Numerical Simulation of Plasma Coupling with an External Matching Network by Phasor and Particle Models

In most of the semiconductor fabrication processes such as plasma doping, plasma-enhanced chemical vapor deposition, and plasma etching, electrical plasma (ions and electrons) are involved. To generate the plasma, it is common to use radio-frequency (RF) coupled inductive or capacitive discharge with an external matching network. Verboncoeur et. al. developed a simultaneous potential and circuit solution for ID bounded plasma particle simulation. Gauss's law was applied to the plasma system and the second order finite difference equation was derived for the second order Kirchoff's voltage law for a general voltage-driven RLC circuit. However, an external matching network is more complicated than a simple RLC circuit. In plasma processes such as plasma doping several matching networks %with different frequencies may be applied to the electrodes [4]. Solving the second order Kirochoff's voltage equations pertinent to these complicated matching networks is very difficult. An RF signal going through a general matching network can be described by its absolute amplitude and phase. Based on the phasor analysis, any complicated circuits can be described by several Linear equations that can be solved directly. In this paper, we describe a model combining the plasma particle model and electrical phasor model, so that the coupling of the plasma with any external matching network can be effectively simulated. The chamber and electrodes, either RF powered or grounded, can be regarded as a good capacitor. In the presence of a semi-ionized plasma, the ions are in room temperature and the electrons are in thermal equilibrium of a few eV in temperature. The accumulated surface charges and consequently the potential are modified by the plasma. Using this model, the amplitude and phase of the voltage/current of the external matching network are numerically estimated, and an auto-matching network is also
simulated.