**5.1 Kinetics of formation of microvoids**

The experimentally determined temperature range of the formation of microvoids in crystals with a large diameter is 1403…1343 K (Kato et al., 1996; Itsumi, 2002). In this respect, the approximate calculations for the solution in terms of the model of point defect dynamics were performed at temperatures in the range 1403…1073 K. The computational model uses the classical theory of nucleation and formation of stable clusters and, in strict sense, represents the size distribution of clusters (microvoids) reasoning from the time process of their formation and previous history.

The calculations were carried out in the framework of the model of point defect dynamics, i.e., for the same crystals with the same parameters as in already the classical work on the simulation of microvoids and interstitial dislocation loops (A-microdefects) (Kulkarni et al., 2004). According to the analysis of the modern temperature fields used when growing crystals by the Czochralski method, the temperature gradient was taken to be *G* = 2.5 K/mm (Kulkarni et al., 2004). The simulation was performed for crystals 150 mm in diameter, which were grown at the rates *Vg* = 0.6 and 0.7 mm/min. These growth conditions correspond to the growth parameter *Vg*/*G* > ξcrit.

Detailed calculations are presented in the articles (V.I. Talanin & I.E. Talanin, 2010b).Our results somewhat differ from those obtained in (Kulkarni et al., 2004). These differences are as follows: (i) the nucleation rate of microvoids at the initial stage of their formation is low and weakly increases with a decrease in the temperature and (ii) a sharp increase in the nucleation rate, which determines the nucleation temperature, occurs at a temperature T ~ 1333 K. These differences result from the fact that the recombination factor in our calculations was taken to be 0 *kIV* . For 0 *kIV* , consideration of the interaction between impurities and intrinsic point defects in the high-temperature range becomes impossible, which is accepted by the authors of the model of point defect dynamics (Kulkarni et al., 2004). In this case, in terms of the model, there arises a contradiction between the calculations using the mathematical model and the real physical system, which manifests itself in the ignoring of the precipitation process (Kulkarni et al., 2004).

### **5.2 Kinetics of formation of dislocation loops (A-microdefects)**

The computational experiment was performed similarly to the calculations of the formation of microvoids. The simulation was performed for crystals 150 mm in diameter, which were grown at the rates *Vg* = 0.10 and 0.25 mm/min for the temperature gradient *G* = 2.5 K/mm. These growth conditions correspond to the growth parameter *Vg*/*G* < ξcrit.

Detailed calculations are presented in the articles (V.I. Talanin & I.E. Talanin, 2010b). The temperature of the formation of A-microdefects corresponds to ~1153 K. An increase in the crystal growth rate weakly decreases the critical radius of A-microdefects and slightly affects the nucleation temperature. An increase in the crystal growth rate leads to an almost twofold decrease in the concentration of introduced defects.

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

The experimentally determined temperature range of the formation of microvoids in crystals with a large diameter is 1403…1343 K (Kato et al., 1996; Itsumi, 2002). In this respect, the approximate calculations for the solution in terms of the model of point defect dynamics were performed at temperatures in the range 1403…1073 K. The computational model uses the classical theory of nucleation and formation of stable clusters and, in strict sense, represents the size distribution of clusters (microvoids) reasoning from the time process of

The calculations were carried out in the framework of the model of point defect dynamics, i.e., for the same crystals with the same parameters as in already the classical work on the simulation of microvoids and interstitial dislocation loops (A-microdefects) (Kulkarni et al., 2004). According to the analysis of the modern temperature fields used when growing crystals by the Czochralski method, the temperature gradient was taken to be *G* = 2.5 K/mm (Kulkarni et al., 2004). The simulation was performed for crystals 150 mm in diameter, which were grown at the rates *Vg* = 0.6 and 0.7 mm/min. These growth conditions

Detailed calculations are presented in the articles (V.I. Talanin & I.E. Talanin, 2010b).Our results somewhat differ from those obtained in (Kulkarni et al., 2004). These differences are as follows: (i) the nucleation rate of microvoids at the initial stage of their formation is low and weakly increases with a decrease in the temperature and (ii) a sharp increase in the nucleation rate, which determines the nucleation temperature, occurs at a temperature T ~ 1333 K. These differences result from the fact that the recombination factor in our calculations was taken to be 0 *kIV* . For 0 *kIV* , consideration of the interaction between impurities and intrinsic point defects in the high-temperature range becomes impossible, which is accepted by the authors of the model of point defect dynamics (Kulkarni et al., 2004). In this case, in terms of the model, there arises a contradiction between the calculations using the mathematical model and the real physical system, which manifests

The computational experiment was performed similarly to the calculations of the formation of microvoids. The simulation was performed for crystals 150 mm in diameter, which were grown at the rates *Vg* = 0.10 and 0.25 mm/min for the temperature gradient *G* = 2.5 K/mm.

Detailed calculations are presented in the articles (V.I. Talanin & I.E. Talanin, 2010b). The temperature of the formation of A-microdefects corresponds to ~1153 K. An increase in the crystal growth rate weakly decreases the critical radius of A-microdefects and slightly affects the nucleation temperature. An increase in the crystal growth rate leads to an almost

itself in the ignoring of the precipitation process (Kulkarni et al., 2004).

These growth conditions correspond to the growth parameter *Vg*/*G* < ξcrit.

**5.2 Kinetics of formation of dislocation loops (A-microdefects)** 

twofold decrease in the concentration of introduced defects.

(V.I. Talanin & I.E. Talanin, 2006a, 2007a).

**5.1 Kinetics of formation of microvoids** 

their formation and previous history.

correspond to the growth parameter *Vg*/*G* > ξcrit.

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 single crystals.

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