Sensitivity Analysis of the Deal‐Grove Model Parameters: Problem Definition and Strategy Considerations

Abstract

In this work the Deal-Grove (DG) oxidation model, being the key ingredient in many current oxidation models, has been examined in detail from the physical and numerical point of view. It is clearly proven that for a rather broad range of (B/A, B) combinations the fitting error is smaller than the experimental error. So there exists a wide range of allowable parameters of the Deal-Grove model to satisfy different physical explanations. This means that a good fit alone is not a sufficient criterion to validate a possible physical model. It is also demonstrated that the existence of breakpoints in the temperature behavior could be seen as an artifact of the model parameters extraction procedure normally used. Therefore, a new and efficient parameter extraction strategy, which carefully considers the influence of experimental errors, is suggested in this work. As an example, a physically acceptable expression for the parameters in the HCl-oxidation is found as a substitute for the look-up table used in SUPREM-3.The Deal-Grove model (1) has been used over more than twenty years to describe the silicon oxidation process. This model is based on an equilibrium process between the. diffusion of the oxidizing species through the oxide layer and the reaction of this species at the interface, resulting in both a parabolic and a linear growth rate component.Although th!s model is widely being used to describe the oxidation process, it suffers from a lack of accuracy mainly for the description of the growth of thin layers. To modify this model in the initial phase of dry oxidation, a lot of models were proposed over the last decade: e.g., oxide charge effects (2), chemisorption (3), structural channels (4), stress effects (5), parallel independent diffusion mechanisms of the oxidizing species in the oxide film (6, 7), and interface potential effects (8).Nearly all of these models exhibit a quite good data fitting, although they are based on some different physical assumptions and in most cases exclude each other implicitly. The .data fitting and the observations of the variation of the linear-rate constant B/A and the parabolic-rate constant B of the Deal-Grove model as a function of some factors (e.g., temperature, 02 pressure, HCI%...), are the usual way to provide evidence for the proposed models. For example, some authors have observed breakpoints in the In (B/A) vs. 1/KT and in (B) vs. 1/KT plots, and to explain these breakpoints the stress effect has been proposed (5).However, in this work, it will be indicated that there exists an inherent problem with the Deal-Grove model parameters (linear-parabolic law) which may raise some questions about a number of physical phenomena which are used to explain some observed effects. In effect, it turns out that the "exact" values of the parameters B/A and B are quite insensitive to the Deal-Grove model, especially when the oxide fitting range is getting smaller. Namely, by allowing a domain of sets of (B/A, B) values, the fitting error for many (B/A, B) combinations is less than th...

The Effect of Aqueous Chemical Cleaning on Si (100) Dry Oxidation Kinetics

“…While many other physical process may be important (27) [i.e., stress (28)(29)(30), oxide structure (31), ionic transport ( 14)], we assume that these additional factors are adequately described by the parameters of our models for the experimental conditions considered herein. Despite the difficulties that can be encountered in extracting rate constants from experimental data (32), and the limitations imposed on data interpretation by the choosing of a particular class of oxidation models (33), we will proceed with an analysis of the kinetics data.…”

The Implementation of the Response Surface Techniques to the Analytical Oxidation Model Evaluation

“…The results are shown in Fig. 7a and 7b, which are taken from our previous paper (15). Therefore, it is insufficient to use the obtained breakpoints as evidence to conclude that this intrinsic stress predominantly affects the kinetics behavior of the diffusion process or the reaction to form SiO2.…”

“…Whenever one uses the parameter values to support or reject some new ideas, it is important to ask how reliable they are. It was indicated (15,26) that one will be misled in the oxidation kinetics studies, if one relies too much on data-fitting. In this work, we introduced mathematical techniques to evaluate the parameter interdependence and uncertainty, and to justify the need for independent measurements of some physical parameters, e.g., diffusivity.…”

“…Since the authors assumed that stress relaxation occurs around T = 950~ (2), the importance of the modified term concerning the stress effect in this model will be decreased if By using the extreme values as defined above in the model, i.e., Eq. [15], without modification, the fitting errors are about 1.82% and 1.83%, respectively.…”

“…It was discussed in our previous paper (15) that there is no good reason to prefer 0* over any other value of 0 for which 0 r O* and JR(O) -R(e*)l-< 9 [2] where 9 is defined as the experimental error. This means that all parameter vectors e with a residual R(O) less than the experimental error 9 should be considered as acceptable.…”