# Fast, kinetically self-consistent simulation of RF modulated plasma boundary sheaths

Mohammed Shihab, Dennis Ziegler, Ralf Peter Brinkmann

J. Phys. D: Appl. Phys 45, 185202 (2012)

## Abstract

A mathematical model is presented which enables the efficient, kinetically self-consistent simulation of RF modulated plasma boundary sheaths in all technically relevant discharge regimes. It is defined on a one-dimensional geometry where a Cartesian -axis points from the electrode or wall at towards the plasma bulk. An arbitrary endpoint is chosen 'deep in the bulk'. The model consists of a set of kinetic equations for the ions, Boltzmann's relation for the electrons and Poisson's equation for the electrical field. Boundary conditions specify the ion flux at and a periodically—not necessarily harmonically—modulated sheath voltage or sheath charge . The equations are solved in a statistical sense. However, it is not the well-known particle-in-cell (PIC) scheme that is employed, but an alternative iterative algorithm termed ensemble-in-spacetime (EST). The basis of the scheme is a discretization of the spacetime, the product of the domain and the RF period . Three modules are called in a sequence. A Monte Carlo module calculates the trajectories of a large set of ions from their start at until they reach the electrode at , utilizing the potential values on the nodes of the spatio-temporal grid. A harmonic analysis module reconstructs the Fourier modes of the ion density from the calculated trajectories. A field module finally solves the Boltzmann–Poisson equation with the calculated ion densities to generate an updated set of potential values for the spatio-temporal grid. The iteration is started with the potential values of a self-consistent fluid model and terminates when the updates become sufficiently small, i.e. when self-consistency is achieved. A subsequent post-processing determines important quantities, in particular the phase-resolved and phase-averaged values of the ion energy and angular distributions and the total energy flux at the electrode. A drastic reduction of the computational effort compared with PIC calculations is achieved. As a first application of the new model, the influence of ion inertia on the dynamics of a collisionless sheath is studied and a comparison of the simulated ion energy distribution with published analytical solutions is performed.

[DOI]