Parameterization of Transient Models of Defect Dynamics in Czochralski Silicion Crystal Growth
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PARAMETERIZATION OF TRANSIENT MODELS OF DEFECT DYNAMICS IN CZOCHRALSKI SILICION CRYSTAL GROWTH Talid Sinno and Thomas Frewen Department of Chemical Engineering, University of Pennsylvania, Philadelphia, PA 19104 Erich Dornberger, Robert Hoelzl, and Christian Hoess Wacker Siltronic AG Burghausen D84479, Germany ABSTRACT A transient model for point defect dynamics during Czochralski silicon crystal growth is parameterized using detailed experimental data generated under varying crystal growth conditions. Simulated Annealing is used to perform the model parameterization because of the complex nature of the defined objective functions. It is shown that the method is robust and despite the computational expense associated with a large number of function evaluations, can be readily used in multiple dataset, multiple objective function environments. INTRODUCTION One of the most critical elements in the development of predictive models for defect and dopant dynamics in silicon crystal growth and wafer processing is the thermophysical property database for point defects. Despite many years of theoretical and experimental attempts at determining accurately the diffusion coefficients and equilibrium concentrations of native point defects in silicon, substantial uncertainty in estimates for these values still exists, rendering defect dynamics models semi-quantitative and without predictive capability. Reasons for these difficulties arise from the fact that self-interstitials and vacancies cannot be directly observed experimentally and must be inferred indirectly by model parameterization [1]. Furthermore, while ab initio calculations of these parameters are increasingly accurate, they are generally not of sufficient accuracy for direct use in continuum models [2]. The latter have been found to be extremely sensitive to the assumed values of the point defect diffusivities, equilibrium concentrations and recombination rate [3]. An important difficulty in the parameterization of point defect dynamics models using experimental data is the number of fitting parameters that must be considered simultaneously. Typically, even the simplest diffusion-reaction models for describing point defect distributions during crystal growth or wafer thermal annealing contain several unknown parameters. The sensitivity of the model “goodness-of-fit” to each of these parameters varies dramatically and makes the task of finding a unique set of parameters difficult. As a result, models are usually sub-optimally parameterized and the estimated values for the point defect properties are not valid for phenomena other than the one used for the model regression. This work describes the use of a transient Czochralski (CZ) crystal growth experiment as a database for estimating point defect thermophysical properties. A stochastic global optimization method, Simulated Annealing [4] (SA), is used to parameterize the model to the experimental data. While the convergence of SA is difficult to ascertain practically [5], it allows for S8.3.1
optimization of non-con
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