Thin film growth is a critical process in the manufacture of integrated circuits and other microelectronic and MEMS devices. Rapid developments in the field of microelectronics have placed increasingly stringent requirements on the quality (properties such as uniformity, composition and micro-structure) of such films. In this context, real-time process control of thin film deposition has gained importance. In this contribution, we describe our efforts at devising an optimal control strategy for controlling the evolution of roughness in thin film deposition.
To be able to effectively control the thin film process using advanced control strategies, it is essential that a detailed dynamic model be available. For the length scales that are prevalent in thin film deposition, the dynamics are defined by atomic-level interactions, and these may be described by the master equation. The master equation is a stochastic partial differential equation (SPDE) describing the probabilities of transitions between different configurations of the system. However, the master equation cannot be directly used for control or optimization, as it is typically high-dimensional, and is not available in a closed form. Approximate solutions to the master equation are often obtained using kinetic Monte Carlo (KMC) models [1]; however, KMC models, too, are not in closed form, and can not be used directly for control. To overcome these difficulties in the application of control and optimization to stochastic thin film growth models, research has mainly focussed on (a) the construction of reduced order approximations of the master equation [2], (b) the construction of estimators and observers for surface roughness using the outputs of multiple KMC models [3], and (c) control using alternate SPDE models [4].
The main difficulty with the control approaches described above is that some form of model developed from first principles is directly used in controller implementation. These models are, in general, difficult to develop, and are too complex to use for online control. An alternate approach is to develop control relevant models directly from input-output data and use them for developing control schemes for managing surface properties. In this paper, we propose a data driven approach (using KMC simulations to generate data) to obtain dynamic models relating surface roughness and manipulated temperature. The identified models have been used for control using the model predictive control (MPC) framework.
We use a one-dimensional KMC model for the simulations. For stochastic processes, each KMC simulation gives a different realization of the master equation, and the average over many simulation runs has been used in the generation of data for identification. The KMC model considered three basic mechanisms - adsorption, surface diffusion and desorption; the latter two mechanisms were considered neighbour dependent [2]. The system behaviour falls into three regimes with fuzzy boundaries in the overall operating range of 600-1200K. In the low temperature range (~600 to 800K), growth by adsorption is dominant, the high temperature range (~950 to 1200K) is dominated by surface diffusion, and the intermediate range is not dominated by a single process. KMC simulations were performed for four different temperature profiles in these operating regimes, and the average behaviour over multiple KMC runs was computed for each of these profiles. Data from three of these profiles were used for system identification, and the fourth data set was used to validate the identified model.
KMC simulations showed that, despite being a stochastic process, the time evolution of surface roughness shows a definite trend, which we capture using the system identification approach.
From the viewpoint of control, the operation is similar to a semi-batch process. The controlled output (surface roughness) is a quality variable, and the manipulated input is temperature. The prime concern is the final surface roughness value after growing a certain specified number of layers. In a manner similar to [2], we assume that measured outputs are available every second, and the manipulated input moves are made at the same frequency. To develop a control relevant dynamic model for this process, we use a time series model with integrated input, i.e., with ARMAIX structure. The proposed modification of the conventional ARMAX structure facilitates modelling of the integrating nature of the process.
The model obtained by system identification can only be used to predict ‘mean profiles' and not individual realizations. In this regard, the model identified may not be the ‘best' model for open loop simulations (and hence for generating an open loop optimal profile). However, when the model is used as an observer, the predictions are reasonably accurate. Thus, the model identified is, in a sense, a control relevant model and not an exact model for the evolution of surface roughness. The identified time series model was cast as an innovation form of state-space model (Kalman predictor) for the development of the MPC scheme. The process dynamics is simulated using KMC simulations. It was observed that the total time taken to grow a given number of layers in a given growth regime is more or less constant. Thus, MPC was formulated with a shrinking prediction horizon with a fixed batch time. All the simulations were performed from a clean substrate (zero layers). There was an initial start-up period, where the surface roughness rapidly increased from zero as atoms were adsorbed. To avoid falling into local minima in the initial start-up phase, the initial growth was in open loop for the growth of about 1 layer, after which MPC was started. When the set points were maintained above the physical limit of roughness for this process, the whole input space was activated during control, and the final set point was achieved. This demonstrates the efficacy of the identification and control scheme for thin film growth.
References:
1. A. Chatterjee and D.G. Vlachos, ‘An overview of spatial microscopic and accelerated kinetic Monte Carlo methods', J. Comp.-Aided Mat. Design, 14(2), 253-308 (2007).
2. Gallivan, M. A., R. M. Murray, and D. Goodwin, 'The dynamics of thin film growth: A modeling study', M. Allendorf, and M. Hitchman, eds., CVD XV: Proceedings of the fifteenth symposium of Chemical Vapor Deposition, volume 616, 168, The Electrochemical Society (2000).
3. Lou, Y., and P. D. Christofides, ‘Feedback control of growth rate and surface roughness in thin film growth', AIChE J., 49(8), 2099-2113 (2003).
4. Y. Lou and P. D. Christofides, ‘Feedback control of surface roughness using stochastic PDEs', AIChE J., 51:345-352 (2005).
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