Capacitive discharges driven by combined dc/rf sources

We have obtained analytic expressions for the sheath voltages and sheath widths for both collisional and collisionless sheaths driven by a combination of dc and rf voltage sources. A “dc/rf sheath” develops on the negatively biased electrode while a typical rf sheath develops on the other electrode. The dc/rf sheath has a dc region with negligible electron density near the negatively biased electrode. Furthermore, if the rf power is held constant (typical for industrial plasmas driven by rf power sources), the sheath voltage drop of the rf sheath is nearly independent of the dc voltage provided the discharge configuration does not lead to a significant increase in the ionization efficiency of the secondary electrons. The analysis is done for both symmetric (equal area) and asymmetric diode discharges, as well as a triode configuration. The analytical results for the symmetric and asymmetric diode discharges are compared to the results of numerical simulations using plane-parallel and cylindrical particle-in-cell (PIC) codes over a wide range of pressures and rf frequencies, finding good agreement. The effect of secondary electrons is also examined with the PIC codes, finding increased discharge efficiency.