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

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).