Stability analysis of reactive sputtering process

In reactive sputtering, the introduction of reactive gas would create a hysteresis transition from metal to compounds (oxides, nitrides and carbides, etc.) in both target and substrate. The hysteresis transition is characterized by a sudden change in partial pressure, cathode voltage, sputtering rate and fraction of compound formation. Therefore, the stability is an important issue of process control. In this article, a simple mathematical model based on Berg's original proposal is used to study the stability of steady state solutions. In order to facilitate the analysis, several non-dimensional parameters are identified and used for formulation. The stability is then investigated in two folds. For a small variation away from the steady state, the linear perturbation method is utilized to reduce the model to an eigenvalue problem. For large deviation from the steady state, a full numerical analysis is employed to obtain the steady state. Results show that an unsteady system converges to steady states relatively fast at low inflow rates and the positive range of eigenvalue are coherent with the presence of hysteresis loop. With the aid of the eigenvalue, it is also found that when the chemical reaction on the substrate is moderate, a higher sputter yield of the compound leads to a more stable steady state at lower inflow rates. Finally, a convergence parameter similar to the Lyapunov number is introduced for full numerical analysis to give a qualitative measure for the stability at steady state.