Skin effect in a small symmetrically driven capacitive discharge

Abstract

The so-called 'electrostatic' approximation postulates that the electric field can be represented by the gradient of a scalar potential, even under dynamical conditions. This assumption reduces the set of Maxwell's equations to the much simpler Poisson equation and is often employed for modeling and simulation of radio frequency driven capacitive low pressure discharges. While it is now widely acknowledged that the neglect of induction phenomena breaks down for large-area plasma sources driven at high frequencies (such as used for VLSI processing), smaller experimental devices excited at moderate frequencies (e.g. 13.56 MHz) are generally thought to be uncritical. This paper demonstrates the opposite: even small plasma reactors of the size of the Gaseous Electronics Conference reference cell exhibit a considerable skin effect in the low pressure, high density regime and render the electrostatic approximation invalid. The point is made, however, that this phenomenon is not 'fully electromagnetic' (in the sense that its analysis requires the full set of Maxwell's equations), but can be understood by means of a simplified model which assumes quasi-neutrality and may therefore be called 'magnetostatic'.

Observations of Surface Mode Influence on Plasma Uniformity in PIC/MCC Simulations of Large Capacitive Discharges

“…In this model the PSR oscillations can only have symmetric vertical (axial) electric field, since for the asymmetric case the bulk vertical electron current must vanish at the midplane. These oscillations have been traditionally considered in the electrostatic approximation along with the lateral uniformity assumption underlying an 1d treatment (e.g., ref kx→0 limit.…”

“…These oscillations have been traditionally considered in the electrostatic approximation along with the lateral uniformity assumption underlying an 1d treatment (e.g., ref. [26] ), which corresponds to the k x ! 0 limit.…”

Simulations of electromagnetic effects in high-frequency capacitively coupled discharges using the Darwin approximation

“…provided all quantities are proportional to e iωt . In (16) and (17) Once one obtains A T and B from solution of (16) and (17), the transversal electric field can be obtained from E T = ∇ × A T and the longitudinal electric field from Ampere's law, Figs. 3 and 4 show the power deposition profiles for two selected cases with similar parameters to the corresponding cases in [14].…”

Modeling and Simulation of Ion Energy Distribution Functions in Technological Plasmas

“…A large number of such models, which are based on different approaches, are available. This phenomenon can be explained by nonuniform plasma distributions at 50 and 150 MHz due to electromagnetic effects as the standing wave effect and the skin effect [29][30][31][32]. A very prominent and elaborated hybrid model is the Hybrid Plasma Equipment Model (HPEM) which has been implemented over the last 30 years by Kushner and co-workers [26].…”

“…One can observe an increasing radial dependence of the maximum of the ion energy as the high-frequency increases. This phenomenon can be explained by nonuniform plasma distributions at 50 and 150 MHz due to electromagnetic effects as the standing wave effect and the skin effect [29][30][31][32]. Fig.…”