arXiv subjects
Publications and source records attributed to Dmytro Sydorenko.
Two-dimensional particle-in-cell simulations of an inductively coupled plasma (ICP) are used to investigate the influence of radio-frequency (RF) current waveform and frequency on plasma characteristics, collision processes, and the electron velocity distribution function driven by asymmetric triangular waveform current.
Low-temperature plasmas are essential to both fundamental scientific research and critical industrial applications. As in many areas of science, numerical simulations have become a vital tool for uncovering new physical phenomena and guiding technological development. Code benchmarking remains crucial for verifying implementations and evaluating performance. This work continues the Landmark benchmark initiative, a series specifically designed to support the verification of low-temperature plasma codes. In this study, seventeen simulation codes from a collaborative community of nineteen international institutions modeled a partially magnetized ExB Penning discharge. The emergence of large scale coherent structures, or rotating plasma spokes, endows this configuration with an enormous range of time scales, making it particularly challenging to simulate. The codes showed excellent agreement on the rotation frequency of the spoke as well as key plasma properties, including time-averaged ion density, plasma potential, and electron temperature profiles. Achieving this level of agreement came with challenges, and we share lessons learned on how to conduct future benchmarking campaigns. Comparing code implementations, computational hardware, and simulation runtimes also revealed interesting trends, which are summarized with the aim of guiding future plasma simulation software development.
We perform extensive 2D Particle-In-Cell (PIC) electromagnetic simulations of low pressure Inductively Coupled Plasma (ICP) discharges with various coil current and driving frequencies. Our simulations show that in low-frequency cases, electrons in the skin region near the coil can be predominantly magnetized by the Radio Frequency (RF) magnetic field. More specifically, the electrons are trapped in the combined potential well formed by the vector and electrostatic potentials, where they oscillate for most of the RF period while drifting perpendicular to the RF magnetic field. When the magnetic field weakens, electrons shortly demagnetize, leading to jet-like currents and periodic bursts of energy deposition. Based on the newly discovered electron trajectories, we develop a new kinetic theory for the plasma skin effect in low-frequency Inductively Coupled Plasma (ICP) discharges, incorporating nonlinear electron motion in an RF magnetic field by integrating Vlasov equation along the unperturbed particle trajectory. This theory successfully predicts the time evolution of electron currents in low-frequency ICP plasmas, as well as a nonlinear relation between electron current and the RF inductive electric field in the new regime we found. Furthermore, by coupling this new kinetic theory with a global model, we provide a straightforward method for estimating equilibrium electron temperature, plasma density and electron current. These analytical predictions match well with our 2D PIC simulations and can be validated through future experimental studies.
Two-dimensional, electromagnetic particle-in-cell simulations are employed to study particle kinetics and power deposition in the skin layer when a Radio Frequency (RF) electromagnetic field penetrates into a background plasma. We identify a new regime at low frequency ($\sim\mathrm{MHz}$) and low pressure, where the motion of electrons can be highly nonlinear in the skin region. Through most of the RF cycle, the electrons are trapped in the effective potential formed by the vector and electrostatic potentials, with energy deposition being small and magnetic moment $μ$ no longer being an adiabatic invariant. However, for a brief period around the null of the oscillating magnetic field, the electrons get detrapped, causing a jet-like current penetrating into the bulk plasma. During these brief periods, the power deposition becomes high, exhibiting a periodic burst nature. Based on kinetic theory, we provide analytical expressions for the plasma current and energy deposition in the new regime. A criterion for transition between the newly identified low-frequency, periodic-burst regime and the usual anomalous non-local skin effect regime is proposed and verified.
A two-dimensional particle-in-cell code for simulation of low-frequency electromagnetic processes in laboratory plasmas has been developed. The code uses the Darwin method omitting the electromagnetic wave propagation. The Darwin method separates the electric field into solenoidal and irrotational parts. The irrotational electric field is the electrostatic field calculated with the direct implicit algorithm. The solenoidal electric field is calculated with a new algorithm based on the equation for the electric field vorticity. The new algorithm is faster and more reliable than the Streamlined Darwin Field Formulation introduced decades ago. The system of linear equations in the new algorithm is solved using a standard iterative method. The code is applied to simulate an inductively coupled plasma with the driving current flowing around the plasma region.
The effect of a uniform magnetic field on particle transport in low-pressure radio frequency (RF) capacitively coupled plasma (CCP) has been studied using a particle-in-cell (PIC) model. Three distinct regimes of plasma behavior can be identified as a function of the magnetic field. In the first regime at low magnetic fields, asymmetric plasma profiles are observed within the CCP chamber due to the effect of E x B drift. As the magnetic field increases, instabilities develop and form self-organized spoke-shaped structures that are distinctly seen within the bulk plasma closer to the sheath. In this second regime, the spoke-shaped coherent structures rotate inside the plasma chamber in the - E x B direction, where E and B are the DC electric and magnetic field vectors, respectively, and the DC electric field exists in the sheath and pre-sheath regions. The spoke rotation frequency is in the MHz range. As the magnetic field strength increases further, the rotating coherent spokes continue to exist near the sheath. The coherent structures are, however, accompanied by new small-scale incoherent structures originating and moving within the bulk plasma region away from the sheath. This is the third regime of plasma behavior. The threshold values of the magnetic field between these regimes were found not to vary with changing plasma reactor geometry (e.g., area ratio between ground and powered electrodes) or the use of an external capacitor between the RF-powered electrode and the RF source. The threshold values of the magnetic field between these regimes shift toward higher values with increasing gas pressure. This paper provides guidance on the upper limit of the magnetic field for instability-free operation in low-pressure CCP-based semiconductor deposition and etch systems that use the external magnetic field for plasma uniformity control.
The process of numerical thermalization in particle-in-cell (PIC) simulations has been studied extensively. It is analogous to Coulomb collisions in real plasmas, causing particle velocity distributions (VDFs) to evolve towards a Maxwellian as macroparticles experience polarization drag and resonantly interact with the fluctuation spectrum. This paper presents a practical tutorial on the effects of numerical thermalization in 2D PIC applications. Scenarios of interest include simulations which must be run for many thousands of plasma periods and contain a population of cold electrons that leave the simulation space very slowly. This is particularly relevant to many low temperature plasma discharges and materials processing applications. We present numerical drag and diffusion coefficients and their associated timescales for a variety of grid resolutions, discussing the circumstances under which the electron VDF is modified by numerical thermalization. Though the effects described here have been known for many decades, direct comparison of analytically derived, velocity-dependent numerical relaxation timescales to those of other relevant processes has not often been applied in practice due to complications that arise in calculating thermalization rates in 1D simulations. Using these comparisons, we estimate the impact of numerical thermalization in several example low temperature plasma applications including capacitively coupled plasma (CCP) discharges, inductively coupled plasma (ICP) discharges, beam plasmas, and hollow cathode discharges. Finally, we discuss possible strategies for mitigating numerical relaxation effects in 2D PIC simulations.
Achieving large-scale kinetic modelling is a crucial task for the development and optimization of modern plasma devices. With the trend of decreasing pressure in applications such as plasma etching, kinetic simulations are necessary to self-consistently capture the particle dynamics. The standard, explicit, electrostatic, momentum-conserving Particle-In-Cell method suffers from restrictive stability constraints on spatial cell size and temporal time step, requiring resolution of the electron Debye length and electron plasma period respectively. This results in a very high computational cost, making the technique prohibitive for large volume device modeling. We investigate the Direct Implicit algorithm and the explicit Energy Conserving algorithm as alternatives to the standard approach, both of which can reduce computational cost with a minimal (or controllable) impact on results. These algorithms are implemented into the well-tested EDIPIC-2D and LTP-PIC codes, and their performance is evaluated via 2D capacitively coupled plasma discharge simulations. The investigation revels that both approaches enable the utilization of cell sizes larger than the Debye length, resulting in reduced runtime, while incurring only minor inaccuracies in plasma parameters. The Direct Implicit method also allows for time steps larger than the electron plasma period, however care must be taken to avoid numerical heating or cooling. It is demonstrated that by appropriately adjusting the ratio of cell size to time step, it is possible to mitigate this effect to an acceptable level.
Electron-beam plasma interaction has long been a topic of great interest. Despite the success of Quasi-Linear (QL) theory and Weak Turbulence (WT) theory, their validities are limited by the requirement of sufficiently dense mode spectrum and small wave amplitude. In this paper, we extensively studied the collective processes of a mono-energetic electron beam emitted from a thermionic cathode propagating through a cold plasma by performing a large number of high resolution two-dimensional (2D) particle-in-cell (PIC) simulations and using analytical theories. We confirm that the initial stage of two-stream instability is saturated due to well-known wave-trapping mechanism. Further evolution occurs due to strong wave-wave nonlinear processes. We show that the beam-plasma interaction can be classified into four different physical regimes in the parameter space for the plasma and beam parameters. The differences between the different regimes are analyzed in detail. For the first time, we identified a new regime in strong Langmuir turbulence featured by what we call Electron Modulational Instability (EMI) that could create a local Langmuir wave packet growing faster than the ion plasma frequency. Ions do not have time to respond to EMI in the initial growing stage. On a longer timescale, the action of the ponderomotive force produces very strong ion density perturbations, and eventually the beam-plasma wave interaction stops being resonant due to strong ion density perturbations. Consequently, in this EMI regime, electron beam-plasma interaction occurs in a periodic (intermittent) process. The beams are strongly scattered by waves, and the Langmuir wave spectrum is significantly broadened, which in turn gives rise to strong heating of bulk electrons. A resulting kappa distribution and a wave-energy spectrum, E^2 (k)~k^(-5), are observed in the strong turbulent regime.
We study collective processes for an electron beam propagating through a background plasma using simulations and analytical theory. A new regime where the instability of a Langmuir wave packet can grow locally much faster than ion frequency is clearly identified. The key feature of this new regime is an Electron Modulational Instability that rapidly creates a local Langmuir wave packet, which in its turn produces local charge separation and strong ion density perturbations because of the action of the ponderomotive force, such that the beam-plasma wave interaction stops being resonant. Three evolution stages of the process and observed periodic burst features are discussed. Different physical regimes in the plasma and beam parameter space are demonstrated for the first time.
The Electron Cyclotron Drift Instability (ECDI) driven by the electron $E\times B$ drift in partially magnetized plasmas is investigated with highly resolved particle-in-cell simulations. The emphasis is on two-dimensional effects involving the parallel dynamics along the magnetic field in a finite length plasma with dielectric walls. It is found that the instability develops as a sequence of growing cyclotron harmonics demonstrating wave breaking and complex nonlinear interactions, being particularly pronounced in ion density fluctuations at short wavelengths. At the same time, nonlinear evolution of fluctuations of the ion and electron density, as well as the anomalous electron current, shows cascade toward long wavelengths. Tendency to generate long wavelength components is most clearly observed in the spectra of the electron density and the anomalous current fluctuations. An intense but slowly growing mode with a distinct eigen-mode structure along the magnetic field develops at a later nonlinear stage enhancing the tendency toward long wavelength condensation. The latter mode having a finite wavelength along the magnetic field is identified as the Modified Two-Stream Instability (MTSI). It is shown that the MTSI mode results in strong parallel heating of electrons.
Nonlinear dynamics of the electron-cyclotron instability driven by the electron $\mathbf{E\times B}$ current in crossed electric and magnetic field is studied. In nonlinear regime the instability proceeds by developing a large amplitude coherent wave driven by the energy input from the fundamental cyclotron resonance. Further evolution shows the formation of the long wavelength envelope akin to the modulational instability. Simultaneously, the ion density shows the development of high-k content responsible for wave focusing and sharp peaks on the periodic cnoidal wave structure. It is shown that the anomalous electron transport (along the direction of the applied electric field) is dominated by the long wavelength part of the turbulent spectrum.
The instability of a monoenergetic electron beam in a collisional one-dimensional plasma bounded between grounded walls is considered both analytically and numerically. Collisions between electrons and neutrals are accounted for the plasma electrons only. Solution of a dispersion equation shows that the temporal growth rate of the instability is a decreasing linear function of the collision frequency which becomes zero when the collision frequency is two times the collisionless growth rate. This result is confirmed by fluid simulations. Practical formulas are given for the estimate of the threshold beam current which is required for the two-stream instability to develop for a given system length, neutral gas pressure, plasma density, and beam energy. Particle-in-cell simulations carried out with different neutral densities and beam currents demonstrate good agreement with the fluid theory predictions for both the growth rate and the threshold beam current.