Spherical shell model of an asymmetric rf discharge

A spherical shell model is used to study ion transport and bias voltage formation in asymmetric, capacitive rf discharges, which have unequal areas A and glow-to-electrode voltages V at the powered (a) and grounded (b) electrodes. Ions are generated by thermal electron ionization and are lost by ambipolar diffusion in the glow. Resonant charge transfer with a constant cross section is assumed to dominate the ion transport. We obtain the density ratio scaling n/sub a//n/sub b/proportional(A/sub b//A/sub a/ )7/sup //24, where n is the density at the glow-sheath edge. Three electrode sheath models are considered: collisionless ions, collisional (constant mobility) ions, and a constant-ion cross-section collisional law. Using these and the continuity of the rf current flow, we obtain the scaling of the electrode voltage ratio with the electrode area ratio: V/sub a//V/sub b/proportional(A/sub b//A/sub a/ )/sup q/. For typical rf materials processing discharges, the constant cross section law yields q≅2.21. The effects of secondary electron ionization and local ionization near the sheaths due to stochastic heating are shown to further reduce the value of q