Thomas Mussenbrock

Simulation benchmarks for low-pressure plasmas: Capacitive discharges

Benchmarking is generally accepted as an important element in demonstrating the correctness of computer simulations. In the modern sense, a benchmark is a computer simulation result that has evidence of correctness, is accompanied by estimates of relevant errors, and which can thus be used as a basis for judging the accuracy and efficiency of other codes. In this paper, we present four benchmark cases related to capacitively coupled discharges. These benchmarks prescribe all relevant physical and numerical parameters. We have simulated the benchmark conditions using five independently developed particle-in-cell codes. We show that the results of these simulations are statistically indistinguishable, within bounds of uncertainty that we define. We, therefore, claim that the results of these simulations represent strong benchmarks, which can be used as a basis for evaluating the accuracy of other codes. These other codes could include other approaches than particle-in-cell simulations, where benchmarking could examine not just implementation accuracy and efficiency, but also the fidelity of different physical models, such as moment or hybrid models. We discuss an example of this kind in the Appendix. Of course, the methodology that we have developed can also be readily extended to a suite of benchmarks with coverage of a wider range of physical and chemical phenomena.

Nonlinear electron resonance heating in capacitive radio frequency discharges

Technological processing plasmas are frequently operated at relatively low gas pressure (<10Pa). A characteristic feature of this regime is that collisions of the electrons with the atoms or molecules of the neutral background are relatively rare, and the so-called collisional or Ohmic heating ceases to be an effective mechanism of energy deposition into the plasma. Experiments indicate that at low pressure an alternative mechanism of electron heating exists which can sustain the plasma. Despite 30years of intense research, the exact nature of this “anomalous” heating mechanism is still under discussion. The two standard models are known as “stochastic heating” and “pressure heating,” respectively. This work proposes a third explanation of anomalous electron heating and suggests that, in the last analysis, all three mechanisms may contribute to the observed effect.

Electron beams in asymmetric capacitively coupled radio frequency discharges at low pressures

The generation of directed energetic electrons by the expanding sheath is observed in asymmetric capacitively coupled radio frequency discharges at low pressures (≤ 1 Pa) in different gases. The phenomenon of such electron beams is investigated by a combination of experimental diagnostics, an analytical model and simulations. At sufficiently low pressures multiple reflections of electron beams at the plasma boundaries are observed. An analytical model shows how these beams lead to an enhanced high energy tail of the electron energy distribution function. Thus, stochastic heating is closely related to electron beams.

The multipole resonance probe: A concept for simultaneous determination of plasma density, electron temperature, and collision rate in low-pressure plasmas

A diagnostic concept is presented which enables the simultaneous determination of plasma density, electron temperature, and collision rate in low-pressure gas discharges. The proposed method utilizes a radio-frequency driven probe of particular spherical design which is immersed in the plasma to excite a family of spatially bounded surface resonances. An analysis of the measured absorption spectrum S(ω) of the probe provides information on the distribution of the plasma in its vicinity, from which the values of the plasma parameters can be inferred. In its simplest realization, the probe consists of two dielectrically shielded, conducting hemispheres, which are symmetrically driven by an radio-frequency source, and the excited resonances can be classified as multipole fields, which allows an analytical evaluation of the measured signal. The proposed method is robust, calibration free, economical, and can be used for ideal and reactive plasmas alike.

The effects of nonlinear series resonance on Ohmic and stochastic heating in capacitive discharges

The flow of electron and ion conduction currents across a nonlinear capacitive sheath to the electrode surface self-consistently sets the dc bias voltage across the sheath. We incorporate these currents into a model of a homogeneous capacitive sheath in order to determine the enhancement of the Ohmic and stochastic heating due to self-excitation of the nonlinear series resonance in an asymmetric capacitive discharge. At lower pressures, the series resonance can enhance both the Ohmic and stochastic heating by factors of 2–4, with the Ohmic heating tending to zero as the pressure decreases. The model was checked, for a particular set of parameters, by a particle-in-cell (PIC) simulation using the homogeneous sheath approximation, giving good agreement. With a self-consistent Child-law sheath, the PIC simulation showed increased heating, as expected, whether the series resonance is important or not.

The multipole resonance probe: characterization of a prototype

The multipole resonance probe (MRP) was recently proposed as an economical and industry compatible plasma diagnostic device (Lapke et al 2008 Appl. Phys. Lett. 93 051502). This communication reports the experimental characterization of a first MRP prototype in an inductively coupled argon/nitrogen plasma at 10?Pa. The behavior of the device follows the predictions of both an analytical model and a numerical simulation. The obtained electron densities are in excellent agreement with the results of Langmuir probe measurements.

Electric field reversals in the sheath region of capacitively coupled radio frequency discharges at different pressures

Electric field reversals in single and dual-frequency capacitively coupled radio frequency discharges are investigated in the collisionless (1Pa) and the collisonal (65Pa) regimes. Phase resolved optical emission spectroscopy is used to measure the excitation of the neutral background gas caused by the field reversal during sheath collapse. The collisionless regime is investigated experimentally in asymmetric neon and hydrogen single frequency discharges operated at 13.56MHz in a GEC reference cell. The collisional regime is investigated experimentally in a symmetric industrial dual-frequency discharge operated at 1.937 and 27.118MHz. The resulting spatio-temporal excitation profiles are compared with the results of a fluid sheath model in the single frequency case and a particle-in-cell/Monte Carlo simulation in the dual-frequency case. The results show that field reversals occur in both regimes. An analytical model gives an insight into the mechanisms causing the reversal of the electric field. In the dual-frequency case a qualitative comparison between the electric fields resulting from the PIC simulation and from the analytical model is performed. The field reversal seems to be caused by different mechanisms in the respective regimes. In the collisionless case it is caused by electron inertia, whereas in the collisional regime it is caused by a combination of the low mobility of electrons due to collisions and electron inertia. Finally, the field reversal during the sheath collapse seems to be a general source for energy gain of electrons in both single and dual-frequency discharges. (Some figures in this article are in colour only in the electronic version)

Ionization by bulk heating of electrons in capacitive radio frequency atmospheric pressure microplasmas

Electron heating and ionization dynamics in capacitively coupled radio frequency (RF) atmospheric pressure microplasmas operated in helium are investigated by particle-in-cell simulations and semi-analytical modeling. A strong heating of electrons and ionization in the plasma bulk due to high bulk electric fields are observed at distinct times within the RF period. Based on the model the electric field is identified to be a drift field caused by a low electrical conductivity due to the high electron?neutral collision frequency at atmospheric pressure. Thus, the ionization is mainly caused by ohmic heating in this ??-mode?. The phase of strongest bulk electric field and ionization is affected by the driving voltage amplitude. At high amplitudes, the plasma density is high, so that the sheath impedance is comparable to the bulk resistance. Thus, voltage and current are about 45? out of phase and maximum ionization is observed during sheath expansion with local maxima at the sheath edges. At low driving voltages, the plasma density is low and the discharge becomes more resistive, resulting in a smaller phase shift of about 4?. Thus, maximum ionization occurs later within the RF period with a maximum at the discharge center. Significant analogies to electronegative low-pressure macroscopic discharges operated in the drift-ambipolar mode are found, where similar mechanisms induced by a high electronegativity instead of a high collision frequency have been identified.

Modeling and simulation of the plasma absorption probe

The plasma absorption probe (PAP) was invented as an economical and robust diagnostic device to determine the electron density distribution in technical plasmas. It consists of a small antenna enclosed by a dielectric tube which is immersed in the plasma. A network analyzer feeds a rf signal to the antenna and displays the frequency dependence of the power absorption. From the absorption spectrum the value of the electron density is calculated. The original evaluation formula was based on the dispersion relation of plasma surface waves propagating along an infinite dielectric cylinder. In this letter the authors present the analysis of a less idealized configuration. The calculated spectra are in good qualitative agreement with their experimental counterparts, but differ considerably from those predicted by the surface wave ansatz. An evaluation scheme which takes our findings into account will improve the performance of the PAP technique further.

A nonlinear global model of a dual frequency capacitive discharge

The behavior of dual frequency capacitively coupled plasmas is investigated. Assuming a realistic reactor configuration represented by effective geometry factors and taking into account two separate sinusoidal voltage sources operating at different frequencies, an ordinary differential equation is derived which describes the nonlinear dynamics of such discharges. An exact analytical solution of the equation is presented and employed for a parameter study of the discharge current characteristics. Simulation results for various gas pressures (=various electron-neutral collision rates), various amplitude ratios of the two independent rf sources, and various integer frequency ratios are shown. When the two frequencies are comparable, surprising nonlinear effects are observed. Particular under study is the heating at the plasma series resonance, either by direct excitation or via the nonlinear electron resonance heating mechanism.