Numerical Studies On Tuned Substrate Self-bias In A Radio-frequency Inductively Coupled Plasma

Numerical studies are focused the elucidation of the physical mechanism underlying continuity and bi-stability experimentally observed in a radio-frequency inductively coupled plasma with a tuned substrate. For the sake of simplicity but feasibility to include key factors influencing the tuned substrate bias, the tedious calculations of inductive-coupling to obtain plasma density and electron temperature is omitted, but discuss tuned substrate self-bias under assumed plasma density and electron temperature. On the other hand, parameters influencing capacitive-coupling are retained in modeling the system via equivalent circuit. The influences on the curve of V_tsb-C_t by the resistance in LCR, inductor, stray capacitor, substrate area, wall sheath area, glass sheath area, electron temperature, plasma density etc. are discussed. It is found that turned-substrate self-bias presents multi-stable identity when the electron temperature is higher or lower, and multi-stable width reaches its lowest value when the electron temperature varied to the critical value. It is found that multi-stability appears when one of the parameters such as resistance in LCR, stray capacitor, substrate area, grounded wall area, plasma density decreased to its critical magnitude, or coil rf voltage, glass sheath area are increased to the critical values. The influences of the above "external" parameters can be universally attributed to the "inner" parameter of substrate sheath voltage drop. In the multi-solution region, not all the solutions appear in advance of the series resonant point of the substrate circuit branch. The two physical factors originating the multi-stability are the nonlinearity of substrate sheath capacitance and the tuning characteristic of the external LCR network connected to substrate. With the combination of the two physical factors, the multi-stability appears when the voltage drop across the substrate sheath is sufficiently large.According to the characteristics of the tuning system, variation of the impendence of the substrate branch are discussed, the influences of the series resonance on the substrate branch about the stray capacitance, the currents in every branch, the characteristics about the amplitude and phase delay angle of the plasma potential. According to the relationship between the voltage on the coil, the plasma density, electron temperature and discharge power, pressure, the numerical results are consistent with the experimental results.