An angular distribution function for the sputter-depositing atoms and general equations describing the initial thickness profile of a thin film deposited inside a via and trench by sputtering

We developed an angular distribution function that describes the distribution of directional sputter-depositing atoms: g(θ) = (k sin2θ + k−1cos2θ)−2, where k is a directionality factor that determines the directionality of depositing atoms and θ is the incident angle of a depositing atom. The thickness profiles of the sputtered films deposited inside a vertical trench were simulated using the ballistic transport reaction model in conjunction with the angular distribution function, g(θ). The simulated thickness profile agreed well with the experimentally measured thickness profile. General equations that describe the thickness profile of sputter-deposited films inside vias and trenches were derived. The initial film thickness profile could be predicted by substituting the directionality factor, k, and the geometric parameters of the via or trench in the general equations. An optimum directionality factor, defined as the directionality factor that maximizes the sidewall coverage of a vertical pattern, was obtained by solving the general equations. An ideally-tapered via that maximizes sidewall coverage was identified by optimizing the directionality factors and aspect ratios of the vias.