(546a) Construction of Stochastic Differential Equation Models for Porous Thin Film Deposition Processes

Hu, G., University of California, Los Angeles
Zhang, X., University of California, Los Angeles

In many industrial applications, structural defects in thin film growth processes, usually expressed in terms of thin film porosity, play an important role in determining electrical and mechanical properties of the thin films. For example, low-k dielectric films of high porosity are being used in current interconnect technologies to meet resistive capacitive (RC) delay goals and minimize cross-talk. However, increased porosity negatively affects the mechanical properties of dielectric films, increasing the risk of thermo-mechanical failures [1]. Furthermore, in the case of gate dielectrics, it is important to reduce thin film porosity as much as possible and eliminate the development of holes close to the interface.

Kinetic Monte Carlo (kMC) methods are widely employed in the modeling of thin film growth and porosity. However, kMC methods are not suitable for the design of model-based control systems owing to their computational demand and the need to carry out calculations in real-time in feedback control systems. Stochastic differential equation (SDE) model provide an attractive alternative, as the basis for control system design, owing to their potential to capture key process behavior from a control point of view and their computational efficiency [2,3]. SDE models can be derived from the corresponding master equations and their coefficients can be estimated from kMC data. Successful applications of SDE modeling of the surface height profile have been done in the context of thin film growth processes, such as SDE modeling of surface height profile in random deposition with surface relaxation deposition processes, ballistic deposition processes and sputtering processes [4]. However, there are no works that have focused on SDE modeling of thin film porosity.

In this work, a method is presented for the construction of lumped and spatially distributed SDE models that describe the evolution of the film porosity in thin film deposition process. Film density is introduced into the SDE model to represent porosity and is defined as the ratio between the numbers of occupied sites and total bulk sites. The data for model identification are generated from a lattice-based kMC simulator of the deposition process, which allows porosity during film growth. Two microscopic events are considered in the process: random deposition and thermal diffusion. Vacancies and overhangs are allowed in the kMC model to introduce porosity, which is mathematically represented by the thin film density. Various lumped and spatially distributed SDE models are constructed. The coefficients of the SDE models are estimated by least-square fitting method to match the profiles of expected values and variances of the film density of the SDE model to that of the kMC data. The dependence of the model coefficients of the SDE on the process parameters, i.e., substrate temperature and deposition rate, is investigated and expressed in terms of explicit functions. Simulation results demonstrate that the constructed SDE models reproduce well the kMC model behavior for a variety of operating conditions.

1. Kloster, G., Scherban, T., Xu, G., Blaine, J., Sun, B. and Zhou, Y. Porosity effects on low-k dielectric film strength and interfacial adhesion. Proceedings of the IEEE 2002 International; 242-244.

2. Edwards, S. F. and Wilkinson, D. R. The surface statistics of a granular aggregate. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1982;381:17-31.

3. Lauritsen, K. B., Cuerno, R. and Makse, H. A. Noisy Kuramoto-Sivashinsky equation for an erosion model. Physical Review E 1996; 54:3577-3580.

4. Hu, G., Lou, Y. and Christofides, P. D. Model Parameter Estimation and Feedback Control of Surface Roughness in a Sputtering Process. Chemical Engineering Science 2008;63:1800-1816.