Analytical model of a dual frequency capacitive sheath

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 E_lf in the sheath is much higher than the high frequency (hf) electric field E_hf. 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 E_hf. 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.