Nonlinear standing wave excitation by series resonance-enhanced harmonics in low pressure capacitive discharges

Abstract

It is well-known that standing waves having radially center-high rf voltage profiles exist in high frequency capacitive discharges. It is also known that in radially uniform discharges, the capacitive sheath nonlinearities excite strong nonlinear series resonance harmonics that enhance the electron power deposition. In this work, we consider the coupling of the series resonance-enhanced harmonics to the standing waves. A one-dimensional, asymmetric radial transmission line model is developed incorporating the wave and nonlinear sheath physics and a self-consistent dc potential, for both conducting and insulating electrode surfaces. The resulting coupled pde equation set is solved numerically to determine the discharge voltages and currents. A 10 mTorr argon plasma is chosen with density m−3, gap width 2 cm and conducting electrode radius 15 cm, driven by a 500 V rf source with resistance 0.5 . We examine a set of frequencies from near 30 MHz up to frequencies more than three times as high. For most frequencies, no harmonics correspond exactly with the series or spatial resonances, which is the generic situation. Nevertheless, nearby resonances lead to a significantly enhanced ratio of the electron power per unit area on axis, compared to the average. Nearly similar results are found for insulating electrodes. Strong effects are seen for varying source resistance: high (50 ) resistance damps out most of the harmonic activity, while zero source resistance leads to a non-steady discharge with bias voltage relaxation oscillations. Stronger harmonic effects are seen for an increased radius of 30 cm, as lower harmonics become spatially resonant at lower frequencies. The radial dependence of electron power with frequency showed significant variations, with the central enhancement and sharpness of the spatial resonances depending in a complicated way on the amplitudes of the nearby series resonance current harmonics and the phase relations among the voltage harmonics driving these current harmonics. Significant center/average electron power per unit area enhancement is found even at the lowest frequencies for both high and low densities: 4.5 at 30 MHz and m−3, and 2.2 at m−3.