Analytical model for ion angular distribution functions at rf biased surfaces with collisionless plasma sheaths

Abstract

The article presents an analytical model for evaluation of ion angular distribution functions (IADFs) at a radio frequency (rf)-biased surface in a high-density plasma reactor. The model couples a unified rf sheath model to an assumed ion velocity distribution function-based formulation for determining the IADF under any general rf-bias condition. Under direct-current (dc) bias conditions the IADF profile shape shows a strong dependence on the bias voltage and the ion temperature is relatively independent of the plasma electron temperature, ion density, and the ion mass. The model establishes the importance of rf-bias frequency in determining the IADF. For conditions where the sheath current wave form is sinusoidal, low bias frequencies result in a large-angle tail contribution to the IADF which can potentially lead to poor anisotropic plasma etching behavior. The large-angle tail is absent at higher bias frequencies. An increase in bias power leads to a general narrowing of the IADF, but the large-angle tail for the IADF at low frequencies persists despite increasing bias powers. Therefore, plasma etch anisotropy can be improved by increasing bias powers only if the bias frequency is sufficiently high. Tangential ion drift velocities introduce azimuthal angle dependence on the IADF and a shift in the peak IADF to off-normal polar angles. While the location of the peak IADF in the azimuthal direction is dictated purely by the direction of the drift velocity, the shift in peak IADF in the polar angle depends on both the drift velocity as well as the bias frequency.