Mathematical Modeling of Cyclic Voltammogram Curves of Copper Deposition

The manufacturing of interconnects and the packaging of integrated circuits are achieved with electrodeposition of copper or other metals. In order to increase the rate of deposition, especially for the large features in packaging, forced convection is provided with certain agitation mechanisms. Although this reduces deposition time, it leads to non-uniform mass transport within each feature and between different features. Special organic additives are used in the solution during the process in order to tune the nucleation and growth of metal, as well as to modify the deposition rate and improve the uniformity. A mathematical model to describe the behavior of organic additives in conjugation with fluid flow and features of various geometry and dimensions is very much desired to facilitate chemistry and process development. In order to achieve this, the physiochemical kinetics of additive and their influence on the Cu deposition rate need to be described precisely. This presentation focuses on a method to extract the kinetic parameters describing the combined effect of multiple additives during copper deposition using rotating disk electrode (RDE). The one-dimensional steady state convection-diffusion equation for each of the chemical species including copper is solved by a semi-analytical method for a range of potentials. The boundary conditions of these differential equations are coupled on the surface of the RDE through the surface coverage of the absorbed species. The steady state of surface coverage of the species represents a dynamic equilibrium of three key processes i.e., adsorption, desorption, and consumption (incorporation). When equilibrium is achieved, the net rate of adsorption and desorption becomes equal to the rate of consumption. At each value of potential, the surface coverage of the additives is solved. At first, the solution is obtained with only one species known as suppressor and it was found that in a specific range of voltage and kinetic parameter multiple solutions of the surface coverage exist at same applied potential. This mathematically explains the S-shaped negative differential resistance (NDR) feature in experimental Cyclic Voltammogram (CV) curves. Figure 1 shows three such experimental S-shaped curves for different concentration of suppressors. The NDR region obtained in the theoretical CV curve is sensitive to the kinetic parameters of the additives. It is possible to match the theoretical and the experimental CV curves by optimizing the kinetic parameters. Determination of the kinetic parameters by particle swarm optimization using experimental data for multiple additive concentration will be discussed in detail in this talk.