Self-consistent nonlinear transmission line model of standing wave effects in a capacitive discharge

It has been shown previously [Lieberman et al., Plasma Sources Sci. Technol. 11, 283 (2002)], using a non-self-consistent model based on solutions of Maxwell’s equations, that several electromagnetic effects may compromise capacitive discharge uniformity. Among these, the standing wave effect dominates at low and moderate electron densities when the driving frequency is significantly greater than the usual 13.56 MHz. In the present work, two different global discharge models have been coupled to a transmission line model and used to obtain the self-consistent characteristics of the standing wave effect. An analytical solution for the wavelength λ was derived for the lossless case and compared to the numerical results. For typical plasma etching conditions (pressure 10–100 mTorr), a good approximation of the wavelength is λ/λ0≈40 V01/10l−1/2f−2/5, where λ0 is the wavelength in vacuum, V0 is the rf voltage magnitude in volts at the discharge center, l is the electrode spacing in meters, and f the driving frequency in hertz.