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Self-Limited Reaction-Diffusion in Nanostructured Substrates: Surface Coverage Dynamics and Analytic Approximations to ALD Saturation Times

Authors

  • Angel Yanguas-Gil,

    1. Energy Systems Division, Argonne National Laboratory, 9700 S Cass Ave, Argonne, IL 60439 (USA)
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  • Jeffrey W. Elam

    Corresponding author
    1. Energy Systems Division, Argonne National Laboratory, 9700 S Cass Ave, Argonne, IL 60439 (USA)
    • Energy Systems Division, Argonne National Laboratory, 9700 S Cass Ave, Argonne, IL 60439 (USA).
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  • This work was supported in part by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Industrial Technologies Program under FWP-4902A. Elam was supported by the Argonne-Northwestern Solar Energy Research (ANSER) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0001059. Argonne is a U.S. Department of Energy Office of Science laboratory, and is operated under Contract No. DE-AC02-06CH11357 by UChicago Argonne, LLC.

Abstract

We present a general model based on a time-dependent, reaction–diffusion equation to determine the dosing times and coverage profiles in structured substrates during atomic layer deposition (ALD). We first derive expressions comprising a non-linear diffusion–reaction equation coupled to a surface kinetic equation. In their non-dimensional forms, these equations show that coverage dynamics during ALD in nanostructured substrates depend only on two non-dimensional parameters, the Damkoler and precursor excess (number of molecules per surface site in the nanostructure) numbers. Using the assumptions of molecular flow in a circular pore, we derive a general, analytic equation to predict saturation exposure times. To demonstrate the utility of our model, we derive additional expressions incorporating a precursor loss term relevant to predicting exposure times during ozone-based ALD. Because our model makes no assumptions about the diffusion coefficient or sample geometry, it can easily be adapted to describe a broad range of ALD systems such as trenches and vias, anodized alumina, or aerogels under almost any conditions including molecular, viscous, and transition flow regimes

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