Theoretical studies of radio-frequency sheath

Abstract

In processing plasma, the ion flux, energy distribution and angular distribution on the wafer surface are crucial to the industrial applications. These ion properties depend on the sheath dynamics. For a radio-frequency biased sheath, it has been found that the rf sheath dynamics is characterized by the ratio of the rf frequency ω and the ion transit frequency crossing the sheath ωtr. Conventionally, the ion transit frequency and the bulk ion plasma frequency ωpi are used interchangeably since they are in the same order for a collisionless sheath. In this research, we study the rf sheath dynamics theoretically as well as computationally based on both the fluid and kinetic models. Based on the fluid model, we develop a one-dimensional code to solve the continuity and momentum equation for electrons and ions and Poisson’s equation. The system of equations is coupled to an external rf circuit model and solved in the different rf frequency regimes. The numerical results are compared with our theoretical models. vi In the low frequency regime where the rf frequency is much lower than the ion plasma frequency, we find that the presheath introduces an additional time scale, the ion transit frequency crossing the presheath ωpre. If the rf frequency ω ≥ ωpre, the ions in the presheath respond instantaneously to the rf field. Consequently, the ion current entering the sheath is no longer timeindependent, but varies with time. The time-varying ion current affects the electric field in the sheath and the ion energy distribution at the electrodes significantly. We find good agreement between our theoretical results and our numerical results. In the intermediate frequency regime where the rf frequency is comparable to the ion plasma frequency, we propose a sheath model to describe the dynamics of highly collisional sheath. We consider two cases: (1) constant collisional frequency, (2) constant ion mean free path. The computed ion velocity and the energy distribution at the electrodes agree well with the numerical results. In the regime where the rf frequency is much higher than the ion plasma frequency, we investigate the effects of the electron dynamics on the plasma and sheath dynamics. We have shown that the assumption that the electrons follow the Boltzmann distribution is invalid in the sheath region. The electron inertia can be neglected and the electron dynamics can be described well by the drift-diffusion model provided the ratio of the plasma current and the ion current is much smaller than q mi/me(here mi and me are the ion and electron mass respectively). Otherwise, the electron inertia affects the plasma and sheath dynamics significantly. Lastly, we investigate the plasma-sheath problem using a hybrid simulation with kinetic ions and the drift-diffusion model for electrons. We solve the vii one-dimensional Vlasov equation for ions by using the cubic interpolated propagation (CIP) scheme to obtain the ion velocity distribution function. For a dc case, the results are in good agreement with fluid and Self’s plasma-sheath theories. We also study the rf sheath dynamics in the different frequency regimes and find that the results of the kinetic and fluid models are different in the intermediate or low frequency regime. In this regime, the ion energy distributions (IEDs) of the kinetic model depend on the ionization term. If the ion production rate is significant in the sheath, a low-energy tail will be formed in the IEDs.