Analytical model for the radio-frequency sheath

A simple analytical model for the planar radio-frequency (rf) sheath in capacitive discharges is developed that is based on the assumptions of a step profile for the electron front, charge exchange collisions with constant cross sections, negligible ionization within the sheath, and negligible ion dynamics. The continuity, momentum conservation, and Poisson equations are combined in a single integro-differential equation for the square of the ion drift velocity, the so called sheath equation. Starting from the kinetic Boltzmann equation, special attention is paid to the derivation and the validity of the approximate fluid equation for momentum balance. The integrals in the sheath equation appear in the screening function which considers the relative contribution of the temporal mean of the electron density to the space charge in the sheath. It is shown that the screening function is quite insensitive to variations of the effective sheath parameters. The two parameters defining the solution are the ratios of the maximum sheath extension to the ion mean free path and the Debye length, respectively. A simple general analytic expression for the screening function is introduced. By means of this expression approximate analytical solutions are obtained for the collisionless as well as the highly collisional case that compare well with the exact numerical solution. A simple transition formula allows application to all degrees of collisionality. In addition, the solutions are used to calculate all static and dynamic quantities of the sheath, e. g., the ion density, fields, and currents. Further, the rf Child-Langmuir laws for the collisionless as well as the collisional case are derived. An essential part of the model is the a priori knowledge of the wave form of the sheath voltage. This wave form is derived on the basis of a cubic charge-voltage relation for individual sheaths, considering both sheaths and the self-consistent self-bias in a discharge with arbitrary symmetry. The externally applied rf voltage is assumed to be sinusoidal, although the model can be extended to arbitrary wave forms, e. g., for dual-frequency discharges. The model calculates explicitly the cubic correction parameter in the charge-voltage relation for the case of highly asymmetric discharges. It is shown that the cubic correction is generally moderate but more pronounced in the collisionless case. The analytical results are compared to experimental data from the literature obtained by laser electric field measurements of the mean and dynamic fields in the capacitive sheath for various gases and pressures. Very good agreement is found throughout.