Bias voltage in finite length, cylindrical and coaxial radio‐frequency discharges

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Abstract
We obtain the ratio Va/Vb of the direct‐current glow‐to‐electrode voltages at the powered (a) and grounded (b) electrodes in low pressure capacitive radio‐frequency (rf) discharges having unequal electrode areas Aa and Ab in finite length cylindrical and coaxial geometry. We use a transport model in the glow in which ions are generated by thermal electron ionization and are lost by ambipolar diffusion in the glow, with a constant diffusion coefficient. Three electrode sheath models are considered: collisionless ions, collisional ions with a constant mean free time, and collisional ions with a constant mean free path. Using these and the continuity of the rf current flow, we obtain the voltage ratio as a function of the electrode area ratio, the length‐to‐radius ratio of the cylinder, and, for coaxial systems, the ratio of inner‐to‐outer radii. The theory is in good agreement with experiments in cylindrical and coaxial geometry.

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