Sheath effects with thermal electrons on the resonance frequency of a DC-biased hairpin probe

The dielectric constant of the sheath—whether ionic or electronic—formed around the cylindrical limbs of a hairpin probe is often assumed to be the same as the vacuum value when used in low-temperature plasmas. However, this assumption does not hold true for electron sheaths and electron-permeating ion sheaths, particularly near the plasma potential. This discrepancy significantly impacts the effective dielectric between the cylindrical limbs of a hairpin probe, thereby affecting the theoretically estimated resonance frequency characteristic curve (i.e., resonance frequency vs probe bias). In this study, we investigate the influence of thermal electrons on the dielectric constant of the sheath and, consequently, on the resonance frequency characteristic curve of a DC-biased hairpin probe. It is found that as the electron temperature increases, the peak of the resonance frequency characteristic curve shifts toward higher positive probe bias values. Additionally, the resonance frequency characteristic curve broadens rather than exhibiting a sharp peak at the plasma potential, which is qualitatively validated by measurements obtained in a low-pressure capacitively coupled argon discharge.