Analysis Of Wave Equation's Gaussian Process Model And Its Applications On Silicon Wafer Slicing Process

Wave equation, as a widely used Partial Differential Equation (PDE), is utilized to describe the vibration performance of elastic bodies. As a deterministic model, wave equation does not take error factors into consideration. On the other hand, the conventional statistical models have the capability to model the error factors, but are limited to incorporate the engineering knowledge and the physical correlations between the variables, which may lead to unreasonable results. These two models have limitations to predict and control the real wave processes.In this study, a Gaussian Process (GP) model is developed for the wave equation, named as"PDE-GP"model, to overcome the limitations of the pure engineering models and conventional statistical models. The proposed PDE-GP model has the capability to eliminate the influence of the error factors and incorporate the characteristics described by the wave equation. The PDE-GP model decomposes the experimental data as four parts, i.e., the mean part, the systematic errors presented as linear combinations of basis functions, the systematic errors modeled as a Gaussian Process, and the random errors caused by the measurement errors. The decomposition improves the prediction accuracy of the model. Independent of the external interference, the basis functions of the PDE-GP model are the intrinsic functions of the wave equation. The physical correlation of the process variables are presented by the basis functions and incorporated into the PDE-GP model. Based on the PDE-GP model, not only the mean part but also the systematic errors could be controlled in the process.Widely used in wafer manufacturing, the multi-wire saw slicing technology is a typical manufacturing process described by the wave equation. Determining the thickness uniformity and kerf loss of the sliced wafers, the Material Removal Rate (MRR) is an important quality concern for the slicing and the down stream processes such as lapping and polishing. In this study, based on the analysis to the wire saw's vibration and the pseudo-hydrodynamic characteristics of the slurry film, an engineering model is developed, containing the theoretical formulation of MRR, the wave equation of the wire saw, and the Reynold's equation of the slurry film. Because the wave process is influenced by the external interference, the wire saw's wave equation has limitations to present the real wave process. Furthermore, it is infeasible to solve the wave equation analytically or numerically because it is coupled with the Reynold's equation. Therefore, a PDE-GP model is developed for the multi-wire saw slicing process based on the Fourier series expansions and Global Galerkin discretization of the wave equation. The model validation is conducted using the experimental data of the multi-wire saw slicing process. With the same sample set, the results of the cross-validation illustrate that the PDE-GP model has better prediction performance than the widely used Universal Kriging model and the regression model. The optimization strategy based on the PDE-GP model is also developed and validated by the experimental data.At last, several forms of the PDE-GP model are proposed for kinds of wave equations. The PDE-GP modeling approach developed for the wave equation is also extended for a common engineering model.