(595a) The Computational Challenges of Simulating Atomic Layer Deposition (ALD) Process Dynamics

Curtisha D. Travis* and Raymond A. Adomaitis

Department of Chemical and Biomolecular Engineering

Institute for Systems Research

University of Maryland

College Park, MD 20742 USA

*Presenting author: cdtravis@umd.edu

Submitted to the 2012 AIChE Annual Meeting, Pittsburgh, PA

A physically based model of atomic layer deposition reaction kinetics is
developed and applied to alumina ALD using water and trimethylaluminum
precursors [1]. The ALD surface reaction models are treated as true
dynamic systems, with continuous ALD reactor operation described by
limit-cycle solutions, numerically computed using a polynomial collocation
technique. We compute a solution by a full discretization of the
differential equations followed by a Newton-Raphson method for the resulting
nonlinear algebraic equations. Because the dynamics of the ALD system are
relatively well-behaved, a low-order polynomial collocation scheme will
work well for the discretization procedure.

To implement the collocation method, we first write the modeling equations
over both half-cycles in vector form, subject to initial conditions for
each full cycle equal to the final conditions at the end of the previous
cycle and a condition requiring that the coverage be continuous between
the two half-cycles. Using a simple polynomial Lobatto collocation
technique over the unit interval with a set number of collocation points
for each half cycle, we compute the discrete-ordinate formulation of the
1st-order differentiation array. We use an orthogonal polynomial collocation
on finite elements discretization procedure for forced-periodic systems,
where the resulting set of nonlinear algebraic equations are solved using
a Newton-Raphson method. It was found that that convergence can be sensitive
to the accuracy of the initial estimate of the solution, therefore, a
predictor-corrector arc-length continuation procedure generally was used
to compute solutions starting from sets of parameters corresponding to
limit-cycle solutions for which accurate solution estimates could be made.

To compute the film growth per cycle (GPC), we compute the number of Al and
O atoms deposited per unit area over one deposition cycle and by numerically
integrating the reaction rates over each half-cycle using quadrature weights
defined at the collocation points. Because the limit-cycle solutions
represent continuous reactor operation, the ALD film composition then can
be computed, which lends to calculation of the growth per cycle. A
representative limit cycle solution computed using the collocation
procedure is shown in Fig. 1 where the state of the surface during each
each exposure cycle (TMA and water half-cycles denoted by A and B,
respectively) is plotted as a function of time. An alternate view of this
limit-cycle solution is shown in Fig. 2, where the changes in surface
-CH3 and -OH groups during the TMA and water exposures are shown. To
illustrate the dependency of GPC as a function of both precursor exposure
levels for a fixed precursor pressure of 1 Torr, a GPC map generated by
the limit cycle solutions is shown in Fig. 2 (right). Model predictions
indicating optimal operating conditions will be discussed.

Figure 1: TMA and water dose dynamics where the red solid curves
correspond to surface -CH3 groups, the blue correspond to surface -OH during
the first A-B cycle starting from a fully hydroxylated surface, and the
dashed line indicates the close-packing limit of methyl groups on the
growth surface. Dotted curves indicate the corresponding limit-cycle
solutions with the dots showing the location of collocation points.

Figure 2: Surface -CH3 and -OH limit-cycle coverage dynamics (left) for
a representative set of alumina ALD operating conditions. The red curve
corresponds to the TMA dose, the blue to water. Alumina GPC (A/cycle) map
(right) as a function of each precursor exposure level, with limit-cycle
conditions marked as +.

Reference

[1] Puurunen, R. L., ``Surface chemistry of atomic later deposition: A case
study of the trimethylaluminum/water system,'' Appl. Phys. Rev. 97 121301 (2005).