Electron Heating in Capacitively Coupled RF Discharges Investigated with the Smooth Step Model

Electron heating in radio-frequency driven capacitively coupled plasmas is studied on the basis of the recently proposed \textit{Smooth Step Model} [R.P.~Brinkmann, \textit{Plasma~Sources Sci.~Technol.} (2015)]. This algebraic model provides an expression for the electric field in a RF modulated plasma sheath transition which yields i) the space charge field in the sheath, ii) the generalized Ohmic and ambipolar field in the plasma, and iii) a smooth interpolation for the rapid transition in between. It was derived via an expansion of an electron fluid model in terms of two smallness parameters, the ratios $\epsilon =\lambda_{\rm D}/l$ of the Debye length $\lambda_{\rm D}$ to the minimum gradient length $l$ and $\eta=\omega_{\rm RF}/\omega_{\rm pe}$ of the RF frequency $\omega_{\rm RF}$ to the electron plasma frequency $\omega_{\rm pe}$. The explicit field formula provided by the Smooth Step Model enables semi-analytic expressions for the phased-resolved and the phase-averaged dissipated power. Comparison with other models of electron heating is made.