Modeling Interfacial Oxidation of Cylindrically Curved Silicon Surfaces Including Dependencies on Stress From Coupled Elastic Analysis

Abstract

Our model, treating oxide as solid annulus freely expanded from the silicon (Si) consumed due to increased molecular volume whose geometry enables closed-form expression of time as a function of thickness in constant-parameters case, was revised in non-dimensional form maintaining the appearance of the original Si radius. While this constant-parameters case describes oxide thickness decreasing with decreasing Si radius in concave cases as reported from the experiment, in convex cases thickness is instead described to increase with decreasing Si radius, contradicting published experimental observations. Performing stress analysis displacing surfaces of expanded oxide and remaining Si back to their shared interface, stress-dependent solubility, diffusivity, and reaction rate were investigated toward resolving this discrepancy between the model and reported experiments. With stress-dependent parameters, closed-form expression of time as a function of oxide thickness is no longer achieved, with numerical integration instead required to compute oxidation times. If considering solubility or diffusivity to increase with hydrostatic stress or reaction rate to decrease with increasing interface pressure radially, as hypothesized, increasing oxide thickness with decreasing original Si radius in convex cases remains predicted, in conflict with experimental reports in the literature. It is shown that the experimental observation of an oxide thickness decreasing with decreasing Si radius in convex cases is possible if considering reaction rate to instead increase with increasing interfacial pressure. The same may be possible if considering solubility or diffusivity to instead decrease with increasing hydrostatic stress, tuning activation energies describing the strength of such dependence.