Standing wave and skin effects in large-area, high-frequency capacitive discharges

TLDR

Large‑area capacitive discharges driven above 13.56 MHz are increasingly used for etching and thin‑film deposition, yet standing‑wave and skin‑effect limitations can compromise plasma uniformity beyond what electrostatic models predict. The authors aim to formulate an electromagnetic theory that captures surface and evanescent wave propagation and the interplay of capacitive and inductive fields in high‑frequency, large‑area capacitive discharges. The theory models a discharge between two circular plates of radius R separated by 2l, incorporating wave propagation from the discharge edge to the centre and the roles of capacitive and inductive fields in power absorption. The model predicts significant standing‑wave and skin‑effect distortions, with uniformity achieved only when λ0 ≫ 2.6 √(l/s) R and δ ≫ 0.45 √(dR); for a 50 cm radius, 4 cm gap, and 2 mm sheath, skin effects dominate at densities ≳10¹⁰ cm⁻³ and standing waves at frequencies ≳70 MHz.

Abstract

Large-area capacitive discharges driven at frequencies higher than the usual industrial frequency of 13.56 MHz have attracted recent interest for materials etching and thin film deposition on large-area substrates. Standing wave and skin effects can be important limitations for plasma processing uniformity, which cannot be described by conventional electrostatic theory. An electromagnetic theory is developed for a discharge having two plates of radius R and separation 2l, which accounts for the propagation of surface and evanescent waves from the discharge edge into the centre and the role of capacitive and inductive fields in driving the power absorption. Examples of discharge fields are given having substantial standing wave and/or skin effects. The conditions for a uniform discharge without significant standing wave and skin effects are found to be, respectively, λ0>>2.6(l/s)1/2R and δ>>0.45(dR)1/2, where λ0 is the free space wavelength, s is the sheath width, δ = c/ωp is the collisionless skin depth, with c the speed of light and ωp the plasma frequency, and d = l-s is the plasma half-width. Taking the equality for these conditions for a discharge radius of 50 cm, plate separation of 4 cm, and sheath width of 2 mm, there is a substantial skin effect for plasma densities ≳1010 cm-3, and there is a substantial standing wave effect for frequencies f≳70 MHz.