Demonstration of Similarity Laws and Scaling Networks for Radio-Frequency Plasmas

We experimentally demonstrate similarity laws for capacitive radio-frequency (rf) plasmas, showing that two rf discharges are scale-invariant in geometrically similar systems in which the gas pressure, gap dimension, and driving frequency are proportionally tuned. Spatiotemporal distributions of the excitation rate are measured based on phase-resolved optical emission spectroscopy, and the tendencies of the excitation dynamics scaling with control parameters are presented and agree well with particle-in-cell simulations. Furthermore, similarity-based scaling networks are established, which extensively correlate the discharge states (i.e., the initial, intermediate, and similarity states), enabling an effective strategy for determining scaling relations with fewer experiments. The framework of the scaling networks is interpreted based on the kinetic Boltzmann equation coupled with Poisson's equation. The present work reveals the nature of discharge similarity and provides an additional knob for the exploration of upscaled rf plasma sources for industrial applications, such as large-area etching facilities.