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Large-area capacitive discharges driven at frequencies higher than the usual industrial frequency of 13.56 MHz have attracted recent interest for materials etching and thin film deposition on large-area substrates. Standing wave and skin effects can be important limitations for plasma processing uniformity, which cannot be described by conventional electrostatic theory. An electromagnetic theory is developed for a discharge having two plates of radius R and separation 2l, which accounts for the propagation of surface and evanescent waves from the discharge edge into the centre and the role of capacitive and inductive fields in driving the power absorption. Examples of discharge fields are given having substantial standing wave and/or skin effects. The conditions for a uniform discharge without significant standing wave and skin effects are found to be, respectively, λ 0 2.6(l/s) 1/2 R and δ 0.45(dR) 1/2 , where λ 0 is the free space wavelength, s is the sheath width, δ = c/ω p is the collisionless skin depth, with c the speed of light and ω p the plasma frequency, and d = l − s is the plasma half-width. Taking the equality for these conditions for a discharge radius of 50 cm, plate separation of 4 cm, and sheath width of 2 mm, there is a substantial skin effect for plasma densities 10 10 cm −3 , and there is a substantial standing wave effect for frequencies f 70 MHz.

Benchmarking is generally accepted as an important element in demonstrating the correctness of computer simulations. In the modern sense, a benchmark is a computer simulation result that has evidence of correctness, is accompanied by estimates of relevant errors, and which can thus be used as a basis for judging the accuracy and efficiency of other codes. In this paper, we present four benchmark cases related to capacitively coupled discharges. These benchmarks prescribe all relevant physical and numerical parameters. We have simulated the benchmark conditions using five independently developed particle-in-cell codes. We show that the results of these simulations are statistically indistinguishable, within bounds of uncertainty that we define. We therefore claim that the results of these simulations represent strong benchmarks, that can be used as a basis for evaluating the accuracy of other codes. These other codes could include other approaches than particle-in-cell simulations, where benchmarking could examine not just implementation accuracy and efficiency, but also the fidelity of different physical models, such as moment or hybrid models.We discuss an example of this kind in an appendix. Of course, the methodology that we have developed can also be readily extended to a suite of benchmarks with coverage of a wider range of physical and chemical phenomena. * Electronic address: miles.turner@dcu.ie 2

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Dual frequency capacitive discharges are designed to offer independent control of the flux and energy of ions impacting on an object immersed in a plasma. This is desirable in applications such as the processing of silicon wafers for microelectronics manufacturing. In such discharges, a low frequency component couples predominantly to the ions, while a high frequency component couples predominantly to electrons. Thus, the low frequency component controls the ion energy, while the high frequency component controls the plasma density. Clearly, this desired behaviour is not achieved for arbitrary configurations of the discharge, and in general one expects some unwanted coupling of ion flux and energy. In this paper we use computer simulations with the particle-in-cell method to show that the most important governing parameter is the ratio of the driving frequencies. If the ratio of the high and low frequencies is great enough, essentially independent control of the ion energy and flux is possible by manipulation of the high and low frequency power sources. Other operating parameters, such as pressure, discharge geometry, and absolute power, are of much less significance.