Numerical studies on the transition of tuned substrate self-bias in a radio-frequency inductively coupled plasma

Numerical studies concentrate on the physical mechanism underlying the continuity and bistability experimentally observed in a radio-frequency inductively coupled plasma with a tuned substrate. For the sake of simplicity, but to feasibly include key factors influencing the tuned substrate bias, the tedious calculations of inductive coupling in order to obtain plasma density and electron temperature are omitted; therefore, discussions on the tuned substrate self-bias are made under an assumed plasma density and electron temperature. Parameters influencing capacitive coupling are retained in modeling the system via an equivalent circuit. It is found that multistability appears when one of the parameters, such as the resistance in the LCR (inductor, capacitor, and resistor) network, the substrate (grounded wall) sheath area or the plasma density is decreased to its critical magnitude, or when the coil radio-frequency voltage and electron temperature are increased to their critical values. The influences of the above-mentioned 'external' parameters can be universally attributed to one 'inner' parameter of substrate sheath voltage drop. In the numerical trisolution region, the middle and high solutions are always entirely located in the inductive region of substrate-branch-circuit impedance. The two physical factors causing the multistability are the nonlinearity of substrate sheath capacitance and the tuning characteristic of the external LCR network connected to the substrate. With the combination of the two physical factors, the multistability appears when the voltage drop across the substrate sheath exceeds the critical magnitude