Similarity Theory and Scaling Networks for Electromagnetic Wave-Driven Plasmas

Abstract

We demonstrate the scale-invariant behavior of electromagnetic wave-driven radio-frequency plasmas across different dimensional scales. Using two-dimensional electromagnetic particle-in-cell simulations, we show that plasma uniformity remains the same in similar discharges. Building on the concept of similarity laws, we develop scaling networks that effectively relate plasma parameters across varying operating conditions. These results establish a generalized similarity theory derived from the Boltzmann equation coupled with the full set of Maxwell equations, extending the theoretical framework of similarity laws into electromagnetic regimes.

Figures (4)

• Figure 1: (a) Schematic of radially nonuniform rf plasma density with electromagnetic wave effects. (b) Dimensions of the 2D computational domain for the electromagnetic wave-driven rf plasmas.
• Figure 2: (a) Distributions of normalized electron density for the base case (000), the six intermediate cases (100), (010), (001), (011), (101), (110), and the similarity case (000). (b) Radial distributions of the electron density at $z=d/2$. The scaled electron density distributions for the base case and similarity case overlaps, which confirms the scale-invariant nature in similar rf discharges.
• Figure 3: (a) Distributions of the normalized electric field for the base case (000), intermediate cases (100), (010), (001), (011), (101) and (110), and similarity case (111). (b) Radial distributions of the electric field at $z=d/2$. The scaled electric field distributions for the base case and similarity case overlap.
• Figure 4: (a) Electron energy probability functions for compared cases. (b) Scaling networks for the plasma nonuniformity parameter $C_\text{v}$ with $[p$, $g$, $f]$ gradually tuned at different states (S1--S2--S3--S4).