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.

• Received 12 September 1991

DOI:https://doi.org/10.1103/PhysRevA.45.5913