Power laws for the spatial dependence of electrical parameters in the high-voltage capacitive RF sheath

Abstract

Following the analytical description of the collisionless high-voltage capacitive RF sheath in Lieberman (1988) and of the collisional high-voltage capacitive RF sheath in Lieberman (1989), electrical sheath parameters are calculated for a square-wave RF discharge current. The interest in the square-wave current lies in the analytical simplicity of the solutions. The spatial dependences of the time-averaged electrostatic potential /spl Phi/~ electric field E~ net-charge carrier density n~, and ion density n/sub i/ are related to the distance x from the plasma/sheath edge by power laws. For the collisionless RF sheath, /spl Phi/~/spl prop/x/sup 3/2/ E~/spl prop/x/sup 1/2/, n~/spl prop/x-/sup -1/2/and n/sub i//spl prop/x/sup -3/4/ (the respective exponents for the collisionless RF sheath are 4/3, 1/3, -2/3, -2/3), For the collisional RF sheath, /spl Phi/~/spl prop/x/sup 2/ E~/spl prop/x/sup 0/ and n/sub i//spl prop/x/sup -1/2/ (the respective exponents for the collisional DC sheath are 5/3, 2/3, -1/3, -1/3). The RF sheath solution is compared with Lieberman's solution for the sinusoidal RF discharge current. The spatial dependences of /spl Phi/~, E~, n~, and n/sub i/ are almost identical for the two current waveforms. For fixed ionic sheath charge and ion current, the sheath width and the time-averaged sheath voltage are similar for the two current waveforms.