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We reproduce the consistently-seen experimental voltage vs pressure (V–p) dependence of DC magnetron sputtering with two-dimensional-RZ particle-in-cell (PIC) simulation. Informed by PIC simulation, we develop a steady-state, 1D-axial fluid model of the sheath and presheath that also reproduces this V–p dependence. The V–p dependence is the relationship between the steady-state voltage needed to maintain a constant discharge current and the neutral gas pressure. V–p dependence is fundamental to device performance, but has not previously been reproduced with simulation or satisfactorily explained. We find that the decrease in voltage with increasing pressure is not due to electron recapture at the cathode. Rather, the constant current dictates a constant global ionization rate, so the voltage decrease compensates for the increase in neutral gas density by lowering the energy of the plasma electrons, which decreases their ionization probability. The PIC simulations also reveal that the presheath and bulk plasma are unaffected by the electron reflection coefficient at the cathode; the only effect of increasing reflection is a reduction in the sheath voltage and width. In addition to the potential structure, we explore how pressure affects the plasma density, particle drifts, and particle energy distributions. 

When the Debye length is not resolved in a simulation using the most common particle-in-cell (PIC) algorithm, the plasma will unphysically heat until the Debye length becomes resolved via a phenomenon known as grid heating. This paper presents detailed numerical measurements of grid heating for several explicit PIC algorithms, including the first systematic (covering the Debye length resolution and drift-velocity parameter space) study of grid-heating growth rates for the most common electrostatic momentum-conserving PIC algorithm. Additionally, we derive and test a cubic-spline-based PIC algorithm that ensures that the interpolated electric field has a continuous first derivative but find that a differentiable electric field has minimal impact on grid-heating stability. We also considered energy-conserving PIC algorithms with linear and quadratic interpolation functions. In all cases, we find that unphysical heating can occur for some combinations of Debye under-resolution and plasma drift. We demonstrate analytically and numerically that grid heating cannot be eliminated by using a higher-order field solve and give an analytical expression for the cold-beam stability limits of some energy-conserving algorithms. 

Turbulent, relativistic non-thermal plasmas are ubiquitous in high-energy astrophysical systems, as inferred from broad-band non-thermal emission spectra. The underlying turbulent non-thermal particle acceleration (NTPA) processes have traditionally been modelled with a Fokker–Planck (FP) diffusion–advection equation for the particle energy distribution. We test FP-type NTPA theories by performing and analysing particle-in-cell simulations of turbulence in collisionless relativistic pair plasma. By tracking large numbers of particles in simulations with different initial magnetization and system size, we first test and confirm the applicability of the FP framework. We then measure the FP energy diffusion (D) and advection (A) coefficients as functions of particle energy $$\gamma m c^2$$, and compare their dependence to theoretical predictions. At high energies, we robustly find $$D \sim \gamma ^2$$ for all cases. Hence, we fit $$D = D_0 \gamma ^2$$ and find a scaling consistent with $$D_0 \sim \sigma ^{3/2}$$ at low instantaneous magnetization $$\sigma (t)$$, flattening to $$D_0 \sim \sigma$$ at higher $$\sigma \sim 1$$. We also find that the power-law index $$\alpha (t)$$ of the particle energy distribution converges exponentially in time. We build and test an analytic model connecting the FP coefficients and $$\alpha (t)$$, predicting $$A(\gamma) \sim \gamma \log \gamma$$. We confirm this functional form in our measurements of $$A(\gamma ,t)$$, which allows us to predict $$\alpha (t)$$ through the model relations. Our results suggest that the basic second-order Fermi acceleration model, which predicts $$D_0 \sim \sigma$$, may not be a complete description of NTPA in turbulent plasmas. These findings encourage further application of tracked particles and FP coefficients as a diagnostic in kinetic simulations of various astrophysically relevant plasma processes like collisionless shocks and magnetic reconnection. 

We present results from particle-in-cell (PIC) simulations of direct current magnetron sputtering (dcMS) in a 2D cylindrically symmetric geometry. The PIC model assumes an electrostatic approximation and includes the Monte Carlo collision method to model collisions between electrons and the neutral gas. A newly implemented explicit energy-conserving PIC algorithm (EC-PIC) is also exercised by the model, and results are compared with the standard momentum-conserving PIC (MC-PIC) method. We use these simulation tools to examine how changes in ion-induced secondary electron yield (SEY) and the external circuit impact the steady-state current, voltage, and plasma density of dcMS discharges. We show that in general, higher ion-induced SEY and lower external resistance values lead to larger currents, smaller voltages, and larger plasma densities. Simulation results presented in this paper related to the ion-induced SEY demonstrate similar trends that have been observed in previous experimental work and theory. Finally, we demonstrate that EC-PIC maintains numerical stability up to cell sizes as large as ten times the electron Debye length. While we have not performed a comprehensive stability study of MC-PIC, this paper demonstrates improved stability over the standard practice often assumed in MC-PIC. We therefore demonstrate that EC-PIC allows for modeling a wide range of plasma currents and densities using modest computational resources compared with PIC models that require resolution of the electron Debye length. 

We present the largest 3D particle-in-cell shearing-box simulations of turbulence driven by the magnetorotational instability, for the first time employing the realistic proton-to-electron mass ratio. We investigate the energy partition between relativistically hot electrons and subrelativistic ions in turbulent accreting plasma, a regime relevant to collisionless, radiatively inefficient accretion flows around supermassive black holes such as those targeted by the Event Horizon Telescope. We provide a simple empirical formula to describe the measured heating ratio between ions and electrons, which can be used for more accurate global modeling of accretion flows with standard fluid approaches such as general-relativistic magnetohydrodynamics. 

Abstract Using 3D particle-in-cell simulation, we characterize energy conversion, as a function of guide magnetic field, in a thin current sheet in semirelativistic plasma, with relativistic electrons and subrelativistic protons. There, magnetic reconnection, the drift-kink instability (DKI), and the flux-rope kink instability all compete and interact in their nonlinear stages to convert magnetic energy to plasma energy. We compare fully 3D simulations with 2D in two different planes to isolate reconnection and DKI effects. In zero guide field, these processes yield distinct energy conversion signatures: ions gain more energy than electrons in 2Dxy(reconnection), while the opposite is true in 2Dyz(DKI), and the 3D result falls in between. The flux-rope instability, which occurs only in 3D, allows more magnetic energy to be released than in 2D, but the rate of energy conversion in 3D tends to be lower. Increasing the guide magnetic field strongly suppresses DKI, and in all cases slows and reduces the overall amount of energy conversion; it also favors electron energization through a process by which energy is first stored in the motional electric field of flux ropes before energizing particles. Understanding the evolution of the energy partition thus provides insight into the role of various plasma processes, and is important for modeling radiation from astrophysical sources such as accreting black holes and their jets. 

Abstract The magnetorotational instability (MRI) is a fundamental mechanism determining the macroscopic dynamics of astrophysical accretion disks. In collisionless accretion flows around supermassive black holes, MRI-driven plasma turbulence cascading to microscopic (i.e., kinetic) scales can result in enhanced angular-momentum transport and redistribution, nonthermal particle acceleration, and a two-temperature state where electrons and ions are heated unequally. However, this microscopic physics cannot be captured with standard magnetohydrodynamic (MHD) approaches typically employed to study the MRI. In this work, we explore the nonlinear development of MRI turbulence in a pair plasma, employing fully kinetic particle-in-cell (PIC) simulations in two and three dimensions. First, we thoroughly study the axisymmetric MRI with 2D simulations, explaining how and why the 2D geometry produces results that differ substantially from 3D MHD expectations. We then perform the largest (to date) 3D simulations, for which we employ a novel shearing-box approach, demonstrating that 3D PIC models can reproduce the mesoscale (i.e., MHD) MRI dynamics in sufficiently large runs. With our fully kinetic simulations, we are able to describe the nonthermal particle acceleration and angular-momentum transport driven by the collisionless MRI. Since these microscopic processes ultimately lead to the emission of potentially measurable radiation in accreting plasmas, our work is of prime importance to understand current and future observations from first principles, beyond the limitations imposed by fluid (MHD) models. While in this first study we focus on pair plasmas for simplicity, our results represent an essential step toward designing more realistic electron–ion simulations, on which we will focus in future work. 

Magnetic reconnection, a plasma process converting magnetic energy to particle kinetic energy, is often invoked to explain magnetic energy releases powering high-energy flares in astrophysical sources including pulsar wind nebulae and black hole jets. Reconnection is usually seen as the (essentially two-dimensional) nonlinear evolution of the tearing instability disrupting a thin current sheet. To test how this process operates in three dimensions, we conduct a comprehensive particle-in-cell simulation study comparing two- and three-dimensional evolution of long, thin current sheets in moderately magnetized, collisionless, relativistically hot electron–positron plasma, and find dramatic differences. We first systematically characterize this process in two dimensions, where classic, hierarchical plasmoid-chain reconnection determines energy release, and explore a wide range of initial configurations, guide magnetic field strengths and system sizes. We then show that three-dimensional (3-D) simulations of similar configurations exhibit a diversity of behaviours, including some where energy release is determined by the nonlinear relativistic drift-kink instability. Thus, 3-D current sheet evolution is not always fundamentally classical reconnection with perturbing 3-D effects but, rather, a complex interplay of multiple linear and nonlinear instabilities whose relative importance depends sensitively on the ambient plasma, minor configuration details and even stochastic events. It often yields slower but longer-lasting and ultimately greater magnetic energy release than in two dimensions. Intriguingly, non-thermal particle acceleration is astonishingly robust, depending on the upstream magnetization and guide field, but otherwise yielding similar particle energy spectra in two and three dimensions. Although the variety of underlying current sheet behaviours is interesting, the similarities in overall energy release and particle spectra may be more remarkable. 

Abstract Relativistic magnetized jets, such as those from AGN, GRBs, and XRBs, are susceptible to current- and pressure-driven MHD instabilities that can lead to particle acceleration and nonthermal radiation. Here, we investigate the development of these instabilities through 3D kinetic simulations of cylindrically symmetric equilibria involving toroidal magnetic fields with electron–positron pair plasma. Generalizing recent treatments by Alves et al. and Davelaar et al., we consider a range of initial structures in which the force due to toroidal magnetic field is balanced by a combination of forces due to axial magnetic field and gas pressure. We argue that the particle energy limit identified by Alves et al. is due to the finite duration of the fast magnetic dissipation phase. We find a rather minor role of electric fields parallel to the local magnetic fields in particle acceleration. In all investigated cases, a kink mode arises in the central core region with a growth timescale consistent with the predictions of linearized MHD models. In the case of a gas-pressure-balanced (Z-pinch) profile, we identify a weak local pinch mode well outside the jet core. We argue that pressure-driven modes are important for relativistic jets, in regions where sufficient gas pressure is produced by other dissipation mechanisms.