Time-varying impedance of the sheath on a probe in an RF plasma

Langmuir probes used in radiofrequency (rf) discharges usually include compensation elements that minimize the effect of high frequency oscillations in plasma potential. The design of these elements requires knowledge of the capacitance of the sheath on the probe tip, a quantity which varies nonlinearly during the rf cycle. Sheath capacitance has been studied previously for capacitively coupled discharges, where the rf is applied to the electrodes. Here the problem is treated from the standpoint of a small probe in a fluctuating discharge. This work differs from existing literature in that (a) no step model is used and the Debye sheath is treated exactly, (b) the treatment is simple and analytic, (c) the time-variation of the capacitance is explicitly shown, (d) the results are applied to probe design and (e) cylindrical geometry is considered. The rf frequency is assumed low enough that electron transit times can be ignored. We find that when the rf excursions bring the sheath from the Child-Langmuir region into the Debye sheath or electron saturation region, its capacitance has a strongly non-linear behaviour.