Diffusion Mechanisms in Silicon nanowires
Authors: Mertens Robert George , Velez Victor H. , Sundaram Kalpathy B 
Published: February 2020

In this study we use Fick’s Laws and the properties of bulk silicon (Si) to calculate the diffusion of the dopants, phosphorus and boron, in silicon nanowires (SiNWs). The morphology of SiNWs does not allow the use of the same formulas that are used in the doping and diffusion of bulk Si. While Fick’s Laws still apply, the law as applied is a differential equation and different solutions are used in order to solve this cylindrical problem. An oxidation drive-in is considered in these solutions
.

This is a study on diffusion doping of Si with boron and phosphorus, fairly well understood processes for the case of planar Si, which could perhaps be improved upon, but for the greater part is mathematically complete (refs). Having a long history, dating back to 185530
,
31, and basis in the science of diffusion, even cylindrical diffusion problems are somewhat well understood. Moving boundary diffusion problems are a bit more complex, as are problems with pre-existing diffusion profiles. Here we conduct an in-depth analysis of the mathematical models of cylindrical diffusion, applying these to the SiNWs. The laws of diffusion are applied to the morphology and physical conditions of the problem to arrive at a better formulation of the calculations than has been provided to date.

Figure 1. Pre-dep doping profiles NcB (cylindrical boron), NcP (cylindrical phosphorus), NbB(planar boron), and NbP(planar phosphorus) using Eqs. (5.11) and (5.12) modeled in Mathcad.

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