A time dependent model for long mean-free-path transport of neutral particles in plasma etching reactors

Abstract

Summary form only given. Plasma processing for semiconductor fabrication is moving toward reactors using lower gas pressure, and higher plasma density. At the pressures of interest (1-20 mTorr), the mean free path of neutral species is commensurate with the dimensions of interest (Knudsen number Kn>0.1). As a result, fluid momentum and continuity equations may not be accurate representations of the transport of these species. One method to address long mean-free-path transport is based on the use of a "transition matrix", or "propagator". The propagator specifies the probability that a particle, having suffered a collision or been produced at a remote site, has its next collision at the location of interest. In this work the propagator method has been extended to a time dependent form by utilizing a retarded time. In doing so, we account for the finite flight time of species from the remote site to the local site. The propagator has been implemented as the kinetic module in a 2-dimensional hybrid plasma equipment model, and has been applied to analysis of inductively coupled plasma (ICP) etching reactors. The propagator is first constructed using Monte Carlo techniques by launching particles having a randomly chosen mean-free-paths and velocities, and recording the locations at the end of their flights. MC methods, as opposed to analytic techniques, were used to construct the propagator so that complex equipment geometries could be addressed. In applying the time dependent propagator, species densities at past times are recorded. The instantaneous particle fluxes at local sites are then obtained by interpolating the past history of densities based on the retarded time.