Calculation of the electrode shape for suppression of the standing wave effect in large area rectangular capacitively coupled reactors

The electromagnetic standing wave becomes one of the main sources of plasma nonuniformity in large area capacitively coupled rf reactors. In cylindrical reactors with a central rf connection or one-dimensional linear reactors with rf connections at both extremities, Gaussian shaped electrodes can be used to suppress this standing wave. In this work, we present a two-dimensional quasiplanar circuit model and a numerical method for calculation of the electrode shape that can suppress the standing wave effect in large area rectangular reactors. It is shown that the calculated shapes are not Gaussian, and are not only a function of the reactor dimensions and excitation frequency, but are also strongly influenced by the position and number of rf connections, as will also be the case for a cylindrical reactor with a noncentral rf connection. However, when a shape has been determined for a given reactor geometry and rf excitation frequency, then it is shown that it remains independent of the plasma provided that the electromagnetic skin depth in the plasma remains large enough such that skin effects remain negligible.