The plasma-sheath-transition in RF-modulated low temperature plasmas: On the existence of a modified Bohm criterion

Abstract

Summary form only given. The plasma-sheath transition in radio frequency modulated low temperature plasmas is investigated for arbitrary levels of collisionality. The model under study contains the equations of continuity and motion for a single ion species, formulated in terms of the phase-averaged field <;E>;(x), the electron Boltzmann equilibrium relation and Poisson's equation in dependence of the phase-resolved field E(x,t), and the phase average relation which connects both fields. Assuming that the electron Debye length λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> is small compared with the ion gradient length l=n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> /∂n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> /∂x, a first order differential equation is established for the ion density ni as a function of the transformed spatial coordinate q=∫n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> dx. A characteristic feature of this novel sheath equation is an internal singularity of the saddle point type which separates the depletion-field dominated sheath part of the solution from the ambipolar diffusion-controlled plasma. The properties of this inner singularity allow the definition, in a nonarbitrary way, of a modified Bohm criterion which recovers Bohm's original expression in the collisionless and stationary limit but also remains meaningful when collisions and modulation are included.