# Beyond the step model: Approximate expressions for the field in the plasma boundary sheath

Ralf Peter Brinkmann

© 2007 American Institute of Physics J. App. Phys. 102, 093303 (2007)

## Abstract

The transition from quasineutrality to charge depletion is one of the characteristic features of the plasma boundary sheath. For modeling purposes, this transition is often described in terms of the so-called step model which assumes a sharp transition point s (electron step) where the electron density ne drops from a value equal to the ion density ni (in the bulk, x>s) to a value of zero (in the sheath, x<s). Inserted into Poisson’s equation, the step model yields an expression for the field which is realistic deep in the sheath (for x<<s) but fails to merge correctly into the ambipolar field of the bulk. This work considers the transition from quasineutrality to charge depletion more rigorously. Within the framework of asymptotic scale analysis, a family of field approximations is derived which, in the limit of weak spatial ion density variation ?ni/?x<<ni/?D, exhibit convergence to the exact solution of the Boltzmann–Poisson equation. The first of the approximations recovers only the step model. Higher order expressions, however, also include the ambipolar field and are valid for all values of x. The simplest of them is proposed as being the best suited for a self-consistent sheath analysis. This statement is backed by numerical experiments which show good results for the ion density distribution provided by the collision dominated sheath model.

[DOI]