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, that 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 an 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. * Electronic address: miles.turner@dcu.ie 2

The increasing need to demonstrate the correctness of computer simulations has highlighted the importance of benchmarks. We define in this paper a representative simulation case to study low-temperature partially-magnetized plasmas. Seven independently developed Particle-In-Cell codes have simulated this benchmark case, with the same specified conditions. The characteristics of the codes used, such as implementation details or computing times and resources, are given. First, we compare at steady-state the time-averaged axial profiles of three main discharge parameters (axial electric field, ion density and electron temperature). We show that the results obtained exhibit a very good agreement within 5% between all the codes. As ExB discharges are known to cause instabilities propagating in the direction of electron drift, an analysis of these instabilities is then performed and a similar behaviour is retrieved between all the codes. A particular attention has been paid to the numerical convergence by varying the number of macroparticles per cell and we show that the chosen benchmark case displays a good convergence. Detailed outputs are given in the supplementary data, to be used by other similar codes in the perspective of code verification. 2D axial-azimuthal Particle-In-Cell benchmark for low-temperature partially ...

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 "Ω-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 in 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.

The kinetic origin of resonance phenomena in capacitively coupled radio frequency plasmas is discovered based on particle-based numerical simulations. The analysis of the spatio-temporal distributions of plasma parameters such as the densities of hot and cold electrons, as well as the conduction and displacement currents reveals the mechanism of the formation of multiple electron beams during sheath expansion. The interplay between highly energetic beam electrons and low energetic bulk electrons is identified as the physical origin of the excitation of harmonics in the current.Capacitively coupled radio frequency (CCRF) discharges are indispensable tools for semiconductor manufacturing and other innovative applications [1,2]. At the same time, they are challenging physical systems due to their complex and nonlinear dynamics. At low neutral gas pressures of a few Pa or less, CCRF discharges are operated in a strongly non-local regime. In the so-called "α-mode", electron heating is dominated by stochastic sheath expansion heating [3] and electric field reversal during sheath collapse [4][5][6][7]. Stochastic heating was modelled extensively in the past in the frame of a hard wall model, as well as pressure heating [8][9][10][11][12][13][14][15][16][17]. During the phase of sheath expansion, energetic electron beams are generated and propagate into the plasma bulk, where they sustain the discharge via ionization and lead to a Bi-Maxwellian electron energy distribution function (EEDF) [18][19][20][21][22][23][24]. At low pressures, resonance effects such as the plasma series resonance (PSR) [27][28][29][30] and the plasma parallel resonance (PPR) [31][32][33] can be self-excited and strongly enhance the electron heating [19,29]. In the presence of a sinusoidal driving voltage waveform, the excitation of the PSR results in a non-sinusoidal RF current, due to the appearance of harmonics of the driving frequency [30]. Although these have been observed in experiments [21], they are usually neglected in most models of electron heating in CCRF plasmas. Existing theories which include resonance effects are zero-dimensional global models based on equivalent electrical circuits [29] as well as spatially resolved models based on the cold plasma approximation [30]. As these models do not include any kinetic effects, such resonances should be investigated on a microscopic kinetic level. A kinetic interpretation is required to clarify some of the most important open questions about electron heating dynamics in CCRF plasmas: What is the kinetic origin of the generation of high frequency (HF) oscillations of the RF current and the generation of multiple electron beams during one phase of sheath expansion such as observed in previous works [37]? In what way is current continuity (∇ · j tot = 0) ensured at all times within the RF period in the presence of electron beams, where the total current density j tot = j d + j c is decomposed into the displacement and conduction current density? Our aim is to provide access to a kinetic inter...

The electrical asymmetry effect (EAE) allows an almost ideal separate control of the mean ion energy, hE i i, and flux, C i , at the electrodes in capacitive radio frequency discharges with identical electrode areas driven at two consecutive harmonics with adjustable phase shift, h. In such geometrically symmetric discharges, a DC self bias is generated as a function of h. Consequently, hE i i can be controlled separately from C i by adjusting the phase shift. Here, we systematically study the EAE in low pressure dual-frequency discharges with different electrode areas operated in argon at 13.56 MHz and 27.12 MHz by experiments, kinetic simulations, and analytical modeling. We find that the functional dependence of the DC self bias on h is similar, but its absolute value is strongly affected by the electrode area ratio. Consequently, the ion energy distributions change and hE i i can be controlled by adjusting h, but its control range is different at both electrodes and determined by the area ratio. Under distinct conditions, the geometric asymmetry can be compensated electrically. In contrast to geometrically symmetric discharges, we find the ratio of the maximum sheath voltages to remain constant as a function of h at low pressures and C i to depend on h at the smaller electrode. These observations are understood by the model. Finally, we study the self-excitation of non-linear plasma series resonance oscillations and its effect on the electron heating. V