High Knudsen number physical vapor deposition: Predicting deposition rates and uniformity

The problem of predicting deposition rates and film thickness variation is relevant to many high-vacuum physical vapor deposition (PVD) processes. Analytical methods for modeling the molecular flow fail when the geometry is more complicated than simple tubular or planar sources. Monte Carlo methods, which have traditionally been used for modeling PVD processes in more complicated geometries, being probabilistic in nature, entail long computation times, and thus render geometry optimization for deposition uniformity a difficult task. Free molecular flow is governed by the same line-of-sight considerations as thermal radiation. Though the existence of an analogy between the two was recognized by Knudsen (1909, Ann. Phys., 4(28), pp. 75-130) during his early experiments, it has not been exploited toward mainstream analysis of deposition processes. With the availability of commercial finite element software having advanced geometry modelers and built-in cavity radiation solvers, the analysis of diffuse thermal radiation problems has become considerably simplified. Hence, it is proposed to use the geometry modeling and radiation analysis capabilities of commercial finite element software toward analyzing and optimizing high-vacuum deposition processes by applying the radiation-molecular flow analogy. In this paper we lay down this analogy and use the commercial finite element software ABAQUS for predicting radiation flux profiles from planar as well as tube sources. These profiles are compared to corresponding deposition profiles presented in thin-film literature. In order to test the ability of the analogy in predicting absolute values of molecular flow rates, ABAQUS was also employed for calculating the radiative flux through a long tube. The predictions are compared to Knudsen's analytical formula for free molecular flow through long tubes. Finally, in order to see the efficacy of using the analogy in modeling the film thickness variation in a complex source-substrate configuration, an experiment was conducted where chromium films were deposited on an asymmetric arrangement of glass slides in a high-vacuum PVD chamber The thickness of the deposited films was measured and the source -substrate configuration was simulated in ABAQUS. The variation of radiation fluxes from the simulation was compared to variation of the measured film thicknesses across the slides. The close agreement between the predictions and experimental data establishes the feasibility of using commercial finite element software for analyzing high vacuum deposition processes.