On the Similarity and Scaling Laws in Low-Pressure Capacitive Radio Frequency Discharges

In the work, we studied the similarity and scaling characteristics of low-pressure capacitive radio frequency (rf) discharges via particle-in-cell/Monte Carlo collision simulations. The similarities of the rf discharge at different length scales are demonstrated while maintaining the combined parameters, i.e., reduced driving frequency f/p and reduced gap length pd the same, with f being the driving frequency, p the gas pressure, and d the gap distance. The electron kinetic invariance in similar discharges is confirmed and the foundation of the similarity principles is interpreted based on the scaling of the electron Boltzmann equation. We further compared the similarity relation for electron density with the traditional frequency scaling and found that the parameter dependence is rather close within certain parameter regimes, which can be understood from the same scaling relation of the electron power absorption. The discrepancy of the similarity relation and frequency scaling is also observed when the sheath width becomes sufficiently large and thus the stationary bulk plasma region no longer exists. The effects of the nonlinear collisional processes in terms of similarity laws, such as stepwise ionization, on the electron density scaling are also examined. The results offer new insights and a more comprehensive understanding for correlating the rf discharge of different dimensions across a wide range of parameter regimes.