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

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/ω_p is the collisionless skin depth, with c the speed of light and ω_p 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.