**5. Diffusion kinetic of formation of the microvoids and dislocation loops**

As mentioned earlier the defect formation processes in a semiconductor crystal, in general, and in silicon, in particular, have been described using the model of point defect dynamics; in this case, the crystal has been considered a dynamic system and real boundary conditions have been specified. However, the model of point defect dynamics has not been used for calculating the formation of interstitial dislocation loops and microvoids under the

The Diffusion Model of Grown-In Microdefects Formation

single crystals.

During Crystallization of Dislocation-Free Silicon Single Crystals 629

The data of the computational experiment on the determination of the microvoid concentration correlate well with the experimentally observed results (104 ... 105 cm–3) (Itsumi, 2002). For the A*-*microdefects, for which the concentration according to the experimental data is ~106…107 cm–3 (Petroff & Kock, 1975; Foll & Kolbesen, 1975), the discrepancy is as large as three orders of magnitude. This can be explained by the fact that, unlike microvoids, which are formed only through the coagulation mechanism, the formation of A-microdefects occurs according to both the coagulation mechanism and the mechanism of prismatic extrusion (deformation mechanism) (V.I. Talanin & I.E. Talanin, 2006a). The results of the calculations suggest that the main contribution to the formation of A-microdefects is made by the mechanism of prismatic extrusion when the formation of interstitial dislocation loops is associated with the relieving of stresses around the growing precipitate. Consequently, the impurity precipitation processes that proceed during cooling of the crystal from the crystallization temperature are fundamental (primary) in character and determine the overall defect formation process in the growth of dislocation-free silicon

The calculations of the formation of microvoids and dislocation loops (A-microdefects) demonstrated that the above assumptions do not lead to substantial differences from the results of the previous calculations in terms of the model of point defect dynamics. This circumstance indicates that the mathematical model of point defect dynamics can be adequately used on the basis of the physical model in which the impurity precipitation process occurs before the formation of microvoids or interstitial dislocation loops. Moreover, the significant result of the calculations is the confirmation of the coagulation mechanism of the formation of microvoids and the deformation mechanism of the formation of interstitial dislocation loops. Therefore the model of the dynamics of point defects can be considered as

Model vacancy coalescence is a simplified model for the analysis of individual parameters of process of the formation microvoids. Detailed calculations are presented in the articles (V.I. Talanin & I.E. Talanin, 2010c). The fundamental interaction between impurities and intrinsic point defects upon crystal cooling under certain thermal conditions (*Т* < 1423 К) leads to impurity depletion and the formation of a supersaturated solid solution of intrinsic point defects. The decay of this supersaturated solid solution causes the coagulation of intrinsic

An analysis of the experimental and calculated data within model of the vacancion coalescence in accordance with the heterogeneous diffusion model of the formation of grown-in microdefects revealed the following reasons for the occurrence of microvoids in dislocation-free silicon single crystals: (i) a sharp decrease in the concentration of background impurity that was not associated into impurity agglomerates (formed in the cooling range of 1683…1423 К); (ii) a large (over 80 mm) crystal diameter (in this case vacancies fail to drain from the central part of the crystal to the lateral surface); (iii) crystals of large diameter generally contain a ring of D*-*microdefects which forms due to the emergence of the (111) face on the crystallization front and which depletes the region inside

component part of the diffusion model for formation grown-in microdefects.

**5.3 Model of the vacancy coalescence** 

point defects in the form of microvoids.

with impurity atoms.

assumption that the recombination of intrinsic point defects is absent in the vicinity of the crystallization front. This fact is evidenced by experimental and theoretical investigations (V.I. Talanin & I.E. Talanin, 2006a, 2007a).
