Denis Eremin

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.

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.

Simulations of electromagnetic effects in high-frequency capacitively coupled discharges using the Darwin approximation

The Darwin approximation is investigated for its possible use in simulation of electromagnetic effects in large size, high-frequency capacitively coupled discharges. The approximation is utilized within the framework of two different fluid models which are applied to typical cases showing pronounced standing wave and skin effects. With the first model it is demonstrated that the Darwin approximation is valid for treatment of such effects in the range of parameters under consideration. The second approach, a reduced nonlinear Darwin approximation-based model, shows that the electromagnetic phenomena persist in a more realistic setting. The Darwin approximation offers a simple and efficient way of carrying out electromagnetic simulations as it removes the Courant condition plaguing explicit electromagnetic algorithms and can be implemented as a straightforward modification of electrostatic algorithms. The algorithm described here avoids iterative schemes needed for the divergence cleaning and represents a fast and efficient solver, which can be used in fluid and kinetic models for self-consistent description of technical plasmas exhibiting certain electromagnetic activity.

Ion energy distribution functions behind the sheaths of magnetized and non-magnetized radio frequency discharges

The effect of a magnetic field on the characteristics of capacitively coupled radio frequency discharges is investigated and found to be substantial. A one-dimensional particle-in-cell simulation shows that geometrically symmetric discharges can be asymmetrized by applying a spatially inhomogeneous magnetic field. This effect is similar to the recently discovered electrical asymmetry effect. Both effects act independently, they can work in the same direction or compensate each other. Also the ion energy distribution functions at the electrodes are strongly affected by the magnetic field, although only indirectly. The field influences not the dynamics of the sheath itself but rather its operating conditions, i.e., the ion flux through it and voltage drop across it. To support this interpretation, the particle-in-cell results are compared with the outcome of the recently proposed ensemble-in-spacetime algorithm. Although that scheme resolves only the sheath and neglects magnetization, it is able to reproduce the ion energy distribution functions with very good accuracy, regardless of whether the discharge is magnetized or not.

Fine-sorting one-dimensional particle-in-cell algorithm with Monte-Carlo collisions on a graphics processing unit

Abstract Particle-in-cell (PIC) simulations with Monte-Carlo collisions are used in plasma science to explore a variety of kinetic effects. One major problem is the long run-time of such simulations. Even on modern computer systems, PIC codes take a considerable amount of time for convergence. Most of the computations can be massively parallelized, since particles behave independently of each other within one time step. Current graphics processing units (GPUs) offer an attractive means for execution of the parallelized code. In this contribution we show a one-dimensional PIC code running on NVIDIA ® GPUs using the CUDA ™ environment. A distinctive feature of the code is that size of the cells that the code uses to sort the particles with respect to their coordinates is comparable to size of the grid cells used for discretization of the electric field. Hence, we call the corresponding algorithm “fine-sorting”. Implementation details and optimization of the code are discussed and the speed-up compared to classical CPU approaches is computed.

Kinetic simulation of the sheath dynamics in the intermediate radio frequency regime

The dynamics of temporally modulated plasma boundary sheaths is studied in the intermediate radio frequency regime where the applied radio frequency and the ion plasma frequency (or the reciprocal of the ion transit time) are comparable. Two fully kinetic simulation algorithms are employed and their results are compared. The first is a realization of the well-known particle-in-cell technique with Monte Carlo collisions and simulates the entire discharge, a planar radio frequency capacitively coupled plasma with an additional ionization source. The second code is based on the recently published scheme Ensemble-in-Spacetime (EST); it resolves only the sheath and requires the time-resolved voltage across and the ion flux into the sheath as input. Ion inertia causes a temporal asymmetry (hysteresis) of the charge‐voltage relation; other ion transit time effects are also found. The two algorithms are in good agreement, both with respect to the spatial and temporal dynamics of the sheath and with respect to the ion energy distributions at the electrodes. It is concluded that the EST scheme may serve as an efficient post-processor for fluid or global simulations and for measurements: it can rapidly and accurately calculate ion distribution functions even when no genuine kinetic information is available. (Some figures may appear in colour only in the online journal)

Nonlocal behavior of the excitation rate in highly collisional RF discharges

The present work focuses on the fundamental aspects of atmospheric pressure plasma electropositive discharges operated in the ohmically heated mode, the electron heating and the excitation (ionization) rate. We find that the two do not necessarily have similar profiles and can show peaks at different locations, the ionization rate being much more sensitive to the electric field compared to the sensitivity to the electric field of the electron heating. This suggests an explanation for the discrepancies between the profiles of the power absorbed by electrons and the excitation patterns previously reported in the literature and observed in the present study. The excitation rate profile can then be explained by analyzing overlapping of the electron heating and the electric field profiles. Surprisingly, it has been discovered that the excitation dynamics exhibits nonlocal behavior having maxima spatially separated from the maxima of the electric field and the electron heating rate, a new effect in discharges operated in the mode. The strong electric field in such discharges leads to large displacements of the electron component. This can produce significant charge separation close to the sheath or even in the bulk plasma because electrons are not able to follow the electric field adiabatically and maintain quasineutrality owing to the high collisionality. In particular, this leads to a significant distortion of the sheath structure and increase in the electric field there.

On the physics of a large CCP discharge

Demands of the plasma processing industry gradually lead to an increase in electrode areas and driving frequency of the commonly used capacitively coupled reactors. This brings about new phenomena which differ from the well known physics of smaller capacitively coupled plasma (CCP) devices. In this work we compare experimental data and results of numerical modeling for a large CCP discharge having a GEC cell-like geometry currently studied in context of a possible use as a sputtering device. Using an electrostatic implicit particle-in-cell code with Monte-Carlo collisions (PIC/MCC), we have been capable of reproducing all main features of the experimental discharges, which have strong relevance for the processing applications, such as the plasma uniformity and the self-bias. The side chamber proves to play an essential role in defining the physics of the whole device, featuring substantial production of plasma particles and participating in establishing the self-bias due to the telegraph effect observed for higher frequencies.

Kinetic interpretation of resonance phenomena in low pressure capacitively coupled radio frequency 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.

Modeling of Resonant Surface Wave Excitation in a Large CCP Reactor

Low-pressure plasmas produced in capacitively coupled plasma (CCP) reactors are used extensively in the plasma processing industry. Although the physics of small scale CCP reactors operated under low pressure is fairly well understood, larger scale CCP reactors constructed to meet the industrial needs often exhibit surprising behavior in experiments, which affects discharge parameters of importance to the industrial applications. This can be explained by the resonant excitation of surface modes that a CCP reactor can support. Limitations of the drift-diffusion approximation in description of the surface mode phenomena in low-pressure plasmas are discussed. It is shown that a particular class of the surface modes can be adequately modeled using the electrostatic approximation if the finite electron inertia is taken into account. Continuing a previous work [1], the present study demonstrates how such modes are triggered and how they affect the radial plasma density profile uniformity. This is done by using a self-consistent 2d3v electrostatic implicit PIC/MCC code. It is shown that in highly collisional plasmas, the nonuniformities caused by the modes disappear due to the large mode damping.