Kinetic theory of rf capacitive discharges

In low pressure capacitive rf plasma discharges, stochastic sheath heating, combined with potentials that exclude low energy electrons from reaching the sheath, produces an electron energy probability function (EEPF) which approximates a two-temperature Maxwellian, as seen in both experiments and particle-in-cell (PIC) MonteCarlo simulations. We have used the fundamental kinetic equation to derive a space- and time-averaged kinetic equation to analytically calculate this EEPF. The effect that low energy electrons cannot be heated by stochastic heating is first modeled by a square well, in which electrons with an x component kinetic energy lower than a certain threshold phi are assumed to be prevented from interacting with the sheath heating fields. The model is improved by allowing the stochastic heating to be gradually turned on over a range of electron energies, instead of abruptly. With these approximations and equilibrium conditions for rf discharges, we obtain a self-consistent solution for the quasi-equilibrium discharge parameters valid for low pressures in argon plasma discharges. The effect of different diffusion time scales in the stochastic heating operator is also discussed. The results are in reasonable agreement with experiments and PIC simulations. The theory is then applied to oxygen plasma discharges, which are electronegative. By coupling the kinetic equation with the equilibrium spatial diffusion equations, we obtain a completely self-consistent solution that agrees well with PIC simulation results.