A Modern Formulation of Knudsen Diffusion with Applications to Nanofabrication

Knudsen diffusion is a transport process characterized by a diffusivity being derived from molecule-geometry interactions instead of molecule-molecule collisions. It is a well-established theory, having been introduced in the early 20th century. Nonetheless, in its earliest days, the development of the Knudsen diffusion theory was troubled with misconceptions regarding its application to geometries with arbitrary cross-sections which were gradually addressed over the subsequent decades. More recently, this theory has been applied to nanoscale processing techniques, most notably CVD and ALD. However, some of the aforementioned historical misconceptions were perpetuated in these recent works. Here, we present a modern formulation of Knudsen diffusion theory which addresses these issue by incorporating a dependence on the view factor. We recover the classical Knudsen diffusivity results for a long cylinder and, subsequently, we discuss the issues which arise when attempting to rigorously derive the commonly-used hydraulic diameter approximation for rectangular trenches. Finally, we apply the Knudsen diffusion theory to two different problems in nanofabrication. We investigate aspect ratio dependent reactive ion etching, highlighting the importance of controlling the interactions with the sidewall passivation layer. Through an innovative integration of Knudsen diffusion, direct visibility contributions, and crystallographic-orientation dependent growth rates within a commercial topography simulator, we are able to reproduce the heteroepitaxial growth of 3C-SiC on Si micro-pillars, showcasing a path for future process optimization.