Determining sheath edge electric field around cylindrical pins of a DC-biased hairpin resonator probe

Abstract

The sheath-edge electric field (E_s) is an important parameter to patch the quasi-neutral pre-sheath and non-neutral sheath regions. The choice of E_s significantly influences the theoretically estimated values of the sheath width, potential, and ion density distribution inside the sheath, as determined by the Poisson equation. The precise nature of E_s has been a persistent subject of investigation, giving rise to the question of whether it should be zero or possess a finite value, as proposed by various authors. In this study, we determine the values of E_s by solving Poisson's equation as a boundary-value problem, utilizing experimentally determined values of sheath radius from a DC-biased hairpin probe. The obtained values of E_s are found to be finite and closely align with the analytical expressions presented by K-U Riemann [J. Phys. D: Appl. Phys. 24 493 (1991)] and Igor D. Kaganovich [Phys. Plasmas 9, 4788 (2002)]. Additionally, the impact of electron-penetrating sheaths and interacting sheaths on the applicability of the hairpin probe in low-pressure plasmas is briefly discussed.