Analytical model of a dual frequency capacitive sheath

TLDR

Capacitively coupled plasma discharges operated with two frequencies have attracted growing interest, and the condition that the low‑frequency electric field dominates the high‑frequency field is typically satisfied in such systems. The study proposes an analytical sheath model for dual‑frequency capacitively coupled RF plasma discharges assuming time‑independent, collisionless ion motion. The model derives expressions for the time‑average sheath potential, the nonlinear motion of the electron sheath boundary, and the instantaneous sheath voltage, assuming the low‑frequency field dominates the high‑frequency field, and it predicts how the maximum sheath width and dc voltage drop vary with the hf/lf current and frequency ratios. The high‑frequency electric field significantly alters the sheath structure through electron response, and the analytical model shows good quantitative agreement with particle‑in‑cell simulations.

Abstract

In recent years, there has been an increased interest in capacitively coupled plasma discharges which are operated with two frequencies. An analytical sheath model for a capacitively coupled radio-frequency plasma discharge operated with two frequencies is proposed and studied under the assumptions of a time-independent, collisionless ion motion. Expressions are obtained for the time-average electric potential within the sheath, nonlinear motion of the electron sheath boundary and nonlinear instantaneous sheath voltage. The derived model is valid under the condition that the low frequency (lf) electric field Elf in the sheath is much higher than the high frequency (hf) electric field Ehf. This condition is fulfilled within typical dual frequency conditions. It is shown, however, that the hf electric field modifies the sheath structure significantly because of the electron response to Ehf. This model has been compared to particle-in-cell plasma simulations, finding good quantitative agreement. We present the dependence of the maximum sheath width and the dc sheath voltage drop on the hf/lf current ratio and on the hf/lf frequency ratio.