Linear electromagnetic excitation of an asymmetric low pressure capacitive discharge with unequal sheath widths

Abstract

It is well-known that standing waves having radially center-high radio frequency (rf) voltage profiles exist in high frequency capacitive discharges. In this work, we determine the symmetric and antisymmetric radially propagating waves in a cylindrical capacitive discharge that is asymmetrically driven at the lower electrode by an rf voltage source. The discharge is modeled as a uniform bulk plasma which at lower frequencies has a thicker sheath at the smaller area powered electrode and a thinner sheath at the larger area grounded electrode. These are self-consistently determined at a specified density using the Child law to calculate sheath widths and the electron power balance to calculate the rf voltage. The fields and the system resonant frequencies are determined. The center-to-edge voltage ratio on the powered electrode is calculated versus frequency, and central highs are found near the resonances. The results are compared with simulations in a similar geometry using a two-dimensional hybrid fluid-analytical code, giving mainly a reasonable agreement. The analytic model may be useful for finding good operating frequencies for a given discharge geometry and power.