Continuum modeling of point defect aggregation during silicon crystal growth and wafer annealing has a long-standing history of success. Numerous studies have shown that these models are able to capture, at least qualitatively, the essential features of point defect diffusion and clustering under a variety of crystal growth and wafer annealing conditions, especially in the case of homogeneous vacancy clustering [1,2,3,4].

On the other hand, process modeling of oxygen precipitation in Czochralski-grown (CZ) crystals has been more challenging due to the inherent complexity of oxygen precipitation and its strong coupling to the point defect distributions (e.g. ref. [5]). Oxygen precipitation is a critically important process because bulk oxide precipitates provide essential gettering, or trapping, sites for metallic atoms that are introduced during wafer processing. It is therefore essential to ensure that these precipitates are present in the bulk region of the wafer before metallic elements are introduced, but also that they are not located at the device-active surface region.

We present a comprehensive continuum model, based on a hybrid Master-Fokker Planck equation formulation, of coupled vacancy aggregation and oxide precipitation. This model is an extension of a recently developed model for vacancy aggregation, which has been demonstrated to provide a quantitatively accurate picture of void formation under a wide range of operating conditions during both crystal growth and wafer annealing [1]. An operator splitting scheme is employed to decouple the point defect and cluster equations. It also gives an added advantage in terms of dimensionality reduction of the cluster sub-problem from 2D to 1D for the crystal growth case and from 1D to 0D for the wafer annealing case. The coupling physics between the oxide precipitation and vacancy aggregation behavior is probed in detail and the model results compared to experimental measurements in both crystal growth and wafer thermal annealing environments. Several model descriptions for oxide precipitation are compared and contrasted.

[1] Thomas A. Frewen, Sumeet S. Kapur, Walter Haeckl, Wilfried von Ammon, Talid Sinno, ``A microscopically accurate continuum model for void formation during semiconductor silicon processing'', J. Crystal Growth 279 (2005) 258-271.

[2] Talid Sinno, Robert A. Brown, ``Modeling microdefect formation in Czochralski silicon'', J. Electrochem. Soc. 146 (1999) 2300-2312.

[3] Milind S. Kulkarni, ``A selective review of the quantification of defect dynamics in growing Czochralski silicon crystals'', Ind. Eng. Res. 44 (2005) 6246-6263.

[4] Masanori Akatsuka, Masahiko Okui, Shigeru Umeno, Koji Sueoka, ``Calculation of size distribution of void defects in CZ silicon'', J. Electrochem. Soc. 150 (2003) G587-G590.

[5] Koji Sueoka, Masanori Akatsuka, Masahiko Okui, Hisashi Katahama, ``Computer simulation for morphology, size, and density of oxide precipitates in CZ silicon'', J. Electrochem. Soc. 150 (2003) G469-G475.