E. Kawamura

Ion energy distributions in rf sheaths; review, analysis and simulation

We present a review and analysis of ion energy distributions (IED) arriving at the target of a radio frequency (rf) discharge. We mainly discuss the collisionless regime, which is of great interest to experimentalists and modellers studying high-density discharges in which the sheath is much thinner than in conventional reactive ion etching systems. We assess what has been done so far and determine what factors influence the shape of the IEDs. We also briefly discuss collisional effects on the IEDs. Having determined the important parameters, we perform some particle-in-cell simulations of a collisionless current-driven rf sheath which show that ion modulations in an rf sheath significantly affect the IEDs when ion/rf<1, where ion is the ion transit time and rf is the rf period.

Stochastic heating in single and dual frequency capacitive discharges

Two electron heating mechanisms in capacitive discharges are ohmic heating due to electron-neutral collisions and stochastic heating at the plasma edge due to momentum transfer from high voltage moving sheaths. In this work, the stochastic heating and its dependence on various parameters are determined, focusing on dual frequency discharges in which the sheath motion is driven by a combination of high and low frequency sources. Particle-in-cell (PIC) simulations are used in order to investigate the electron heating. For a uniform fixed-ion discharge in which the ions are held fixed in a uniform density profile, there is no stochastic heating, as expected. For a two-step fixed-ion discharge in which the ions are held fixed in a two-step density profile with bulk density nb and sheath density nsh<nb, the stochastic heating is nearly proportional to (1−nsh∕nb)2. For a self-consistent discharge with mobile ions, the stochastic heating is well described by a “hard wall model” provided that the bulk oscillation...

The effects of nonlinear series resonance on Ohmic and stochastic heating in capacitive discharges

The flow of electron and ion conduction currents across a nonlinear capacitive sheath to the electrode surface self-consistently sets the dc bias voltage across the sheath. We incorporate these currents into a model of a homogeneous capacitive sheath in order to determine the enhancement of the Ohmic and stochastic heating due to self-excitation of the nonlinear series resonance in an asymmetric capacitive discharge. At lower pressures, the series resonance can enhance both the Ohmic and stochastic heating by factors of 2–4, with the Ohmic heating tending to zero as the pressure decreases. The model was checked, for a particular set of parameters, by a particle-in-cell (PIC) simulation using the homogeneous sheath approximation, giving good agreement. With a self-consistent Child-law sheath, the PIC simulation showed increased heating, as expected, whether the series resonance is important or not.

Physical and numerical methods of speeding up particle codes and paralleling as applied to RF discharges

We demonstrate the means, both physical and numerical, for speeding up particle-in-cell (PIC) simulations of RF discharges. These include implicit movers, longer ion timesteps, lighter-mass ions, different weights for electrons and ions, and improved initial density profiles. By using these methods (singly or together) on Ar and O2 RF discharges we were able to achieve speedups of six to 30 times with single-processor machines. In electrostatic 1d3v PIC simulations of RF discharges, the field solve is typically less than 1% of the work load. Even for 2d3v PIC simulations, the field solve can be a small percentage of the work load, especially when fast Fourier transform methods are used to solve the field. Thus, we can obtain significant gains by just paralleling particle processing (e.g., pushing/accumulating) without paralleling the field solve. We applied this simple scheme to conduct 1d3v and 2d3v PIC simulations of Ar RF discharges on two- and four-CPU symmetric multiprocessor machines and on a distributed network of workstations. For a fixed number of grid points, the speedup for this parallel particle processing became more linear with increasing number of particles. The combination of single-processor methods and paralleling makes run times for PIC codes more competitive with other types of codes.

Fast 2D hybrid fluid-analytical simulation of inductive/capacitive discharges

A fast two-dimensional (2D) hybrid fluid-analytical transform coupled plasma reactor model was developed using the finite elements simulation tool COMSOL. Both inductive and capacitive coupling of the source coils to the plasma are included in the model, as well as a capacitive bias option for the wafer electrode. A bulk fluid plasma model, which solves the time-dependent plasma fluid equations for the ion continuity and electron energy balance, is coupled with an analytical sheath model. The vacuum sheath of variable thickness is modeled with a fixed-width sheath of variable dielectric constant. The sheath heating is treated as an incoming heat flux at the plasma–sheath boundary, and a dissipative term is added to the sheath dielectric constant. A gas flow model solves for the steady-state pressure, temperature and velocity of the neutrals. The simulation results, over a range of input powers, are in good agreement with a chlorine reactor experimental study.

Capacitive discharges driven by combined dc/rf sources

We have obtained analytic expressions for the sheath voltages and sheath widths for both collisional and collisionless sheaths driven by a combination of dc and rf voltage sources. A “dc/rf sheath” develops on the negatively biased electrode while a typical rf sheath develops on the other electrode. The dc/rf sheath has a dc region with negligible electron density near the negatively biased electrode. Furthermore, if the rf power is held constant (typical for industrial plasmas driven by rf power sources), the sheath voltage drop of the rf sheath is nearly independent of the dc voltage provided the discharge configuration does not lead to a significant increase in the ionization efficiency of the secondary electrons. The analysis is done for both symmetric (equal area) and asymmetric diode discharges, as well as a triode configuration. The analytical results for the symmetric and asymmetric diode discharges are compared to the results of numerical simulations using plane-parallel and cylindrical particle-...

Particle in cell simulations of low pressure small radius positive column discharges

Improved positive column simulation techniques are needed because of the non-local nature of typical low-pressure discharges used for lighting. In a local model, the power balance between Joule heating and collisional losses must hold for each volume element of the discharge separately while a non-local model requires only a global power balance. The departure from locality increases as either gas density ng or radius R is decreased. Despite this, most current fluorescent lamp software is based on the local concept. We present a non-local kinetic particle-in-cell Monte Carlo collisions (PIC-MCC) code to simulate low-pressure, small-radius, positive column discharges. This code is also compared to a non-local fluid code, a non-local kinetic Monte Carlo code and to experimental data. The PIC-MCC code made the least approximations and assumptions and was accurate and stable over a wider parameter regime than the other codes. Also, 1d3v PIC-MCC simulation speeds are quite competitive even on moderate workstations. Finally, we analyse the PIC-MCC simulation results in detail, especially the power balance and the radial electron kinetic energy flux Hr(r). We found that for low ngR< 1×1015 cm-2, the electron kinetic energy flux is directed radially outward while for higher ngR, it is directed radially inward except right near the wall.

Solvated electrons at the atmospheric pressure plasma–water anodic interface

We present results from a particle-in-cell/Monte Carlo model of a dc discharge in argon at atmospheric pressure coupled with a fluid model of an aqueous electrolyte acting as anode to the plasma. The coupled models reveal the structure of the plasma–electrolyte interface and near-surface region, with a special emphasis on solvated or hydrated electrons. Results from the coupled models are in generally good agreement with the experimental results of Rumbach et al (2016 Nat. Commun. 6 7248). Electrons injected from the plasma into the water are solvated, then lost by reaction with water within about 10–20 nm from the surface. The major reaction products are OH− and H2. The solvated electron density profile is controlled by the injected electron current density and subsequent reactions with water, and is relatively independent of the external plasma electric field and the salt concentration in the aqueous electrolyte. Simulations of the effects of added scavenger compounds (H2O2, , and H+) on near-surface solvated electron density generally match the experimental results. The generation of near-surface OH− following electron-water decomposition in the presence of bulk acid creates a highly basic region (pH ~ 11) very near the surface. In the presence of bulk solution acidity, pH can vary from a very acidic pH 2 away from the surface to a very basic pH 11 over a distance of ~200 nm. High near-surface gradients in aqueous solution properties could strongly affect plasma-liquid applications and challenge theoretical understanding of this complex region.

Fast 2D fluid-analytical simulation of ion energy distributions and electromagnetic effects in multi-frequency capacitive discharges

A fast 2D axisymmetric fluid-analytical plasma reactor model using the finite elements simulation tool COMSOL is interfaced with a 1D particle-in-cell (PIC) code to study ion energy distributions (IEDs) in multi-frequency capacitive argon discharges. A bulk fluid plasma model, which solves the time-dependent plasma fluid equations for the ion continuity and electron energy balance, is coupled with an analytical sheath model, which solves for the sheath parameters. The time-independent Helmholtz equation is used to solve for the fields and a gas flow model solves for the steady-state pressure, temperature and velocity of the neutrals. The results of the fluid-analytical model are used as inputs to a PIC simulation of the sheath region of the discharge to obtain the IEDs at the target electrode. Each 2D fluid-analytical-PIC simulation on a moderate 2.2 GHz CPU workstation with 8 GB of memory took about 15–20 min. The multi-frequency 2D fluid-analytical model was compared to 1D PIC simulations of a symmetric parallel-plate discharge, showing good agreement. We also conducted fluid-analytical simulations of a multi-frequency argon capacitively coupled plasma (CCP) with a typical asymmetric reactor geometry at 2/60/162 MHz. The low frequency 2 MHz power controlled the sheath width and sheath voltage while the high frequencies controlled the plasma production. A standing wave was observable at the highest frequency of 162 MHz. We noticed that adding 2 MHz power to a 60 MHz discharge or 162 MHz to a dual frequency 2 MHz/60 MHz discharge can enhance the plasma uniformity. We found that multiple frequencies were not only useful for controlling IEDs but also plasma uniformity in CCP reactors.

Nonlinear standing wave excitation by series resonance-enhanced harmonics in low pressure capacitive discharges

It is well-known that standing waves having radially center-high rf voltage profiles exist in high frequency capacitive discharges. It is also known that in radially uniform discharges, the capacitive sheath nonlinearities excite strong nonlinear series resonance harmonics that enhance the electron power deposition. In this work, we consider the coupling of the series resonance-enhanced harmonics to the standing waves. A one-dimensional, asymmetric radial transmission line model is developed incorporating the wave and nonlinear sheath physics and a self-consistent dc potential, for both conducting and insulating electrode surfaces. The resulting coupled pde equation set is solved numerically to determine the discharge voltages and currents. A 10 mTorr argon plasma is chosen with density m−3, gap width 2 cm and conducting electrode radius 15 cm, driven by a 500 V rf source with resistance 0.5 . We examine a set of frequencies from near 30 MHz up to frequencies more than three times as high. For most frequencies, no harmonics correspond exactly with the series or spatial resonances, which is the generic situation. Nevertheless, nearby resonances lead to a significantly enhanced ratio of the electron power per unit area on axis, compared to the average. Nearly similar results are found for insulating electrodes. Strong effects are seen for varying source resistance: high (50 ) resistance damps out most of the harmonic activity, while zero source resistance leads to a non-steady discharge with bias voltage relaxation oscillations. Stronger harmonic effects are seen for an increased radius of 30 cm, as lower harmonics become spatially resonant at lower frequencies. The radial dependence of electron power with frequency showed significant variations, with the central enhancement and sharpness of the spatial resonances depending in a complicated way on the amplitudes of the nearby series resonance current harmonics and the phase relations among the voltage harmonics driving these current harmonics. Significant center/average electron power per unit area enhancement is found even at the lowest frequencies for both high and low densities: 4.5 at 30 MHz and m−3, and 2.2 at m−3.