Analysis of a kinematic model for ion transport in rf plasma sheaths

Abstract

An idealized model for ion transport across an oscillating plasma sheath is analyzed to obtain insight into qualitative features of the ion energy distributions observed in low-pressure rf discharges. The sheath is characterized by a constant electric field over an extent that varies sinusoidally with time, and ions incident on it correspond to a monoenergetic flux independent of phase φ in the rf cycle. The dimensionless parameters α=${\mathit{qV}}_{\mathit{s}}$/m${\mathrm{\omega }}^2$ ${\mathit{d}}^2$ and β=${\mathit{v}}_0$/ωd (where d and ${\mathit{V}}_{\mathit{s}}$ are the mean sheath thickness and potential drop, ω is the excitation frequency, and ${\mathit{v}}_0$ and q/m are the incoming ion speed and charge-to-mass ratio) govern the ion trajectories, which are found to divide into groups, delimited by two ‘‘critical’’ values of φ, that undergo N and N+1 encounters with the field. The first critical phase depends only weakly on β, whereas the second is sensitive to both α and β and cycles continuously as these parameters diminish. Correspondingly, within the ‘‘transition regime’’ where α and β are neither very small nor greater than (or comparable to) unity, the precise form of the incident-ion energy spectrum exhibits rapid variations, superposed on a systematic narrowing, as the frequency ω is increased.