Hybrid model for the calculation of ion distribution functions behind a direct current or radio frequency driven plasma boundary sheath

Abstract

A hybrid fluid dynamic/kinetic model is presented which describes the sheath and the presheath regions of dc or rf driven low pressure gas discharges in a realistic and self-consistent way. The model assumes an infinite extended sheath parallel to the electrode, allowing a one-dimensional spatial description. It provides for the presence of multiple positive ion species and their collisional interactions with the neutral background, and takes into account the possibility of a nonharmonic modulation of the sheath potential and the application of an external dc bias; in this work, the model is applied to a two-species capacitively coupled argon and oxygen plasma. The input required by the model consists of the fluxes of the incoming ions, of the modulating current, and of the pressure, the composition, and the temperature of the background gas. On output, the model provides the values of the electric field and of the particle densities within the sheath and the presheath, the total voltage drop across the sheath, and also the energetically and angularly resolved distributions of the positive ions and the energetic neutrals which impinge the material substrate at the boundary. In general, the model is able to treat dc discharges as well as capacitively and/or inductively coupled rf discharges, it thus covers most of the plasmas used in very large scale integration microelectronics manufacturing and other surface modification techniques. Using the model, studies of the energy distributions of the incoming ions have been performed for a wide range of parameters, and the effects of varying process conditions have been investigated. At low and intermediate pressures (p<50 mTorr), the distribution functions of rf driven discharges exhibit a characteristic bimodal structure; this structure disappears with increasing pressure as ion-neutral collisions become significant. A comparison of calculated ion energy distributions with experimental measurements on capacitively coupled argon and oxygen discharges shows excellent quantitative agreement. In addition to the ion energy distribution, the angular distributions of the incident ions at various energies are also discussed as a function of the neutral gas pressure. It turns out that the details of the angular distribution not only depend on the field structure of the sheath itself but also on that of the presheath. The results of the presented model are therefore more reliable than those of previous models which restricted themselves to the sheath region. This high physical accuracy of the presented model, together with its flexibility and its high execution speed, allows its use as a tool for technology-oriented computer-aided design in the microelectronics industry.