From electron depletion to quasi-neutrality: the sheath–bulk transition in RF modulated discharges

摘要

The boundary sheaths of all plasmas are characterized by a gradual transition from unipolarity',electron depletion, ne n,) to ambipolarity (quasi-neutrality, ne n,). Capacitively drivensheaths exhibit a transition which is expanded by the RF modulation and smoothed by thermaleffects, i.e. by the finiteness of the electron temperature Te and the Debye lengthki) = 1E0 Te/ e2n,. Sheath models which neglect thermal effects (`step models') are restrictedto strongly modulated high voltage sheaths with VRF Te/e and fail when this condition isnot met. This work presents an improved analysis of the sheath–bulk transition which takesboth modulation and thermal effects into account. Based on a previously found asymptoticsolution of the Boltzmann–Pbisson equation (Brinkmann 2007 J. Appl. Phys. 102 093393),approximate algebraic (i.e. closed) expressions for the phase-resolved electrical field E andelectron density ne in RF sheaths are derived. Under the assumption that the modulation isperiodic (not necessarily harmonic) with coRF coo, also the phase averages of the field E andthe electron density fie can be expressed in closed form. These results—together referred to asthe advanced algebraic approximation (AAA)—make it possible to formulate efficient andaccurate models for RF driven boundary sheaths for all ratios of VRF to Te/e. As an example, aharmonically RF modulated, collision-dominated single species sheath is studied. Theoutcome is compared both with the numerically constructed exact solution and with thewell-known step model approach of Lieberman (1989 IEEE Trans. Plasma Sci. 17 338). It isfound that the AAA can reproduce the exact numerical solution within a few per cent for allratios of VRF to Te/e. The step model, in contrast, exhibits strong deviations even for largee VRF/ Te and fails completely in the case of weak modulation.