Development of magnetron sputtering simulator with GPU parallel computing

A particle-in-cell Monte Carlo collision (PIC-MCC) model is presented, parallelized to be suitable for magnetron sputtering simulations on general purpose graphics processing units (GPGPUs). To reduce the large computation time generally required to calculate particle-electromagnetic field interactions, a numerical algorithm is optimized to obtain the best performance with GPGPU parallelization. The efficiency and accuracy of the GPGPU parallelized PIC-MCC model are examined by comparing the calculated results and the corresponding computational time with analytical solutions and the computation time of the serial code, respectively. The erosion and deposition rates during sputtering predicted using this model are in good agreement with experimental results. The newly developed PIC-MCC model with GPGPU parallelization is thereby shown to be both efficient and accurate.

The particle-in-cell Monte Carlo collision (PIC-MCC) model is an essential way to investigate the kinetic behaviors of low-temperature plasmas, but usually, it is hugely time consuming in simulating atmospheric radio frequency (RF) plasmas. In this study, a deep neural network (DNN) with multiple hidden layers is developed to predict the kinetic discharge characteristics of atmospheric RF plasmas. The results obtained from the PIC-MCC model are used as the training dataset for the DNN, and the well-trained DNN is able to efficiently yield various kinetic behaviors of atmospheric RF discharges with very high precision. The validation of the results predicted by the DNN algorithm is performed by comparing them with the simulation results directly from the PIC-MCC model. Compared with the time-consuming PIC-MCC simulations, the well-trained DNN takes only 0.01 s to yield the essential kinetic characteristics of atmospheric RF discharges, which saves seven orders of magnitude of computation time compared with the traditional PIC-MCC simulation. The predicted data show that the discharge current of atmospheric RF discharges increases monotonically with the driving frequency at a given applied voltage, and as the driving frequency increases, the electric field in the sheath region is strongly enhanced with the sheath region shrinks and the bulk plasma region expanding. Additionally, the electron energy distribution function (EEDF) can be accurately predicted by DNN; moreover, as the driving frequency increases, the low-energy electrons can be transformed into medium-energy electrons, leading to a transition of the EEDF from a three-temperature distribution to a Maxwellian distribution from the DNN prediction. The study indicates that the well-trained DNN is a promising tool for plasma simulation with very high efficiency and accuracy compared with the PIC-MCC model with huge computational costs, which could provide enough kinetic results to further understand the discharge behaviors in atmospheric RF discharges.

Special Issue on Numerical Modelling of Low‐Temperature Plasmas for Various Applications – Part I: Review and Tutorial Papers on Numerical Modelling Approaches

In the past three decades, first principles-based fully kinetic particle-in-cell Monte Carlo collision (PIC/MCC) simulations have been proven to be an important tool for the understanding of the physics of low pressure capacitive discharges. However, there is a long-standing issue that the plasma density determined by PIC/MCC simulations shows quantitative deviations from experimental measurements, even in argon discharges, indicating that certain physics may be missing in previous modeling of the low pressure radio frequency (rf) driven capacitive discharges. In this work, we report that the energetic electron-induced secondary electron emission (SEE) and excited state atoms play an important role in low pressure rf capacitive argon plasma discharges. The ion-induced secondary electrons are accelerated by the high sheath field to strike the opposite electrode and produce a considerable number of secondary electrons that lead to additional ionizing impacts and further increase of the plasma density. Importantly, the presence of excited state species even further enhances the plasma density via excited state neutral and resonant state photon-induced SEE on the electrode surface. The PIC/MCC simulation results show good agreement with the recent experimental measurements in the low pressure range (1–10 Pa) that is commonly used for etching in the semiconductor industry. At the highest pressure (20 Pa) and driving voltage amplitudes 250 and 350 V explored here, the plasma densities from PIC/MCC simulations considering excited state neutrals and resonant photon-induced SEE are quantitatively higher than observed in the experiments, requiring further investigation on high pressure discharges.

In this work, we present a fully kinetic particle-in-cell - Monte Carlo Collision (PIC-MCC) computer model developed to calculate the dynamic electric field inside an ion engine discharge chamber. This model self- consistently tracks primary electrons, secondary electrons, singly charged and doubly charged xenon ions, and xenon neutrals inside the discharge chamber. Both electric and magnetic field effects are included in the particle tracking. The plasma potential results are evaluated at every time step based on the charged particle distribution. This model avoids using an artificial inflated permittivity assumption and also avoids the use of the experimentally measured plasma potential data in the electric field calculations. Instead, a self-similar scaling system is considered. The computer model has been applied to study NASA's Next Generation Xenon Thruster (NEXT) operating at 5.3 kW, 3.1 A beam current, and 1567 V beam voltage. We discuss the recent algorithm development and present preliminary results such as the electric potential maps, the particle number density distributions, and the particle energy distributions from the NEXT ion thruster discharge chamber simulations.

Summary form only given. A large-area vacuum chamber for capacitively coupled plasmas is required in the semiconductor industry because it saves production cost and time. As the wavelength of the external driving waveform decreases to the order of the radius of the vacuum chamber, the electromagnetic effect is significantly observed. It causes plasma non-uniformity, which makes it hard to control plasma processing. The electromagnetic effect for large-area capacitively coupled plasma has been studied in order to resolve non-uniformity of plasma density in CCPs for a wafer size larger than 300 mm in diameter. Computer simulations have been performed for several decades, but most studies have not solved electromagnetic equations directly because a finite difference time domain method (FDTD) requires a small times step, which makes the simulation time very long to reach a steady state. Recently, alternative methods have been introduced to implement the electromagnetic effect in the plasma simulation. S. Rauf et al. solved magnetic vector potential in frequency domain to couple it with a fluid simulation. They observed significant electromagnetic effect in a large-area CCP. In order to access realistic results with minimizing assumption, a two-dimensional particle-in-cell Monte Carlo collision (PIC-MCC) simulation is developed in this study to be coupled with the electromagnetic field solver in a frequency domain. The benchmark and validation of the electromagnetic PIC-MCC code is presented in this study including the comparison with a fluid model. The electromagnetic field solver in the frequency is coupled with the electrostatic PIC-MCC simulation.

In this paper simulation results for the plasma in the NASA's Evolutionary Xenon Thruster (NEXT) ion engine discharge chamber are presented. The unique aspect of these results is that the Poisson equation solution of the electric field is fully coupled with the particle tracking portion of the particle- in-cell Monte Carlo collision (PIC-MCC) model. This means the effects of charged particles on the electric field are accounted for in a precise, detailed manner. This fidelity of a simulation has never been performed for the plasma in the discharge chamber of an ion engine, until now. In the past, the present authors have presented results where the particle tracking portion of the PIC-MCC solution was weakly coupled to the electric field solution. This approximation was made to reduce the computational time from years to weeks. In this work, full coupling is simulated. The reason this full coupling can be performed in weeks of computational time, instead of years, is the self-similar scaling routine used and the convergence routines used. Many results for the plasma in the discharge chamber are presented in this paper including neutral, first ion, second ion, primary electron, and secondary electron number density distributions; as well as electron energy distributions and, of course, the electric potential distribution. These results are presented for the NEXT throttling level TL35. Our simulation results have been validated against experimental plasma measurements made on the laboratory model NEXT ion thruster at the University of Michigan. In addition, self-consistent ion bombardment sputter yield calculations were performed with our PIC-MCC model to compute the erosion profile of the cathode keeper face plate.

High-power impulse magnetron sputtering (HiPIMS) is a new magnetron sputtering technique which can produce high-density plasmas with a high ionization rate and prepare coatings with a good performance such as large density and high adhesion. To obtain stable discharge and universal materials’ ionization rates, a cylindrical cathode is proposed based on the hollow cathode effect. However, the unusual plasma transport results in a large loss of ions and a low deposition rate. To solve these problems, an expanding electromagnetic field is proposed to control the plasma transport in this work. The particle in cell/Monte Carlo collision (PIC/MCC) method and the plasma diffusion model are used to simulate the plasma transport in and out of the cylindrical cathode with different currents in the electromagnetic coils, respectively. The simulation results reveal that different electromagnetic fields can achieve different plasma density distributions, resulting in different accumulated positions and different diffusion paths. When the coil current is positive, the resistance to axial motion of electrons is small but the resistance to radial motion is large, so that the hollow cathode effect is weakened and the plasma beam tends to output uniformly. When the coil current is negative, the resistance to axial motion of electrons is large but the resistance to radial motion is small, so that the hollow cathode effect is enhanced and the plasma tends to gather on the central axis and then diffuses outward. To verify the simulation results, Ar/Cr HiPIMS discharge experiments are carried out with the cylindrical cathode in a homemade vacuum system. The experiment results indicate that the threshold voltage, the plasma flow shape, the optical emission spectrum (OES) intensity, and the deposition distribution are determined by the electromagnetic coil current. The variation tendency is in coincidence with the prediction of the simulation. Consequently, by adding an expanding electromagnetic field, the plasma discharge in the cylindrical cathode can be easily controlled and the deposition rate is greatly enhanced. This electromagnetic control strategy not only realizes the enhancement and effective control of plasma, but also improves the homogeneity and the deposition rate of the coatings, thus laying a foundation for the industrial application of HiPIMS.

This paper presents the status of three techniques that have been studied for the purpose of reducing the computational time required by a fully coupled, particle tracking and electric field, PIC-MCC (particle-in-cell Monte Carlo collision) computer model that simulates the plasma in the discharge chamber of an ion engine. The three techniques discussed in this paper are a particle fragmentation and merging technique, a semi-implicit multiple Poisson solves technique, and a semi-implicit fourth order electric field technique. While the particle fragmentation and merging technique successfully shows lower computational times, the semi-implicit techniques developed by the authors have not shown computational time improvements. The multiple Poisson solves technique does show increased stability over an explicit PIC-MCC technique for larger time step sizes.

Numerical simulation of plasma etching process with AAO mask based on PIC/MCC model

Particle-in-cell Monte Carlo collision (PIC/MCC) simulations are important for understanding low pressure low-temperature discharge dynamics, since no assumptions are needed to determine the electron energy distribution function. Benchmark work is needed to build a solid base in the correctness of PIC/MCC codes. In this work, the basic object-oriented plasma device PIC/MCC code (oopd1-v1) is strictly benchmarked against the well-established xpdp1 code over a wide range of pressures (0.03 - 15 Torr), and varying size of the blocking capacitor in the external circuit (5 – 10 5 nF). An excellent agreement between codes is obtained. By upgrading oopd1-v1, incorporating excited argon atoms, metastable and radiative Ar m , Ar r , and Ar(4p), their role in rf capacitive discharges, especially in the collisional regime, was explored. It is found that the presence of the metastable atoms Ar m enhances the plasma density by a factor of 3 - 5 at 1.6 Torr and even higher at pressure up to 5 Torr. The proportion of metastable pooling ionization and step-wise ionization as an ionization source increases with increasing pressure and becomes more important than direct electronneutral ionization at 5 – 15 Torr. Also. as the pressure is increased the ionization occurs near the plasma-sheath interfaces, with little ionization within the plasma bulk region.

The electron energy distribution function (EEDF), predicted by the Boltzmann equation solver BOLSIG+ based on the two-term approximation, is introduced into the fluid model for simulating the high-power microwave (HPM) breakdown in argon, nitrogen, and air, and its validity is examined by comparing with the results of particle-in-cell Monte Carlo collision (PIC/MCC) simulations as well as the experimental data. Numerical results show that, the breakdown time of the fluid model with the Maxwellian EEDF matches that of the PIC/MCC simulations in nitrogen; however, in argon under high pressures, the results from the Maxwellian EEDF were poor. This is due to an overestimation of the energy tail of the Maxwellian EEDF in argon breakdown. The prediction of the fluid model with the BOLSIG+ EEDF, however, agrees very well with the PIC/MCC prediction in nitrogen and argon over a wide range of pressures. The accuracy of the fluid model with the BOLSIG+ EEDF is also verified by the experimental results of the air breakdown.

In this paper, the effects of macroparticle (MP) weighting on the plasma discharge, particularly near the centerline, are investigated using a two-dimensional axisymmetric particle-in-cell Monte Carlo collision (PIC/MCC) model. A variable MP weight according to the radial position of the MPs is employed to maintain sufficient number of MPs near the centerline of the plasma source. The plasma density obtained from the PIC/MCC simulations for low-pressure (25–100 mTorr) capacitively coupled plasmas is found to be artificially large when the MP weight near the centerline is not well resolved, demonstrating the need for particle convergence studies for axisymmetric PIC/MCC simulations.

Nanosecond pulsed atmospheric pressure discharge becomes an active research topic because of its promising prospect of applications in many areas. To understand its dynamics, the discharge process was studied by one-dimensional implicit particle-in-cell Monte Carlo collision (PIC-MCC) simulation coupled with a particle renormalization algorithm, in which multiple electron Monte Carlo collisions per step were considered to improve the computation efficiency. In the simulation, the effects of discharge conditions such as plateau voltage, pulse rise time, and initial charged particle density were investigated. It is found that the plateau voltage in the pulse waveform is a major factor controlling the final charged particle density in the plasma bulk, and the built-up time and steady thickness of cathode sheath are proportional to the pulse rise time, whereas reversely proportional to the initial charged particle density.

A parallel 1d3v Particle in Cell/Monte Carlo Collision (PIC/MCC) code was derived and applied for the investigation of the formation of photoplasma in sodium vapor. The effects of particle weighting and the Courant number on the computed plasma properties were examined, and the convergence of the numerical method with respect to these parameters was demonstrated. Simulations were carried out for the stepwise spatial profile of the resonant sodium atoms density. The basic plasma parameters (such as eedf, iedf, atomic and molecular ion and electron densities, and electric field and potential) were computed. The results of the PIC/MCC simulations were compared to those obtained from the fluid model. Simulations revealed a strong spatial non-uniformity in the electron density and the electric potential over the computational domain that provides evidence in favour of photovoltaic conversion of light energy into electrical energy.