Study on the similarity laws for capacitively coupled radio frequency discharges across pressure and dimension scales

Capacitive radio frequency (rf) plasma sources designed with various dimensions and gas pressure conditions have been widely utilized for etching and thin film deposition during chip fabrication in the semiconductor industry. The discharges at varied dimensional and pressure scales can be correlated via similarity theory, which has been demonstrated for rf plasmas with highly nonlocal electron kinetics but simplified chemical reactions and limited gas pressure regimes. In this work, we investigated the validity of similarity laws for capacitive rf discharges across a wide range of pressures and dimensional scales via two-dimensional fluid simulations. The product of gas pressure p and gap dimension d spans from 0.6 to 12.0 Pa m; 18 collisional reactions for argon discharge are considered, including both linear and nonlinear collision processes in terms of similarity theory. The simulation results show that the similarity laws for the densities of electrons (e), atomic ions (Ar+), excited species (Ar∗), and excimers (Ar2∗) are rigorously satisfied at smaller pd values. However, violations of similarity relations are observed with increasing pd, which is related to pronounced contributions from nonlinear collision processes. The two-dimensional spatial distributions of the electron density with consideration of edge effects are shown to be the same when the gas pressure, gap dimension, and driving frequency are simultaneously tuned. Furthermore, a similarity relation for the excimer Ar2∗ density was theoretically derived, which aligns with the numerical simulations. The results of this study suggest potential applications of similarity laws for rf plasmas across a broader range of parameter regimes.