Non-planar silicon oxidation: an extension of the Deal-Grove model

Abstract

Silicon oxidation has been the cornerstone of the semiconductor industry for many years, so understanding and being able to predict the oxidation process is paramount. The most popular model to date is the Deal-Grove model for the thermal oxidation of planar silicon surfaces. The Deal-Grove model owes its popularity to the overall simplicity in which it was derived and the accuracy in which it predicts the oxidation of planar silicon geometries. Due to this popularity and accuracy it is desirable to extend the Deal-Grove model beyond flat surfaces to other geometries such as cylinders and spheres. Extending the Deal-Grove model to these types of geometries would allow the prediction of the oxidation of silicon nano-wires and silicon nanocrystals. Being able to predict the oxidation is attractive due to the recent progress of integration of silicon nano-wires and silicon nano-crystals into microelectronic devices. Prediction of the oxidation of silicon cylinders (nano-wires) and spheres (nano-crystals) by simply utilizing the established planar Deal-Grovel model results in highly exaggerated oxide thicknesses compared with empirical data. This exaggeration for small silicon cylinders and spheres is due to the effects of the reduction in the available surface area for oxidation along with the stress induced due to the volumetric expansion and viscous flow of the oxide on non-planar surfaces. These stress effects retard the oxidation rate in non-planar silicon geometries with respect to flat surfaces. This reduction in the oxidation rate reduction is caused by the normal compressive stress which is normal to the SiO[subscript]2/Si interface due to the volumetric expansion during oxidation. This compressive stress reduces the reaction rate constant at the SiO[subscript]2/Si interface and thus retards the overall oxidation rate for silicon cylinders and spheres with respect to planar silicon. The focus of this paper will be to contrast cylindrical and spherical versions of the Deal-Grove model to the well established planar version. Surface area and stress effects will also be explored as they help explain the reduction in the oxidation rate for non-planar silicon geometries.