Electron heating by sheaths in radio frequency discharges

The detailed interaction of individual electrons with oscillating radio frequency (rf) sheaths is considered in this reexamination of the sheath heating process. We develop an analytic expression for the energy delivered to the discharge through sheath heating and that lost due to electron escape to the electrode. Using a simple model of the time-dependent sheath electric field, we find that electron energy gain due to this process, averaged over an rf period is roughly proportional to the square of the maximum sheath speed and drops slightly with increasing electron temperature. Energy loss due to escaping electrons on the other hand, rises with temperature and ultimately outweighs gain when the average electron velocity exceeds the maximum sheath velocity. The requirement of a net power input to the plasma places upper bounds on bulk electron temperature and density attainable by plasmas to be maintained solely by this mechanism, independent of other aspects of container geometry. However, by increasing discharge frequency, power input can be increased for a fixed sheath voltage while reducing electron escape.