Many-particle charged-particle plasma simulations using spatial meshes for the electromagnetic field solutions, particle-in-cell (PIC) merged with Monte Carlo collision (MCC) calculations, are coming into wide use for application to partially ionized gases. The author emphasizes the development of PIC computer experiments since the 1950s starting with one-dimensional (1-D) charged-sheet models, the addition of the mesh, and fast direct Poisson equation solvers for 2-D and 3-D. Details are provided for adding the collisions between the charged particles and neutral atoms. The result is many-particle simulations with many of the features met in low-temperature collision plasmas; for example, with applications to plasma-assisted materials processing, but also related to warmer plasmas at the edges of magnetized fusion plasmas. >

Particle-in-Cell Charged Particle Simulations, plus Monte Carlo Collisions with Neutral Atoms, PIC-MCC

A low-frequency capacitively coupled radio-frequency (rf) discharge in Ar excited at 1.76 MHz is studied both experimentally and theoretically. Experimental measurements of electron concentration, discharge voltage and current are presented for a wide range of rf input powers. The rf current shape is nonsinusoidal, close to the triangle one. The evolution of Ar(2p1) emission excitation function in the interelectrode gap during an rf cycle is measured using the phase-resolved optical emission spectroscopy technique. Theoretical study is based on the particle-in-cell Monte Carlo collision numerical simulation. Specific dynamic features of the low-frequency discharge are discussed. The important role of secondary electrons in discharge maintenance and power balance is shown. This study is crucial for understanding dual-frequency discharges with a corresponding value of low frequency.

/pdf/experimental-and-theoretical-study-of-dynamic-effects-in-low-1koxlm5t2w.pdf

Experimental and theoretical study of dynamic effects in low-frequency capacitively coupled discharges

The effects of the simultaneous presence of two different types of plasma asymmetry, viz, geometric and electrical, on low-pressure capacitively coupled argon discharges are studied by 2D3V graphics-processing-unit-based particle-in-cell/Monte Carlo simulations. The geometric asymmetry originates from the different powered vs grounded electrode surface areas, while the electrical asymmetry is established by applying peaks/valleys and sawtooth-up/-down driving voltage waveforms. While in geometrically symmetric discharges, the {peaks ↔ valleys} and the {sawtooth-down ↔ sawtooth-up} switching of the waveforms is equivalent to exchanging the powered and grounded electrodes, this transformation is violated when the geometric symmetry is broken. Under such conditions, the plasma characteristics and the DC self-bias generation behave differently, compared to the geometrically symmetric case. This leads to different sheath dynamics and, therefore, strongly influences the electron power absorption dynamics. For identical peak-to-peak voltages, the plasma density obtained for such tailored voltage waveforms is found to be higher compared to the classical single-frequency waveform case. Reduced plasma densities are found in the valleys- and sawtooth-down waveform cases, compared to the peaks- and sawtooth-up waveforms. By including realistic energy and material-dependent secondary electron emission (SEE) coefficients in the simulations, the electron-induced SEE is found to be reduced in the valleys- and sawtooth-down waveform cases, which explains the behaviour of the plasma density. Using such tailored waveforms in geometrically asymmetric discharges is also found to lead to the formation of different charged particle energy distributions at the boundary surfaces, compared to those in geometrically symmetric plasma sources.

2D particle-in-cell simulations of geometrically asymmetric low-pressure capacitive RF plasmas driven by tailored voltage waveforms

Surface processes in low-pressure capacitive radio frequency discharges driven by tailored voltage waveforms

Low pressure single- or dual-frequency capacitively coupled radio frequency (RF) plasmas are frequently used for high-aspect ratio (HAR) dielectric etching due to their capability to generate vertical ion bombardment of the wafer at high energies. Electrons typically reach the wafer at low energies and with a wide angular distribution during the local sheath collapse. Thus, in contrast to positive ions, electrons cannot propagate deeply into HAR etch features and the bottom as well as the sidewalls of such trenches can charge up positively, while the mask charges negatively. This causes etch stops and distortion of profile shapes. Here, we investigate low pressure, high voltage capacitively coupled RF argon gas discharges by Particle-In-Cell/Monte Carlo collisions simulations and demonstrate that this problem can be solved by Voltage Waveform Tailoring, i.e. the velocity and angular distribution of electrons impacting on the electrodes can be tuned towards high velocities and small angles to the surface-normal, while keeping the energies of the impacting ions high. The applied voltage waveforms consist of a base frequency of 400 kHz with 10 kV amplitude and a series of higher harmonics. A high frequency component at 40 or 60 MHz is used additionally. Square voltage waveforms with different rise-times are examined as well. We show that high fluxes of electrons towards the wafer at normal velocities of up to 2.2 × 107 m s−1 (corresponding to 1.4 keV energy) can be realized.

https://iopscience.iop.org/article/10.1088/1361-6463/abf229/pdf

Control of electron velocity distributions at the wafer by tailored voltage waveforms in capacitively coupled plasmas to compensate surface charging in high-aspect ratio etch features

The ion-induced secondary electron emission coefficient (γ) is a vital parameter in the modeling of low temperature RF plasmas. Often, the value of γ drastically affects the electron power absorption dynamics, the plasma parameters and the quality of the separate control of ion flux and mean ion energy at the electrodes. Experimental results for γ under plasma exposure are difficult to obtain. Therefore, γ is either assumed to be a constant chosen with some uncertainty, or is approximated as a quantity that is a function of the ion energy and cleanliness of the electrode surface. It is hypothesized that these assumptions are not valid for all materials and plasma conditions. In this work, Hagstrum’s theory on Auger emission is suggested as a robust, ab initio model for accurately predicting γ for metal surfaces with a wide range of surface conditions and for a variety of ion species. To demonstrate the effect of the choice of γ on modeling results, we carry out particle-in-cell/Monte Carlo collision simulations of 13.56 MHz, single-frequency argon and helium capacitive discharges. Simulations are run assuming that: (i) γ is a constant, (ii) γ is an energy and surface condition dependent quantity that is independent of the electrode material, and (iii) γ is obtained from the ab initio model for different clean metals. The energy distribution of the emitted electrons resulting from Hagstrum’s theory is also implemented as a uniform, metal dependent distribution with physically accurate energy domain. It is found that this is important for some metals in both helium and argon. Lastly, it is observed that, depending on the assumed surface conditions, the plasma properties change dramatically. Based on these results we conclude that a realistic, material dependent implementation of γ is required to obtain realistic simulation results and that Hagstrum’s model suits this purpose.

https://iopscience.iop.org/article/10.1088/1361-6595/ab094f/pdf

Material dependent modeling of secondary electron emission coefficients and its effects on PIC/MCC simulation results of capacitive RF plasmas

The particle-in-cell/Monte Carlo collisions (PIC/MCC) simulation approach has become a standard and well-established tool in studies of capacitively coupled radio frequency (RF) plasmas. While code-to-code benchmarks have been performed in some cases, systematic experimental validations of such simulations are rare. In this work, a multi-diagnostic experimental validation of 1d3v electrostatic PIC/MCC simulation results is performed in argon gas at pressures ranging from 1 Pa to 100 Pa and at RF (13.56 MHz) voltage amplitudes between 150 V and 350 V using a custom built geometrically symmetric reference reactor. The gas temperature, the electron density, the spatio-temporal electron impact excitation dynamics, and the ion flux-energy distribution at the grounded electrode are measured. In the simulations, the gas temperature and the electrode surface coefficients for secondary electron emission and electron reflection are input parameters. Experimentally, the gas temperature is found to increase significantly beyond room temperature as a function of pressure, whereas constant values for the gas temperature are typically assumed in simulations. The computational results are found to be sensitive to the gas temperature and to the choice of surface coefficients, especially at low pressures, at which non-local kinetic effects are prominent. By adjusting these input parameters to specific values, a good quantitative agreement between all measured and computationally obtained plasma parameters is achieved. If the gas temperature is known, surface coefficients for different electrode materials can be determined in this way by computationally assisted diagnostics. The results show, that PIC/MCC simulations can describe experiments correctly, if appropriate values for the gas temperature and surface coefficients are used. Otherwise significant deviations can occur.

https://iopscience.iop.org/article/10.1088/1361-6595/ac2222/pdf

Multi-diagnostic experimental validation of 1d3v PIC/MCC simulations of low pressure capacitive RF plasmas operated in argon

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Material dependent modeling of secondary electron emission coefficients and its effects on PIC/MCC simulation results of capacitive RF plasmas

Multi-diagnostic experimental validation of 1d3v PIC/MCC simulations of low pressure capacitive RF plasmas operated in argon