Particle-in-cell Monte Carlo Collision Research Articles

Particle-in-cell simulations of electron beam-plasma discharge in a narrow gap with varying transverse boundary conditions

Abstract In this paper, the effects of transverse boundary conditions, specifically the bias voltage on the transverse wall and the gap width, on electron beam-generated plasmas (EBPs) confined in a narrow gap, are investigated using the particle-in-cell/Monte Carlo collision (PIC/MCC) simulations. Simulation results reveal that the application of bias voltage causes beam deflections, leading to the formation of band structures in the beam electron velocity space. Three branches of electrostatic waves, including electron beam mode, Langmuir wave, and electron acoustic mode, are identified. Increasing the bias voltage and reducing gap width intensify beam deflections, resulting in the suppression of waves. Both wave excitation and beam deflection significantly modify beam electron transport, leading to the plasma non-uniformity. These findings enhance the understanding of beam transport and plasma behavior in discharges confined in a narrow gap.

On the in-situ determination of the effective secondary electron emission coefficient in low pressure capacitively coupled radio frequency discharges based on the electrical asymmetry effect

Abstract We present a method for the in-situ determination of the effective secondary electron emission coefficient (SEEC, γ) in a capacitively coupled plasma (CCP) source based on the γ-dependence of the DC self-bias voltage that develops over the plasma due to the electrical asymmetry effect (EAE). The EAE is established via the simultaneous application of two consecutive radio-frequency harmonics (with a varied phase angle) for the excitation of the discharge. Following the measurement of the DC self-bias voltage experimentally, particle-in-cell/Monte Carlo collision simulations coupled with a diffusion-reaction-radiation code to compute the argon atomic excited level dynamics are conducted with a sequence of SEEC values. The actual γ for the given discharge operating conditions is found by searching for the best match between the experimental and computed values of the DC self-bias voltage. The γ ≈ 0.07 values obtained this way are in agreement with typical literature data for the working gas of argon and the electrode material of stainless steel in the CCP source. The method can be applied for a wider range of conditions, as well as for different electrode materials and gases to reveal the effective SEEC for various physical settings and discharge operating conditions.

Stochastic weighted particle control for electrostatic particle-in-cell Monte Carlo collision simulations in an axisymmetric coordinate system

Stochastic weighted particle control for electrostatic particle-in-cell Monte Carlo collision simulations in an axisymmetric coordinate system

Influence of the edge effect of the solenoid magnetic field on the discharge process of magnetron sputtering in long and narrow chambers

The fourth-generation light source or future particle accelerators have compact vacuum systems, resulting in a small flow conductance of the beam chamber. The development of NEG (Non-Evaporable Getters) films technology with distributed pumping is a good solution. This technology can obtain a pressure distribution with a small pressure gradient and an ultra-high vacuum of 10-8 Pa inside the vacuum chambers. In the magnetron sputtering process for NEG coating, the inhomogeneity of the magnetic field at the edge of the solenoid is a key parameter for affecting the stability of the magnetron sputtering discharge process. The influence mechanism of the edge magnetic field on the plasma discharge process and discharge uniformity was studied to optimize the coating parameters, which is very important. The Particle-In-Cell/Monte Carlo Collision (PIC/MCC) method was employed to simulate plasma discharge in a narrow vacuum chamber under various edge magnetic field conditions. The potential distribution, charged particle density distribution, and particle density distribution on the target were acquired for four sets of discharge parameters, followed by data processing and analysis. The electric field distribution during the discharge process is closely linked to the non-uniformity of the magnetic field at the solenoid's edge. The smaller Br (the radial component of the magnetic field) component of the magnetic field, the more uniform distribution of Ar+ on the target surface, and the smaller the difference along the length of the vacuum chamber. A smaller radial component (Br) of the magnetic field results in a more uniform distribution of Ar+ on the target surface and reduces variations in film thickness along the length of the vacuum chamber. This is attributed to the small distance between the inner wall of the vacuum chamber and the cathode target. The change in the edge magnetic field will result in the rapid migration of electrons to the inner wall of the vacuum chamber and insufficient collision with the discharge gas, making it challenging to sustain the discharge process. Moreover, fluctuations in the edge magnetic field of the solenoid induce gradient drift and curvature drift in charged particles, resulting in rapid electron loss and alterations in the density and potential distribution of charged particles at the ends of the vacuum chamber, ultimately leading to discharge instability. The research results show the importance of the edge magnetic field effect on the magnetron sputtering coating of the slender vacuum chambers, which provides an important reference for the segmented coating process of similar vacuum chambers in future accelerators.

Hysteresis between gas breakdown and plasma discharge

In direct-current (DC) discharge, it is well known that hysteresis is observed between the Townsend (gas breakdown) and glow regimes. Forward and backward voltage sweep is performed using a one-dimensional particle-in-cell Monte Carlo collision (PIC-MCC) model considering a ballast resistor. When increasing the applied voltage after reaching the breakdown voltage (Vb), transition from Townsend to glow discharges is observed. When decreasing the applied voltage from the glow regime, the discharge voltage (Vd) between the anode–cathode gap can be smaller than the breakdown voltage, resulting in a hysteresis, which is consistent with experimental observations. Next, the PIC-MCC model is used to investigate the self-sustaining voltage (Vs) in the presence of finite initial plasma densities between the anode and cathode gap. It is observed that the self-sustaining voltage coincides with the discharge voltage obtained from the backward voltage sweep. In addition, the self-sustaining voltage decreases with increased initial plasma density and saturates above a certain initial plasma density, which indicates a change in plasma resistivity. The decrease in self-sustaining voltage is associated with the electron heat loss at the anode for the low pd (rarefied) regime. In the high pd (collisional) regime, the ion energy loss toward the cathode due to the cathode fall and the inelastic collision loss of electrons in the bulk discharge balance out. Finally, it is demonstrated that the self-sustaining voltage collapses to a singular value, despite the presence of a initial plasma, for microgaps when field emission is dominant, which is also consistent with experimental observations.

The detachment-induced mode in electronegative capacitively coupled radio-frequency plasmas

Insights into the spatio-temporally resolved electron power absorption dynamics in capacitively coupled radio-frequency plasmas are essential for understanding the fundamentals of their operation and as a basis for knowledge-based plasma process development. Similar to the γ-mode, an ionization maximum is observed at the sheath edge around the time of maximum sheath voltage in electronegative oxygen discharges at a pressure of 300 Pa. Based on Particle-in-Cell/Monte Carlo Collisions (PIC/MCC) simulations, we demonstrate that this maximum is not only caused by secondary electrons emitted at the electrode and collisionally multiplied inside the sheath. In fact, it also occurs in the complete absence of secondary electrons in the simulation, and is caused by the generation of O− ions by electron attachment close to the electrode during the local sheath collapse. These negative ions are accelerated towards the plasma bulk by the sheath electric field during sheath expansion. By electron detachment from these negative ions, electrons are generated inside the sheath and are accelerated towards the plasma bulk by the instantaneous sheath electric field—similarly to secondary electrons. Ionization is also observed in the plasma bulk and caused by electrons generated by detachment and accelerated by the high drift-and ambipolar electric fields. This detachment-induced electron power absorption is found to have significant effects on the discharge in the presence and absence of secondary electron emission. Its fundamentals are understood based on an analysis of the spatio-temporal electron and O− power absorption dynamics as well as the trajectory of selected O− ions close to the electrode.

Numerical Analysis of the Breakdown Process of CF3I at Low Pressure

The breakdown of CF3I gas at low pressure is of significant importance for applications in fields such as aerospace and microelectronics. However, the DC low-pressure breakdown characteristics of CF3I remain underexplored. In this work, we utilize a one-dimensional implicit particle-in-cell/Monte Carlo collision (PIC/MCC) algorithm to investigate the complete DC breakdown process of low-pressure CF3I. Our model accounts for ion–molecule collisions, recombination reactions, and external circuit influences. The breakdown process is delineated into three stages: before breakdown, breakdown, and after breakdown. In the before-breakdown stage, both the density and energy of particles are low. In the breakdown stage, the rapid increase in electron density and energy accelerates ionization reactions, leading to successful breakdown. The circuit behavior transitions from capacitive to resistive, sharing voltage with the external resistance. In the after-breakdown stage, continued positive ion growth leads to the formation of a thin anode sheath and a negative plasma potential. Energy production, including heating power and secondary electron emission (SEE) power, balances with energy loss through collision and boundary absorption. Specifically, 62% of the total heating power comes from positive ions, 1.5% from negative ions, and approximately 85% of electron energy is lost via boundary absorption. Finally, we compare the Paschen curves of CF3I with those of SF6, providing insights that are beneficial for the application of CF3I as an SF6 alternative.

Modelling of plasma discharge in a filament negative ion source

Most of the currently operating Negative ion Neutral Beam Injectors (N-NBIs) exploit filament powered sources for the generation of beam ions. Being widely used in fusion experiments, the filament arc technology has been thoroughly investigated and optimized over the years, allowing to achieve excellent performances in terms of extracted beam optics. The source geometry, the magnetic field topology and the arc power strongly influence both the plasma discharge and the background gas properties and, consequently, the beam features. In this framework, this contribution describes a numerical investigation of the plasma properties in a filament-powered negative ion source, performed by means of a 2D3V Particle In Cell-Monte Carlo Collisions (PIC-MCC) code. Specifically, we discuss plasma formation by the thermionic electrons emitted by the filaments, investigating their interaction with the background gas. We also study the plasma diffusion through the magnetic filter field and how the latter modifies plasma density, electron temperature and plasma potential along the axial direction when approaching the plasma facing electrode.

A comparative study on the acceleration techniques for solving finite difference discretization poisson’s equation in the PIC/MCC Method

A wide variety of plasma phenomena have been investigated during the past decades using the particle-in-cell/Monte Carlo collisions (PIC/MCC) method. As an important component of the PIC/MCC method, solving Poisson’s equation is crucial for the accuracy and efficiency of calculations. Different acceleration techniques for solving finite difference discretization Poisson’s equation are investigated and compared, including direct method, iterative method, multigrid (MG) method, parallel computing and inherited initial value. The charge density distribution with a known analytical solution is used to validate the algorithm and code. The optimal relaxation factor for the successive over-relaxation (SOR) method in 2D Poisson’s equation with unequal grid node numbers in different dimensions is derived, which is only related to the dimension with the largest grid number. Although there will be a ‘more optimal’ relaxation factor deviated from in some simulation cases, selecting the optimal relaxation factor derived always leads to a not slow solving speed. However, when SOR is used in MG for smoothing, the optimal relaxation factor will shift to 0.5–1.2 from the theoretical optimal value derived with the increase of MG levels. By comparing the convergence order under different relaxation factors and MG levels, the suitable MG level is proposed as log2[min(N x , N y )]−2. Combining the optimal SOR relaxation factor, MG, parallel computing and inherited initial values, the computational cost may decrease by 5 orders of magnitude than that by the simple Gaussian elimination (GE). Based on the optimal acceleration techniques mentioned above, a benchmark simulation case electron cyclotron drift instability (ECDI) in magnetized plasmas was run to further validate the developed PIC/MCC code. The distributions of electric field in the x-direction, electron density and electron temperature are all consistent with the literatures. This paper provides a reference for the acceleration strategy selection for solving Poisson’s equation quickly in plasma simulations.

Effect of desorbed gas on microwave breakdown on vacuum side of dielectric window

The gas desorbed from the dielectric surface has a great influence on the characteristics of microwave breakdown on the vacuum side of the dielectric window. In this paper, the dielectric surface breakdown is described by using the electromagnetic particle-in-cell-Monte Carlo collision (PIC-MCC) model. The process of desorption of gas and its influence on the breakdown characteristics are studied. The simulation results show that, due to the accumulation of desorbed gas, the pressure near the dielectric surface increases in time, and the breakdown mechanism transitions from secondary electron multipactor to collision ionization. More and more electrons generated by collision ionization drift to the dielectric surface, so that the amplitude of self-organized normal electric field increases in time and sometimes points to the dielectric surface. Nevertheless, the number of secondary electrons emitted in each microwave cycle is approximately equal to the number of primary electrons. In the early and middle stages of breakdown, the attenuation of the microwave electric field near the dielectric surface is very small. However, the collision ionization causes a sharp increase in the number density of electrons, and the microwave electric field decays rapidly in the later stage of breakdown. Compared with the electromagnetic PIC-MCC simulation results, the mean energy and number of electrons obtained by the electrostatic PIC-MCC model are overestimated in the later stage of breakdown because it does not take into account the attenuation of microwave electric field. The pressure of the desorbed gas predicted by the electromagnetic PIC-MCC model is close to the measured value, when the number of gas atoms desorbed by an incident electron is taken as 0.4.

Plasma density enhancement in radio-frequency hollow electrode discharge

The plasma density enhancement outside hollow electrodes in capacitively coupled radio-frequency (RF) discharges is investigated by a two-dimensional (2D) particle-in-cell/Monte-Carlo collision (PIC/MCC) model. Results show that plasma exists inside the cavity when the sheath inside the hollow electrode hole is fully collapsed, which is an essential condition for the plasma density enhancement outside hollow electrodes. In addition, the existence of the electron density peak at the orifice is generated via the hollow cathode effect (HCE), which plays an important role in the density enhancement. It is also found that the radial width of bulk plasma at the orifice affects the magnitude of the density enhancement, and narrow radial plasma bulk width at the orifice is not beneficial to obtain high-density plasma outside hollow electrodes. Higher electron density at the orifice, combined with larger radial plasma bulk width at the orifice, causes higher electron density outside hollow electrodes. The results also imply that the HCE strength inside the cavity cannot be determined by the magnitude of the electron density outside hollow electrodes.

The dynamical behavior of the capacitively coupled argon plasma driven by very high frequency

The dynamical behavior of the capacitively coupled Ar plasma driven by four frequencies (54, 80, 94 and 100[Formula: see text]MHz) at fixed pressure are simulated by the particle-in-cell/Monte-Carlo collisions (PIC/MCC) model based on the electromagnetic field. The magnetic field is generated by plasma current itself. The results show that the electron density, charge density, and electron temperature increase with the frequency increase when radio frequency (RF) power is 40[Formula: see text]W. The electron heating rate and argon ion heating rate near the sheath increase with the frequency increase. The high-energy electron density ([Formula: see text] [Formula: see text]eV) varies nonlinearly with the frequency increase. The high-energy electron density is highest at 80[Formula: see text]MHz among the four frequencies. The electron density, charge density, and high-energy electron density increase with RF power (30, 45, 60, and 75[Formula: see text]W) increase at 80[Formula: see text]MHz. The electron temperature decreases with RF power increase at 80[Formula: see text]MHz. The electron heating rate and argon ion heating rate near the sheath have no change with RF power increase at 80[Formula: see text]MHz. The electron energy probability distributions (EEPF) show that the number of high-energy electrons ([Formula: see text] [Formula: see text]eV) is highest at 80[Formula: see text]MHz. The reason may be the electron resonant heating that occurs in the plasma when the electron cyclotron frequency equals source frequency at 80[Formula: see text]MHz. The electron dynamics of capacitively coupled plasma is important for microelectronics etching and membrane deposition.

A neural-network-based model of radio-frequency hollow cathode discharge characterized using particle-in-cell/Monte Carlo collision simulation

An understanding of the plasma dynamics of radio-frequency (RF) hollow cathode discharges (HCDs) at low to moderate pressures is important due to their wide range of applications. A HCD consists of a hollow cylindrical cavity in the RF-powered cathode separated from a grounded electrode by a dielectric. In RF HCDs, RF sheath heating can play a significant role in plasma production in addition to secondary electrons. In this study, a single hollow cathode hole is modeled using the particle-in-cell/Monte Carlo collision (PIC-MCC) technique at low pressure, where kinetic effects are important. Characterization of a single hollow cathode using PIC-MCC simulation is, however, computationally expensive. For improved computational efficiency, a neural network modeling framework has been developed using the temporal variations of applied RF voltages as input and the electrode current as output. A space-filling design for computational experiments is used, where the variables include the RF voltage at the fundamental frequency, RF voltage at the second harmonic, and their phase difference. The predictions of the electrode current using the trained neural network model compare well with the results of the PIC/MCC simulations, but at a significantly lower computational cost. The neural network model predicts the current very well inside the training domain, and reasonably well even outside the training domain considered in this study.

An experimental and computational study on the ignition process of a pulse modulated dual-RF capacitively coupled plasma operated at various low-frequency voltage amplitudes

The ignition process of a pulse modulated capacitively coupled argon discharge driven simultaneously by two radio frequency voltages [12.5 MHz (high frequency) and 2.5 MHz (low frequency, LF)] is investigated by multifold experimental diagnostics and particle in cell / Monte Carlo collision simulations. In particular, (i) the effects of the LF voltage amplitude measured at the end of the pulse-on period, VL,end , on the spatiotemporal distribution of the electron impact excitation rate determined by phase resolved optical emission spectroscopy, and (ii) the electrical parameters acquired by analyzing the measured waveforms of the plasma current and voltage, are studied. Computed electrical parameters and spatio-temporal excitation maps show a good qualitative agreement with the experimental results. Especially, various breakdown mechanisms are found at different VL,end . At low values of VL,end , the ‘RF-avalanche’ mode (volume process) dominates the electron multiplication process. By increasing VL,end , the ionization caused by the volume electrons is suppressed and the electron loss at the electrodes is enhanced, leading to a delayed ignition. At higher values of VL,end , the avalanche ionization is significantly enhanced by ion-induced secondary electron emission at the electrodes, so that the ignition is significantly advanced.

3D simulations of positive streamers in air in a strong external magnetic field

We study how external magnetic fields from 0 to 40 T influence positive streamers in atmospheric pressure air, using 3D PIC-MCC (particle-in-cell, Monte Carlo collision) simulations. When a magnetic field B is applied perpendicular to the background electric field E , the streamers deflect towards the +B and −B directions which results in a branching into two main channels. With a stronger magnetic field the angle between the branches increases, and for the 40 T case the branches grow almost parallel to the magnetic field. Due to the E×B drift of electrons we also observe a streamer deviation in the opposite −E×B direction, where the minus sign appears because positive streamers propagate opposite to the electron drift velocity. The deviation due to this E×B effect is smaller than the deviation parallel to B . In both cases of B perpendicular and parallel to E , the streamer radius decreases with the magnetic field strength. We relate our observations to the effects of electric and magnetic fields on electron transport and reaction coefficients.

Frequency-dependent electron power absorption mode transitions in capacitively coupled argon-oxygen plasmas

Phase Resolved Optical Emission Spectroscopy (PROES) measurements combined with 1d3v Particle-in-Cell/Monte Carlo Collisions simulations are performed to investigate the excitation dynamics in low-pressure capacitively coupled plasmas (CCPs) in argon-oxygen mixtures. The system used for this study is a geometrically symmetric CCP reactor operated in a 70% Ar-30% O2 mixture at 120 Pa, applying a peak-to-peak voltage of 350 V, with a wide range of driving RF frequencies (2 MHz ⩽f⩽ 15 MHz). The measured and calculated spatio-temporal distributions of the electron impact excitation rates from the Ar ground state to the Ar 2p1 level show good qualitative agreement. The distributions show significant frequency dependence, which is generally considered to be predictive of transitions in the dominant discharge operating mode. Three frequency ranges can be distinguished, showing distinctly different excitation characteristics: (i) in the low frequency range ( f⩽ 3 MHz), excitation is strong at the sheaths and weak in the bulk region; (ii) at intermediate frequencies (3.5 MHz ⩽f⩽ 5 MHz), the excitation rate in the bulk region is enhanced and shows striation formation; (iii) above 6 MHz, excitation in the bulk gradually decreases with increasing frequency. Boltzmann term analysis was performed to quantify the frequency-dependent contributions of the Ohmic and ambipolar terms to the electron power absorption.

Numerical characterization of dual-frequency capacitively coupled plasmas modulated by electron beam injection

The modulated approach of electron beam (EB) injection can achieve favorable parameters for capacitive coupled plasmas (CCP). In this work, a one-dimensional particle-in-cell/Monte Carlo collision (PIC/MCC) model is used to simulate the stable dual-frequency CCP with EB injection. First, when the parameters of EB are kept constant at 0.01 A and 30 eV, the results demonstrate significant enhancements in electron density, self-bias voltage, and ion flux. Furthermore, the electron energy probability function (EEPF) appears to have a transition from a typical bi-Maxwellian distribution to a Maxwellian distribution, and the dominant heating mode shifts from the α-mode to the α-γ-mode. Secondly, when the EB current and energy are all changed, the basic parameters of DF-CCP can be achieved by different modulations. Furthermore, we also discuss the transition of the electron heating mode as the current increases from 0.001 A to 1 A and the energy increases from 10 eV to 490 eV. In particular, we conduct a comparative study among different cases of EB injection. According to these results, the modulation capability of EB injection in DF-CCP is thoroughly investigated, which can greatly benefit atom-scale etching in practical applications.

Study of nanosecond pulsed hollow-cathode electron beam generation and transportation based on PIC-MCC simulation

Study of nanosecond pulsed hollow-cathode electron beam generation and transportation based on PIC-MCC simulation

Hollow cathode effect in radio frequency hollow electrode discharge in argon

Radio frequency capacitively coupled plasma source (RF-CCP) with a hollow electrode can increase the electron density through the hollow cathode effect (HCE), which offers a method to modify the spatial profiles of the plasma density. In this work, the variations of the HCE in one RF period are investigated by using a two-dimensional particle-in-cell/Monte-Carlo collision (PIC/MCC) model. The results show that the sheath electric field, the sheath potential drop, the sheath thickness, the radial plasma bulk width, the electron energy distribution function (EEDF), and the average electron energy in the cavity vary in one RF period. During the hollow electrode sheath’s expansion phase, the secondary electron heating and sheath oscillation heating in the cavity are gradually enhanced, and the frequency of the electron pendular motion in the cavity gradually increases, hence the HCE is gradually enhanced. However, during the hollow electrode sheath’s collapse phase, the secondary electron heating is gradually attenuated. In addition, when interacting with the gradually collapsed hollow electrode sheaths, high-energy plasma bulk electrons in the cavity will lose some energy. Furthermore, the frequency of the electron pendular motion in the cavity gradually decreases. Therefore, during the hollow electrode sheath’s collapse phase, the HCE is gradually attenuated.

First-principles simulation of optical emission spectra for low-pressure argon plasmas and its experimental validation

Various spectral line emissions are often used for the experimental characterization of low-temperature plasmas. For a better understanding of the relation between the plasma characteristics and optical emission spectra, first-principle numerical simulations for low-pressure radio-frequency driven capacitively-coupled plasmas (CCPs) of argon have been performed by coupling one-dimensional particle-in-cell/Monte Carlo collision (PIC/MCC) simulations with a global collisional-radiative model (CRM). The only ionization and excitation mechanisms included in the PIC/MCC simulations of this study are the electron-impact ionization and excitations of the ground-state Ar atoms, as done commonly, whereas the electron-impact ionization of metastable states and other ionization mechanisms are also included in the CRM to account for the optical emission spectra. The PIC/MCC coupled CRM provides the emission spectra, which are then compared with experimental data obtained from the corresponding Ar CCPs with a gas pressure ranging from 2 Pa to 100 Pa. The comparison has shown good agreement for pressures up to about 20 Pa but increasingly notable deviations at higher pressures. The deviation is ascribed to the missing consistency between the PIC/MCC simulations and CRM at higher pressures, where the ionization from the metastable states is more dominant than that from the ground states, indicating a significant change in the electron energy distribution function due to the electron collisions with excited Ar atoms at higher pressures.