Analytical plasma impedance model of dual frequency capacitive discharges with ion dynamics

Abstract

Based on a fluid simulation, an analytical equivalent circuit model was developed for collisional plasmas with dual frequency (DF) excitation. Primary focus is on a fully collisional sheath and a low frequency (LF) below the ion plasma frequency. In the analytical model, the high frequency (HF) component is expressed as a series RC circuit, which matches the classical result for the limit of single frequency radio frequency discharges. The LF circuit is expressed as a parallel RC circuit to consider the response of ions to the LF voltage. The model requires three main parameters: the magnitudes of the HF and LF voltages, which are known during discharge operation and the space and time averaged ion density, which is in general unknown and needs to be determined by other means. Explicit formulas for the calculation of the HF and LF impedances are provided. Good agreement is found between the new analytical model and the fluid simulation. It is worth noting that previously published analytical models fail to reproduce the impedances, predominantly because the dynamic ion behavior and the parallel nature of the resulting ion current in respect to the LF displacement current are not considered. A practical application of the model is the calculation of the ion density in the discharge from measured impedances. For this purpose both the LF and the HF impedances can be used. A comparison of the densities estimated from the different impedances and the average density in the simulation show again good agreement.