Stochastic electron heating in bounded radio-frequency plasmas

Plasma Sources Science and Technology, 2014

We investigate the electron heating dynamics in electropositive argon and helium capacitively coupled RF discharges driven at 13.56 MHz by particle-in-cell simulations and by an analytical model. The model allows one to calculate the electric field outside the electrode sheaths, space and time resolved within the RF period. Electrons are found to be heated by strong ambipolar electric fields outside the sheath during the phase of sheath expansion in addition to classical sheath expansion heating. By tracing individual electrons we also show that ionization is primarily caused by electrons that collide with the expanding sheath edge multiple times during one phase of sheath expansion due to backscattering toward the sheath by collisions. A synergistic combination of these different heating events during one phase of sheath expansion is required to accelerate an electron to energies above the threshold for ionization. The ambipolar electric field outside the sheath is found to be time modulated due to a time modulation of the electron mean energy caused by the presence of sheath expansion heating only during one half of the RF period at a given electrode. This time modulation results in more electron heating than cooling inside the region of high electric field outside the sheath on time average. If an electric field reversal is present during sheath collapse, this time modulation and, thus, the asymmetry between the phases of sheath expansion and collapse will be enhanced. We propose that the ambipolar electron heating should be included in models describing electron heating in capacitive RF plasmas.

Physics of Plasmas, 2015

The electron residual energy originated from the stochastic heating in underdense field-ionized plasma is here investigated. The optical response of plasma is initially modeled by using the concept of two counter-propagating electromagnetic waves. The solution of motion equation of a single electron indicates that by

Jetp Letters, 1999

It is shown theoretically and experimentally that stochastic heating of plasma electrons is highly efficient. Calculations have shown that over the course of 100 periods of an external microwave field the kinetic energy of the particles reaches values of around 1.0 MeV and the average energy reaches values of the order of 0.3 MeV in the field of two oppositely propagating characteristic ͑eigen͒ waves of a cylindrical waveguide, with amplitudes 24 kV/cm in a 1 kG stationary magnetic field. Stochastic instability develops as a result of overlapping of nonlinear cyclotron resonances. The experimental results agree with the theory: When these waves are excited by a 0.9 MW external source, above a threshold of 0.45 MW one obtains x rays with a photon energy corresponding to a maximum electron energy of the order of 1 MeV over about 800 periods of the external microwave field.

ANU, 1990

A one-dimensional, electrostatic particle-in-cell code with non-periodic boundary conditions is used to simulate a low pressure capacitive rf plasma created between two planar electrodes. Ion and electron motion is included and ionising collisions by energetic electrons allow a steady state to be reached and maintained. Realistic values of mi/me are used but there is no attempt to model a real gas and, except for ionisation, no binary collision processes are considered. The simulation plasma is generated by driving one boundary with a sinusoidal rf voltage at a frequency of 10 MHz. The effects of scaling on the steady state, the structure and the impedance of the resulting discharge are investigated. Changes resulting from varying the amplitude of the driving voltage are examined and scaling laws for the plasma potential, electron density and power loss obtained. Sheath heating is shown to be the main electron heating process and power balance is checked. The structure of the rf sheath obtained in the simulation is compared to theoretical models of both the current driven and the voltage driven sheath. Disagreement in the maximum sheath width between the simulation and the model is ascribed to neglect of the period of sheath collapse and the use of an idealised electron density profile in the model. Sheath scaling is shown to underlie the variation of electron density and temperature with rf voltage. The electron sheath interaction is examined and found to differ considerably from current theoretical models. In the range of parameters investigated, it is essential to consider the distortion of the electron velocity distribution in the sheath. A beam-like distribution is observed when the sheath velocity changes rapidly near the time of sheath collapse and an instability develops when electrons are accelerated into the plasma as the sheath expands.

Microwave Heating - Electromagnetic Fields Causing Thermal and Non-Thermal Effects, 2021

In this chapter, the results of theoretical and experimental studies of the interaction of an electromagnetic field with a plasma (fundamental interaction of the wave-particle type) both in the regime of standing waves (in the case of a resonator) and in the case of traveling waves in a waveguide are presented. The results of computer modeling the distribution of a regular electromagnetic field for various designs of electrodynamic structures are considered. The most attractive designs of electrodynamic structures for practical application are determined. A brief review and analysis of some mechanisms of stochastic plasma heating are given as well as the conditions for the formation of dynamic chaos in such structures are determined. Comparison analysis of microwave plasma heating in a regular electromagnetic field (in a regime with dynamical chaos) with plasma heating by random fields is considered. It is shown, that stochastic heating of plasma is much more efficient in comparison...

Plasma Physics and Controlled Fusion, 2012

Capacitive radio frequency (RF) discharge plasmas have been serving hi-tech industry (e.g. chip and solar cell manufacturing, realization of biocompatible surfaces) for several years. Nonetheless, their complex modes of operation are not fully understood and represent topics of high interest. The understanding of these phenomena is aided by modern diagnostic techniques and computer simulations. From the industrial point of view the control of ion properties is of particular interest; possibilities of independent control of the ion flux and the ion energy have been utilized via excitation of the discharges with multiple frequencies. 'Classical' dual-frequency (DF) discharges (where two significantly different driving frequencies are used), as well as discharges driven by a base frequency and its higher harmonic(s) have been analyzed thoroughly. It has been recognized that the second solution results in an electrically induced asymmetry (electrical asymmetry effect), which provides the basis for the control of the mean ion energy. This paper reviews recent advances on studies of the different electron heating mechanisms, on the possibilities of the separate control of ion energy and ion flux in DF discharges, on the effects of secondary electrons, as well as on the non-linear behavior (self-generated resonant current oscillations) of capacitive RF plasmas. The work is based on a synergistic approach of theoretical modeling, experiments and kinetic simulations based on the particle-in-cell approach.

Physical Review A, 2012

Several mechanisms by which an external electromagnetic field influences the temperature of a plasma are studied analytically and specialized to the system of an ultracold plasma (UCP) driven by a uniform radio frequency (RF) field. Heating through collisional absorption is reviewed and applied to UCPs. Furthermore, it is shown that the RF field modifies the three body recombination process by tunnel-ionizing electrons from intermediate high-lying Rydberg states, whereby the three body recombination rate and associated plasma heating is suppressed. Heating through collisionless absorption associated with the finite plasma size is calculated in detail, revealing a temperature threshold below which collisionless absorption is ineffective. It is found that, for large RF amplitude, the ponderomotive potential replaces the electron temperature in functional dependencies of several plasma quantities.

Physics of Fluids, 1985

Physical Review Letters, 1998

Significant frequency dependence of the electron energy distribution has been found in a lowpressure inductive discharge experiment. The observed frequency dependence reveals the presence of collisionless electron heating and appears as a result of finite electron-transit time through the skin layer. The energy distributions calculated from a kinetic equation accounting for nonlocality of electron kinetics and electrodynamics are in good agreement with the experiment. [S0031-9007(98)06549-1]

Journal of Physics D: Applied Physics, 2008

The generation of directed energetic electrons by the expanding sheath is observed in asymmetric capacitively coupled radio frequency discharges at low pressures ( 1 Pa) in different gases. The phenomenon of such electron beams is investigated by a combination of experimental diagnostics, an analytical model and simulations. At sufficiently low pressures multiple reflections of electron beams at the plasma boundaries are observed. An analytical model shows how these beams lead to an enhanced high energy tail of the electron energy distribution function. Thus, stochastic heating is closely related to electron beams.

Physics of Plasmas, 2018

To control the temperature during a plasma treatment, an understanding of the link between the plasma parameters and the fundamental process responsible for the heating is required. In this work, the power supplied by the plasma onto the surface of a glass substrate is measured using the calorimetric method. It has been shown that the powers deposited by ions and electrons, and their recombination at the surface are the main contributions to the heating power. Each contribution is estimated according to the theory commonly used in the literature. Using the corona balance, the Modified Boltzmann Plot (MBP) is employed to determine the electron temperature. A correlation between the power deposited by the plasma and the results of the MBP has been established. This correlation has been used to estimate the electron number density independent of the Langmuir probe in considered conditions.

Physics of Plasmas, 2016

2011

Theoretical and experimental study of the microwave cut-off probe for electron density measurements in lowtemperature plasmas J. Appl. Phys. 110, 073308 (2011) Electron density measurement of inductively coupled plasmas by terahertz time-domain spectroscopy (THz-TDS) J. Appl. Phys. 110, 073303 (2011) Measurements of electron avalanche formation time in W-band microwave air breakdown Phys. Plasmas 18, 080707 Direct thrust measurements and modelling of a radio-frequency expanding plasma thruster Phys. Plasmas 18, 080701 Intermediate frequency band digitized high dynamic range radiometer system for plasma diagnostics and realtime Tokamak control Rev. Sci. Instrum. 82, 063508 (2011)

1985

It is shown that a distribution of electrons in resonance with traveling waves, but colliding with background distributions of electrons and ions, evolves to a steady state* Details of the steady state are given analytically in the asymptotic limit of high electron energy and are compared with numerical solutions. The asymptotic analytic solution may be useful for quickly relating emission data to likely excitations and is more reliable than conventional numerical solutions at high energy. A method of improving numerics at high energy is suggested.

Physical Review E, 2014

Microwave plasmas excited at electron-cyclotron resonance were studied in the 0.5-15 mTorr pressure range. In contrast with low-limit pressure conditions where the plasma emission highlights a fairly homogeneous spatial structure, a periodic spatial modulation (period ß6.2 cm) appeared as pressure increased. This feature is ascribed to a local power deposition (related to the electron density) due to the presence of a standing electromagnetic wave created by the feed electromagnetic field (2.45 GHz) in the cavity formed by the reactor walls. Analysis of the electron energy probability function by Langmuir probe and optical emission spectroscopy further revealed the presence of a high-energy tail that showed strong periodic spatial modulation at higher pressure. The spatial evolution of the electron density and of the characteristic temperature of these high-energy electrons coincides with the nodes (maximum) and antinodes (minimum) of the standing wave. These spatially-modulated power deposition and electron heating mechanisms are then discussed.