(453a) Thin-Film Deposition Process Dynamics on Manifolds and Graphs

There is a long history associated with the mathematical analysis of
chemical reaction dynamics to generate reduced-order models,
particularly in homogeneous reaction systems. Combustion reaction
dynamics research, for example, has produced such studies as Maas and
Pope (1992) and Lam and Goussis (1994); the process systems community
has contributed significantly as well (e.g., Vlachos, 1996; Vora and
Daoutidis, 2001). By comparison, heterogeneous reacting systems have
received less attention, especially in the realm of thin-film
processing.

Effective modeling of heterogeneous reaction dynamics is crucial to
Atomic Layer Deposition (ALD) high-throughput system process design
and optimization. In ALD, thin films are deposited in discrete
(usually sub-monolayer) increments by exposing the growth surface to
the gas-phase reactive precursors in an alternating manner. Unlike
chemical vapor deposition, ALD does not possess steady state growth
modes; instead, continuous ALD operation consists of limit-cycle
behavior.

Generating accurate experimental data useful for studying intrinsic
ALD reaction rates is challenging because of the dynamic nature of the
process, the wide time-scale range of the dynamic processes, and the
small surface concentrations of some of the reaction intermediate
species. Therefore, researchers have turned to quantum chemical
computations to model and compare ALD reaction mechanisms (Elliott,
2012). Because these methods primarily produce static information
regarding reaction energetics and transition state configurations, the
transitions between surface states can only be described in terms of
equilibrium relations. However, conventional transition-state theory
does provide rate expressions for the activated surface reactions, and
rate equations for precursor species adsorption and desorption
processes likewise can be formulated.

Given the modeling situation described, we should expect the overall
ALD reaction network (RN) model to consist of a differential-algebraic
equation (DAE) system; a key objective of our work in this area has
been to develop rational methods to formulate ALD reaction kinetics
models in the form of a well-posed DAE system. By writing all
reactions in terms of their net-forward rates, including equilibrium
processes by adding a fictitious time constant $\epsilon << 1$ s, the
pure differential-equation system then can be factored to decouple the
reaction terms (Remmers, et. al, 2015; Vora and Daoutidis, 2001).
Success of this factorization procedure indicates that the outer
solution to this singular perturbation problem retains the dynamics
required to accurately model the deposition process. A number of
reaction-independent modes also are normally produced by this process,
reflecting the conserved quantities of the process and elimination of
the redundant dynamic modes. Computing the span of the species space
associated with the slow and conserved modes while ignoring minor
surface species can further reduce the dynamic dimension of the
deposition system by identifying combinations of finite-rate processes
that approximate new equilibrium relationships.

The stoichiometry array that premultiplies the reaction-rate array of
the model described is a adjacency matrix connecting reaction rates to
the time-rate of change of the reaction species, and so can be used to
create a (bipartite) species-reaction (SR) graph. The SR graph has
been used previously to analyze RN for the potential of multiplicity
(Craciun and Feinberg, 2006). Here we will use SR graphs to provide
additional insight into our factorization procedure and to diagnose
structural problems within thin-film surface RN models.

References

Craciun, G. and M. Feinberg, Multiple equilibria in complex chemical
reaction networks II: The species-reaction graph, SIAM J. Appl. Math.
66 1320-1338 (2006).

Elliott, S. D., Atomic-scale simulation of ALD chemistry, Semicond.
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Lam, S. H. and D. A. Goussis, The CSP method for simplifying kinetics,
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Maas, U. and S. B. Pope, Simplifying chemical kinetics: Intrinsic
low-dimensional manifolds in composition space, Combustion and Flame
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Remmers, E. M., C. D. Travis, and R. A. Adomaitis, Reaction
factorization for the dynamic analysis of atomic layer deposition
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Vlachos, D. G., Reduction of detailed kinetic mechanisms for ignition
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