Mathematical modeling of chemical vapor deposition processes and its application to thin film technology

A number of workers in the field of Chemical vapor deposition (CVD) have presented mathematical models in the literature. Some workers were able to produce analytical expressions for the interwafer concentration profile. These analytical expressions were based entirely on zero or first order chemical reaction rates. Until now, it appears that a chemical reaction rate expression that is not zero or first order directly, must be handled by a numerical scheme.; Presented herein is a mathematical model with an analytical interwafer concentration profile. This concentration profile is neither zero nor first order but shifts from zero to first order as the reactor is axially traversed.; The approach used avoids the sometimes cumbersome numerical schemes, while dealing effectively with non-integer order rate expressions characteristic to CVD kinetics. This approach is also amenable to higher order rate expressions such as kC{dollar}sp{lcub}rm n{rcub}{dollar}, n {dollar}>{dollar} 1.; We employ a boundary perturbation technique to reduce a nonlinear system of partial differential equations that was otherwise non-tractable analytically. Essentially, analytical expressions are derivable for the concentration profile in the interwafer region regardless of the kinetic expression's non-linearity.; The proposed model was tested with independently published experimental data. In each case the model predictions compare favorable with the experimental data.; Results show that deposition rates of: silicon nitride from dichlorosilane and ammonia, silicon from silane and silicon dioxide from tetraethylorthosilicate can be explained using a shifting order reaction. Further, the neglect of gas phase reactions did not affect the predicted deposition rates.; Concurrence with experimental results on thickness uniformity (radial) is achieved using this model. Control of nonuniformity on the wafers during a CVD process depends on the magnitude of the Sherwood number. Both experimental data and the proposed model show that surface uniformity improves with diminishing Sherwood numbers.; In this work, it is demonstrated (at least qualitatively) that surface chemical reaction provides the controlling resistance. For the range of concentrations and low pressures used in CVD the interwafer Damkohler number is smaller than unity. If the ratio of reaction velocity to diffusion velocity is larger than unity, uniform surface deposition cannot be expected. This implies the surface process is controlling.