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

作者：Shihab Mohammed^*; Ziegler Dennis; Brinkmann Ralf Peter

来源：Journal of Physics D: Applied Physics , 2012, 45(18): 185202.

DOI：10.1088/0022-3727/45/18/185202

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 x-axis points from the electrode or wall at x(E) equivalent to 0 towards the plasma bulk. An arbitrary endpoint x(B) is chosen %26apos;deep in the bulk%26apos;. The model consists of a set of kinetic equations for the ions, Boltzmann%26apos;s relation for the electrons and Poisson%26apos;s equation for the electrical field. Boundary conditions specify the ion flux at xB and a periodically-not necessarily harmonically-modulated sheath voltage V (t) or sheath charge Q(t). 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 [x(E), x(B)] and the RF period [0, T]. Three modules are called in a sequence. A Monte Carlo module calculates the trajectories of a large set of ions from their start at x(B) until they reach the electrode at x(E), utilizing the potential values on the nodes of the spatio-temporal grid. A harmonic analysis module reconstructs the Fourier modes n(im)(x) of the ion density n(i)(x, t) 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.

• 出版日期2012-5-9