Saturation of front propagation in a reaction-diffusion process describing plasma damage in porous low-k materials

TL;DR

This paper models the propagation of plasma damage in porous low-k materials using a reaction-diffusion system, revealing a transition from diffusive to logarithmic interface motion, with implications for plasma processing in microelectronics.

Contribution

It introduces a reaction-diffusion model that captures the saturation of plasma damage propagation, linking theoretical interface dynamics to experimental plasma damage timescales.

Findings

Interface motion transitions from  t^{1/2} to  \, ln t over time.

Saturation of plasma damage occurs around 1 minute in typical conditions.

Dependencies on porosity and reaction rates are theoretically predicted.

Abstract

We propose a three-component reaction-diffusion system yielding an asymptotic logarithmic time-dependence for a moving interface. This is naturally related to a Stefan-problem for which both one-sided Dirichlet-type and von Neumann-type boundary conditions are considered. We integrate the dependence of the interface motion on diffusion and reaction parameters and we observe a change from transport behavior and interface motion \sim t^1/2 to logarithmic behavior \sim ln t as a function of time. We apply our theoretical findings to the propagation of carbon depletion in porous dielectrics exposed to a low temperature plasma. This diffusion saturation is reached after about 1 minute in typical experimental situations of plasma damage in microelectronic fabrication. We predict the general dependencies on porosity and reaction rates.