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

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 Φ¯ electric field E¯ net-charge carrier density n¯, and ion density n_i are related to the distance x from the plasma/sheath edge by power laws. For the collisionless RF sheath, Φ¯∝x^3/2 E¯∝x^1/2, n¯∝x-^-1/2and n_i ∝x^-3/4 (the respective exponents for the collisionless RF sheath are 4/3, 1/3, -2/3, -2/3), For the collisional RF sheath, Φ¯∝x² E¯∝x⁰ and n_i∝x^-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 Φ¯, E¯, n¯, and n _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