Abstract

The paper reviews physical principles and computational methods of solving the multi-dimensional Fokker-Planck equation (FPE) in application to electron kinetics in gas discharge plasmas and semiconductor devices. The four-dimensional (3 spatial coordinates + energy) FPE is obtained from the 6D Boltzmann Transport Equation (BTE) for the case when momentum relaxation occurs faster than energy relaxation. The FPE- based methods offer a very good compromise between physical accuracy and numerical efficiency. We have developed a general purpose 4D FPE solver coupled to electromagnetic, chemistry and other models for self-consistent kinetic simulations of weakly ionized plasmas and semiconductor devices. The FPE describes the Electron Energy Probability Function (EEPF) which provides macroscopic characteristics (electron density, fluxes, rates of electron induced chemical reactions, etc). Using these quantities, the transport of ions and holes is simulated using continuum models. The electromagnetic fields are calculated by solving Maxwell equations for scalar electric and vector magnetic potentials. This paper presents several examples of hybrid kinetic simulations of plasma reactors for semiconductor manufacturing and silicon based semiconductor devices. It also outlines direct methods of solving the BTE to compute the distribution functions with arbitrary anisotropy such as electron and ion beams and ballistic carrier transport in semiconductors.

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