic measurement. An extensive and consistent set of electron impact excitation cross sections and Einstein coefficients obtained through semi-relativistic B-Spline R-Matrix calculations is employed, and estimates for uncertainties are included in the analysis. The results are consistent with theoretical modelling and reference measurements.

Absolute argon excited-state population measurements from emission spectroscopy in an inverted cylindrical magnetron plasma

An inverted cylindrical magnetron sputtering system operating at 6.5 kW, 85 sccm argon, and 0.27 Pa was used to produce a mid-frequency AC (40 kHz) Al–Ar plasma. Optical emission spectroscopy was used to monitor the plasma emission from 680 nm to 830 nm. Absolute calibration of the emission data was performed to find the population density of excited-state argon atoms in the 4p electron energy level for six transitions. The excited-state densities were on the order of 10 13 m − 3 for each transition. For comparison, an argon collisonal-radiative model was used to simulate the excited-state densities. The model used non-Maxwellian electron energy distribution functions obtained from the commercially available software ELENDIF. Good agreement between the measured and predicted Ar 4p excited states was observed.

Self-consistent modeling of nonlocal inductively coupled plasmas

In low-pressure radio-frequency (RF) discharges, the electron-energy distribution function (EEDF) is typically non-Maxwellian for low plasma density. The nonlocal plasma conductivity, plasma density profiles, and EEDF are all nonlinear and nonlocally coupled. For accurate calculation of the discharge characteristics, the EEDF needs to be computed self-consistently. The method of fast self-consistent one-dimensional of planar inductively coupled discharges driven by a RF electromagnetic field is presented. The effects of a non-Maxwellian EEDF, plasma nonuniformity, and finite size, as well as the influence of the external magnetic field on the plasma properties are considered and discussed

Electron transport in the downstream region of planar unbalanced magnetron discharge

In this study, we will investigate the electron transport in the downstream region of a planar and unbalanced (type II) magnetron discharge. The effects of the anode sheath boundary and diverging magnetic field on the electron kinetics such as the electron loss mechanism at plasma-sheath boundary and the electron distribution function will be examined through the probe measurements. The spatially resolved probe measurements reveal the existence of an electron drift from the cathode fall region to the downstream region. It is found that this drift is caused by the axial gradient of magnetic field (the magnetic mirror force) and then derives an electron current to the grounded substrate on which the potential of the sheath is very low; so the current balance between the cathode and anode currents is kept. The experimental results show that the electron transport in the downstream region is not governed by the classical diffusion (mobility and diffusion dominated) but is dominated by the modified diffusion including the electron drift caused by the magnetic mirror force. Additionally, the mechanism and the experimental evidence on the presence of a non-Maxwellian electron energy distribution function (in particular, bi-Maxwellian distribution) in magnetron discharge will be presented showing that the non-Maxwellian electron energy distribution function is due to the combined effects of the electron drift toward the substrate and the sheath boundary condition.

Landau damping and anomalous skin effect in low-pressure gas discharges: Self-consistent treatment of collisionless heating

In low-pressure discharges, where the electron mean free path is larger or comparable with the discharge length, the electron dynamics is essentially nonlocal. Moreover, the electron energy distribution function (EEDF) deviates considerably from a Maxwellian. Therefore, an accurate kinetic description of the low-pressure discharges requires knowledge of the nonlocal conductivity operator and calculation of the non-Maxwellian EEDF. The previous treatments made use of simplifying assumptions: a uniform density profile and a Maxwellian EEDF. In the present study a self-consistent system of equations for the kinetic description of nonlocal, nonuniform, nearly collisionless plasmas of low-pressure discharges is reported. It consists of the nonlocal conductivity operator and the averaged kinetic equation for calculation of the non-Maxwellian EEDF. This system was applied to the calculation of collisionless heating in capacitively and inductively coupled plasmas. In particular, the importance of accounting for the nonuniform plasma density profile for computing the current density profile and the EEDF is demonstrated. The enhancement of collisionless heating due to the bounce resonance between the electron motion in the potential well and the external rf electric field is investigated. It is shown that a nonlinear and self-consistent treatment is necessary for the correct description of collisionless heating.

Self-consistent system of equations for a kinetic description of the low-pressure discharges accounting for the nonlocal and collisionless electron dynamics.

In low-pressure discharges, where the electron mean free path is larger or comparable with the discharge length, the electron dynamics is essentially nonlocal. Moreover, the electron energy distribution function (EEDF) deviates considerably from a Maxwellian. Therefore, an accurate kinetic description of the low-pressure discharges requires knowledge of the nonlocal conductivity operator and calculation of the non-Maxwellian EEDF. The previous treatments made use of simplifying assumptions: a uniform density profile and a Maxwellian EEDF. In the present study, a self-consistent system of equations for the kinetic description of nonlocal, nonuniform, nearly collisionless plasmas of low-pressure discharges is derived. It consists of the nonlocal conductivity operator and the averaged kinetic equation for calculation of the non-Maxwellian EEDF. The importance of accounting for the nonuniform plasma density profile on both the current density profile and the EEDF is demonstrated.

Substantiation of the two-temperature kinetic model by comparing calculations within the kinetic and fluid models of the positive column plasma of a dc oxygen discharge

Results from kinetic and fluid simulations of the positive column plasma of a dc oxygen discharge are compared using commercial CFDRC software ( http://www.cfdrc.com/˜cfdplasma ), which enables one to perform numerical simulations in an arbitrary 3D geometry with the use of both the fluid equations for all the components (fluid model) and the kinetic equation for the electron energy distribution function (kinetic model). It is shown that, for both the local and nonlocal regimes of the formation of the electron energy distribution function (EEDF), the non-Maxwellian EEDF can satisfactorily be approximated by two groups of electrons. This allows one to take into account kinetic effects within the conventional fluid model in the simplest way by using the proposed two-temperature approximation of the nonequilibrium and nonlocal EEDF (2T fluid model).

Effect of electron energy distribution function on power deposition and plasma density in an inductively coupled discharge at very low pressures

A self-consistent one-dimensional model was developed to study the effect of the electron energy distribution function (EEDF) on power deposition and plasma density profiles in a planar inductively coupled plasma (ICP) in the non-local regime (pressure ⩽10 mTorr). The model consisted of three modules: (1) an EEDF module to compute the non-Maxwellian EEDF, (2) a non-local electron kinetics module to predict the non-local electron conductivity, radio frequency (RF) current, electric field and power deposition profiles in the non-uniform plasma, and (3) a heavy species transport module to solve for the ion density and velocity profiles as well as the metastable density. Results using the non-Maxwellian EEDF model were compared with predictions using a Maxwellian EEDF, under otherwise identical conditions. The RF electric field, current and power deposition profiles were different, especially at 1 mTorr, for which the electron effective mean-free-path was larger than the skin depth. The plasma density predicted by the Maxwellian EEDF was up to 93% larger for the conditions examined. Thus, the non-Maxwellian EEDF must be accounted for in modelling ICPs at very low pressures.

Effect of non-local electron conductivity on power absorption and plasma density profiles in low pressure inductively coupled discharges

A self-consistent one-dimensional model was developed to study the effects of non-local electron conductivity on power absorption and plasma density profiles in a planar inductively coupled argon discharge at low pressures (⩽10 mTorr). The model consisted of three modules: (1) an electron energy distribution function (EEDF) module to compute the non-Maxwellian EEDF, (2) a non-local electron kinetics module to predict the non-local electron conductivity, radio frequency (RF) current, electric field and power deposition profiles in the non-uniform plasma, and (3) a heavy species transport module to solve for the ion density and velocity profiles as well as the metastable density. Results using the non-local electron conductivity model were compared with predictions of a local theory (Ohm's law), under otherwise identical conditions. The RF current, electric field, and power deposition profiles were very different, especially at 1 mTorr for which the effective electron mean free path was larger than the skin depth. However, the plasma density profiles were almost identical (within 10%) for the same total power deposition in the plasma. This result suggests that, for computing plasma density profiles, a local conductivity model (Ohm's law), with much reduced computational expense, may be employed even in the non-local regime.

Simulation of hot-electron oxide tunneling current based on a non-Maxwellian electron energy distribution function

For the simulation of gate oxide tunneling currents in sub-quarter-micron devices, the correct modeling of the electron energy distribution function is crucial. Our approach is based on a recently presented transport model which accounts for six moments of the Boltzmann transport equation. A corresponding analytical model for the electron energy distribution function shows good agreement with Monte Carlo data. Using this model, we show that the gate current behavior of short-channel devices can be reproduced correctly. This is not the case for the heated Maxwellian approximation which leads to a massive overestimation of gate currents especially for devices with small gate lengths. We develop a formalism to distinguish between cases where the heated Maxwellian distribution delivers correct results and cases where it overestimates the tunneling current at low drain bias and find that for oxide thicknesses around 2 nm, the heated Maxwellian approximation is only valid for electron temperatures below about 1000 K.

Thomson-scattering observation of non-Maxwellian electron energy distribution in inductively coupled C4F8/Ar and CF4/Ar plasmas and the effect of vibrational excitation and nonlocal electron kinetics

Laser Thomson-scattering diagnostics of electron energy distribution function (EEDF) was carried out for inductively coupled cyclo–C4F8/Ar and CF4/Ar plasmas at 25 mTorr. Care was taken to ascertain that the observed Thomson spectra are unaffected by other laser-induced effects, in particular by Raman scattering. Admixing of C4F8 and CF4 with Ar resulted in marked deviation of EEDF from Maxwellian. The mechanism leading to non-Maxwellian EEDF was also studied by a Monte Carlo simulation. The results suggest that the major factors responsible are rapid electron energy loss due to vibrational excitation and the effect nonlocal electron kinetics associated with localized electron heating.

Energy coupling efficiency of a hydrogen microwave plasma reactor

Zero-dimensional and two-dimensional plasma models and optical emission spectroscopy are used in tandem to investigate the power coupling efficiency for a pure hydrogen microwave plasma. The zero-dimensional model accounts for the vibrational kinetics of H2, the chemistry of H2 and H excited states, and the kinetics of ground-state species. The set of species conservation equations are then coupled to the electron Boltzmann equation (to account for the non-Maxwellian electron energy distribution function) and the total energy equation for solution. The two-dimensional model makes use of a simpler thermochemical description of the plasma. The chemistry is described with nine species and thirty chemical reactions. Three energy modes are considered to describe the plasma’s thermal nonequilibrium, and Maxwellian distribution functions for kinetic and vibrational modes are assumed. The non-Maxwellian nature of the electron energy distribution function is separately accounted for. Experimentally, the absolute line emission intensity is utilized to obtain number densities of up to five hydrogen excited states using the following transitions: Hα (6563 Å), Hβ (4861 Å), Hγ (4340 Å), Hδ (4102 Å), and Hε (3970 Å). The first three transitions were used for a 38 Torr, 1000 W hydrogen discharge, and all five transitions were used for a 121 Torr, 4000 W hydrogen discharge. The absolute continuum emission from the plasma was compared to numerical predictions. The comparison of the numerical and experimental data indicates that 90%–100% of the input power is deposited in the plasma and that both the line and continuum emission match within a factor of 3, with the exception of the high energy excited states for the 4000 W plasma. A control volume heat transfer analysis validates the energy coupling.