A Markov chain approach to simulate Atomic Layer Deposition chemistry and transport inside nanostructured substrates

  • Angel Yanguas-Gil
  • Jeffrey W. Elam
Regular Article
Part of the following topical collections:
  1. Modeling Chemical Vapor Deposition and Atomic Layer Deposition


In this work, we present a new theoretical framework to model the transport and surface chemistry under molecular (Knudsen) flow. Our approach is based on casting the transport inside nanostructures as a single-particle discrete Markov chain process. One of the advantages of this approach is that it allows us to decouple the complexity of the surface chemistry from the transport model, thus allowing its application under general surface chemistry conditions, including atomic layer deposition (ALD) and chemical vapor deposition (CVD). Our model also allows us to determine statistical information of the trajectory of individual molecules, such as the average interaction time or the number of wall collisions for molecules entering the nanostructures as well as to track the relative contributions to thin-film growth of different independent reaction pathways at each point of the feature. This offers a straightforward way of incorporating into ALD simulations non-ideal surface processes, such as parasitic CVD or surface recombination. By studying the asymptotic behavior of the Markov chain process, we were also able to establish a direct link between ballistic models, kinetic Monte Carlo simulations, and continuous models based on the use of the diffusion equation under Knudsen conditions. Finally, we show that, under certain approximations, the coverage profile inside a nanostructure under ALD conditions is controlled by the total exposure, and not by the details of the surface flux dependence with time during the exposure, as long as the reaction probabilities are pressure independent.


Atomic Layer Deposition Chemical Vapor Deposition Ballistic transport Nanostructured features Conformality Step-coverage 



This work was sponsored in part by the U.S. DOE, EERE-Industrial Technologies Program under FWP-4902A. JWE was supported as part of 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. An implementation of the Markov chain model of ballistic transport will be made available at

Supplementary material

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Supplementary material 1 (PDF 1731 kb)


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Copyright information

© Springer-Verlag Berlin Heidelberg (outside the USA) 2014

Authors and Affiliations

  1. 1.Energy Systems DivisionArgonne National LaboratoryArgonneUSA

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