The effect of changing the driving frequency on the plasma density and the electron dynamics in a capacitive radio-frequency argon plasma operated at low pressures of a few Pa is investigated by particle-in-cell/Monte-Carlo collision simulations and analytical modeling. In contrast to previous assumptions, the plasma density does not follow a quadratic dependence on the driving frequency in this non-local collisionless regime. Instead, a step-like increase at a distinct driving frequency is observed. Based on an analytical power balance model, in combination with a detailed analysis of the electron kinetics, the density jump is found to be caused by an electron heating mode transition from the classical α-mode into a low-density resonant heating mode characterized by the generation of two energetic electron beams at each electrode per sheath expansion phase. These electron beams propagate through the bulk without collisions and interact with the opposing sheath. In the low-density mode, the second beam is found to hit the opposing sheath during its collapse. Consequently, a large number of energetic electrons is lost at the electrodes resulting in a poor confinement of beam electrons in contrast to the classical α-mode observed at higher driving frequencies. Based on the analytical model this modulated confinement quality and the related modulation of the energy lost per electron lost at the electrodes is demonstrated to cause the step-like change of the plasma density. The effects of a variation of the electrode gap, the neutral gas pressure, the electron sticking and secondary electron emission coefficients of the electrodes on this step-like increase of the plasma density are analyzed based on the simulation results.

Abstract. We investigate the electron heating dynamics in electropositive argon and helium capacitively coupled RF discharges driven at 13.56 MHz by Particle in Cell simulations and by an analytical model. The model allows to calculate the electric field outside the electrode sheaths, space and time resolved within the RF period. Electrons are found to be heated by strong ambipolar electric fields outside the sheath during the phase of sheath expansion in addition to classical sheath expansion heating. By tracing individual electrons we also show that ionization is primarily caused by electrons that collide with the expanding sheath edge multiple times during one phase of sheath expansion due to backscattering towards the sheath by collisions. A synergistic combination of these different heating events during one phase of sheath expansion is required to accelerate an electron to energies above the threshold for ionization. The ambipolar electric field outside the sheath is found to be time modulated due to a time modulation of the electron mean energy caused by the presence of sheath expansion heating only during one half of the RF period at a given electrode. This time modulation results in more electron heating than cooling inside the region of high electric field outside the sheath on time average. If an electric field reversal is present during sheath collapse, this time modulation and, thus, the asymmetry between the phases of sheath expansion and collapse will be enhanced. We propose that the ambipolar electron heating should be included in models describing electron heating in capacitive RF plasmas.

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...

We investigate the electron heating dynamics in capacitively coupled radio frequency plasmas driven by customized voltage waveforms and study the effects of modifying this waveform and the secondary electron emission coefficient of the electrodes on the spatio-temporal ionization dynamics by particle-in-cell simulations. We demonstrate that changes in the electron heating dynamics induced by voltage waveform tailoring strongly affect the dc self-bias, the ion flux, i , and the mean ion energy, E i , at the electrodes. The driving voltage waveform is customized by adding N consecutive harmonics (N 4) of 13.56 MHz with specific harmonics' amplitudes and phases. The total voltage amplitude is kept constant, while modifying the number of harmonics and their phases. In an argon plasma, we find a dc self-bias, η, to be generated via the electrical asymmetry effect for N 2. η can be controlled by adjusting the harmonics' phases and is enhanced by adding more consecutive harmonics. At a low pressure of 3 Pa, the discharge is operated in the α-mode and E i can be controlled by adjusting the phases at constant i . The ion flux can be increased by adding more harmonics due to the enhanced electron-sheath heating. E i does not remain constant as a function of N at both electrodes due to a change in η. These findings verify previous results of Lafleur et al. At a high pressure of 100 Pa and using a high secondary electron emission coefficient of γ = 0.4, the discharge is operated in the γ -mode and mode transitions are induced by changing the driving voltage waveform. Due to these mode transitions and the specific ionization dynamics in the γ -mode, i is no longer constant as a function of the harmonics' phases and decreases with increasing N .

We investigate the energy and angular distributions of the ions reaching the electrodes in low-pressure, capacitively coupled oxygen radio-frequency discharges. These distributions, as well as the possibilities of the independent control of the ion flux and the ion energy are analysed for different types of excitation: single-and classical dual-frequency, as well as valleys-and sawtooth-type waveforms. The studies are based on kinetic, particle-based simulations that reveal the physics of these discharges in great details. The conditions cover weakly collisional to highly collisional domains of ion transport via the electrode sheaths. Analytical models are also applied to understand the features of the energy and angular distribution functions.