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

Ralf Peter Brinkmann

J. Phys. D: Appl. Phys. 42, 194009 (2009)

## Abstract

The boundary sheaths of all plasmas are characterized by a gradual transition from unipolarity (electron depletion, ) to ambipolarity (quasi-neutrality, ). Capacitively driven sheaths exhibit a transition which is expanded by the RF modulation and smoothed by thermal effects, i.e. by the finiteness of the electron temperature and the Debye length . Sheath models which neglect thermal effects ('step models') are restricted to strongly modulated high voltage sheaths with and fail when this condition is not met. This work presents an improved analysis of the sheath–bulk transition which takes both modulation and thermal effects into account. Based on a previously found asymptotic solution of the Boltzmann–Poisson equation (Brinkmann 2007 *J. Appl. Phys.* **102** 093393), approximate algebraic (i.e. closed) expressions for the phase-resolved electrical field E and electron density ne in RF sheaths are derived. Under the assumption that the modulation is periodic (not necessarily harmonic) with , also the phase averages of the field and the electron density can be expressed in closed form. These results—together referred to as the advanced algebraic approximation (AAA)—make it possible to formulate efficient and accurate models for RF driven boundary sheaths for all ratios of to . As an example, a harmonically RF modulated, collision-dominated single species sheath is studied. The outcome is compared both with the numerically constructed exact solution and with the well-known step model approach of Lieberman (1989 *IEEE Trans. Plasma Sci.* **17** 338). It is found that the AAA can reproduce the exact numerical solution within a few per cent for all ratios of to . The step model, in contrast, exhibits strong deviations even for large and fails completely in the case of weak modulation.