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three-dimensional spatial variation, and they are in general not at steady state. Modeling this system therefore requires a treatment of
Kinetics and transport of charged species (electrons, positive and negative ions);
Kinetics and transport of neutral species in the gas phase; and
Interaction of charged and neutral species with surfaces.
The nonequilibrium nature of the electrons and ions means that these species not only do not have the same average thermal energies, but in addition do not generally share the same form for the distribution of their velocities. Neither electrons nor ions in general follow the Maxwell-Boltzmann form, characteristic of species at local thermal equilibrium. This means that for a rigorous treatment of electron and ion transport some kind of kinetic scheme is necessary: solution of the Boltzmann equation, for example, or a particle simulation such as a Monte Carlo scheme. Also, neutral transport is often complicated by the fact that the collisional mean free path for neutrals (for some low-pressure etching equipment) is on the same order as the dimensions of the reaction chamber. This brings into question the use of continuum or fluid descriptions of neutral transport. An additional complication comes from the fact that neutral species may be in excited vibrational or electronic states. In some cases, then, a rigorous treatment of neutral transport also requires a kinetic approach. In spite of this, schemes that assume some form for distribution functions (i.e. fluid models) have proven useful for conditions under which their limitations are well understood. An increasingly popular approach is to combine fluid and kinetic schemes into a hybrid model, in which, ideally, the strengths of both approaches can be combined, while minimizing their weaknesses.
In the last 5 years, there have been impressive developments in plasma modeling algorithms. These algorithms use fluid, kinetic, and hybrid methods to treat plasma and neutral transport and kinetics. Electromagnetic modules have been coupled successfully to the transport and kinetics codes and have been applied to systems of industrial interest. The availability of engineering workstations with high performance at modest cost has made these developments possible. In addition, optimizing compilers, convenient "debuggers," "windowing" capabilities, and excellent graphics are widely available and increase the productivity of modelers. An example of a two-dimensional, axisymmetric calculation for an inductively coupled (radio-frequency) plasma discharge is shown in Figure 2.1.
The major missing ingredient in further exploitation of large-scale plasma modeling and simulation is the availability of a physical and chemical database for a large and diverse set of species present in the discharge, interacting through a variety of collisional processes in the gas phase and at surfaces. In
An example of a tool-scale simulation of an inductively coupled plasma reactor and various discharge spatial profiles: (top) contour plot of two important discharge characteristics, electron density and plasma potential; (bottom) plot of electron temperature and the source terms for creation of electrons. (Courtesy of Mark Kushner, University of Illinois at Urbana-Champaign.)