Oxygen Precipitation in Silicon: Numerical Models
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OXYGEN PRECIPITATION IN SILICON:
NUMERICAL MODELS
J. P. LAVINE, G. A. HAWKINS, C. N. ANAGNOSTOPOULOS, AND L. RIVAUD Research Laboratories, Eastman Kodak Company, Rochester, NY 14650 ABSTRACT We present a numerical model that simulates the evolution oxygen in of precipitates and the diffusion of interstitial The growth and/or dissolution of each Czochralski silicon. oxygen precipitate and the local concentration of interstitial with which the precipitates interact are followed as a function of time. We treat realistic densities of discrete, interacting precipitates and determine how the precipitate density influThe model also treats ences the extent of the precipitation. oxygen outdiffusion and the formation of precipitate-free or denuded zones. We apply the model to previous experimental data on the time dependence of precipitate growth and to the development of denuded zones during intrinsic gettering. INTRODUCTION We have developed a numerical model that simulates interstitial oxygen diffusion and precipitation in Czochralski We use this model to study intrinsic gettering [1-31 silicon. and the formation of defect-free regions or denuded zones at The motivation for the model is threefold: the wafer surface. (1) to understand a new intrinsic gettering procedure [4] that is highly effective in suppressing the formation of nearsurface defects; (2) to establish a means of estimating the effects of process variations such as ramping [5]; (3) to investigate the extent to which the assumed laws of growth and dissolution can be uniquely determined from experimental data. Precipitate growth has been described for isolated precipHowitates [6] and for regular arrays of precipitates [7]. ever, the realistic situation of a random distribution of Our model treats this case precipitates has not been treated. with simple assumptions for the forms of the growth and dissolution laws. These laws can be varied to allow a realistic We estimate of their ability to predict experimental data. note that our numerical model treats a set of discrete, interacting precipitates. This removes the restrictions of earlier approaches to oxygen precipitation in silicon that assumed either a continuum [8-10] or noninteracting precipitates [11]. of We describe our numerical model and present the results 0 a study of oxygen outdiffusion and precipitation at 1200 C. NUMERICAL MODEL Our numerical model treats the diffusion of interstitial oxygen Oi by dividing space into cubic cells of edge d and oxygen between nearest-neighbor cells transferring interstitial This procedure amounts to solving at each discrete time step. the diffusion equation
Mat. Res. Soc. Symp. Proc. Vol. 59. @1986 Materials Research Society
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a ___
_20 __2_ý
(1)
In eq. (1) t is the time, D is the by an explicit scheme. oxygen diffusion coefficient [12], x is the interstitial spatial coordinate parallel to the wafer surface, and y is the spatial coordinate perpendicular to the wafer surface, which is The diffusion time is divided into discrete time y = 0.0. steps of (2)
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