Nonlinear series resonance and standing waves in dual-frequency capacitive discharges

It is well-known that the nonlinear series resonance in a high frequency capacitive discharge enhances the electron power deposition and also creates standing waves which produce radially center-high rf voltage profiles. In this work, the dynamics of series resonance and wave effects are examined in a dual-frequency driven discharge, using an asymmetric radial transmission line model incorporating a Child law sheath. We consider a cylindrical argon discharge with a conducting electrode radius of 15 cm, gap length of 3 cm, with a base case having a 60 MHz high frequency voltage of 250 V and a 10 MHz low frequency voltage of 1000 V, with a high frequency phase shift phi(H) = pi between the two frequencies. For this phase shift there is only one sheath collapse, and the time-averaged spectral peaks of the normalized current density at the center are mainly centered on harmonic numbers 30 and 50 of the low frequency, corresponding to the first standing wave resonance frequency and the series resonance frequency, respectively. The effects of the waves on the series resonance dynamics near the discharge center give rise to significant enhancements in the electron power deposition, compared to that near the discharge edge. Adjusting the phase shift from p to 0, or decreasing the low frequency from 10 to 2 MHz, results in two or more sheath collapses, respectively, making the dynamics more complex. The sudden excitation of the perturbed series resonance current after the sheath collapse results in a current oscillation amplitude that is estimated from analytical and numerical calculations. Self-consistently determining the dc bias and including the conduction current is found to be important. The subsequent slow time variation of the high frequency oscillation is analyzed using an adiabatic theory.