Nonlinear harmonic excitations in collisional, asymmetrically-driven capacitive discharges

Abstract

The standing wave effect, which may lead to center-high density profiles in high frequency capacitive discharges, can be enhanced by nonlinearly excited harmonics. In this work, a nonlinear transmission line model, which solves for the electromagnetic fields in the time domain, is coupled to a two-dimensional bulk plasma fluid model to study nonlinear effects in asymmetric cylindrical capacitive argon discharges. An analytical collisional or collisionless (ion) sheath model is used to determine the stochastic and ohmic sheath heating and the nonlinear dependence of sheath voltage on sheath charge. We first examine a base case of a 20 mTorr argon discharge driven with an electron power P e = 40 W at a frequency f = 60 MHz, using collisionless and collisional sheath models. For the collisionless sheath model, the nonlinearly excited harmonics near the series and spatial resonance frequencies significantly enhance the on-axis power deposition and lead to a sharp peak of electron density at the discharge center. The collisional sheath model gives a smaller sheath width, leading to lower series and spatial resonance frequencies and a smaller source voltage for the fixed electron power. As a result, lower harmonics with broader spatial profiles and decreased magnitude are excited, reducing the center-high plasma nonuniformity. Then, we examine the discharge in a pressure range of 20–100 mTorr at fixed P e = 40 W and f = 60 MHz, using the collisional sheath model. As pressure increases, the harmonics gradually damp out, and the enhancement of on-axis power deposition becomes less significant. At the same time, more power is localized near the powered electrode edge due to decreased skin depth and smaller energy diffusion. As a result, the density peak shifts from the radial center to the powered electrode edge.