**3. The diffusion model for formation of grown-in microdefects in dislocationfree silicon single crystals**

We propose a new diffusion model of the formation and transformation of grown-in microdefects. It is based on the experimental studies of undoped dislocation-free Si single crystals grown by the floating zone and Czochralski methods. The diffusion model combines the physical model (the heterogeneous mechanism for the formation of grown-in microdefects), the physical classification of grown-in microdefects, and mathematical models of the formation of primary and secondary grown-in microdefects (Fig. 1).

Physical model based on the assumption about the absence of recombination intrinsic point defects at high temperatures. This assumption was confirmed in several experimental works (Talanin et al., 2002a; Talanin et al., 2002b; Talanin et al., 2003). In paper (V.I. Talanin & I.E. Talanin, 2007a) we first theoretically proved the absence of recombination of intrinsic point defects at high temperatures and fast recombination at low temperatures. The experimental data and the results obtained from thermodynamic calculations have demonstrated that the process of aggregation of point defects dominates over the process of recombination of intrinsic point defects. At high temperatures, the process of recombination makes an insignificant contribution to the process of aggregation. Consequently, vacancies and intrinsic interstitial atoms coexist in thermal equilibrium. As a result, intrinsic point defects of both types are simultaneously involved in the process of aggregation. The decomposition of a supersaturated solid solution of point defects occurs upon cooling through two mechanisms, namely, the vacancy and interstitial mechanisms, with the formation of oxygen-vacancy and carbon- interstitial agglomerates.

Absence of recombination intrinsic point defects at high temperatures allows us to propose the physical model of the formation grown-in microdefects. The basic concepts of the physics model for the formation of grown-in microdefects imply the following (V.I. Talanin & I.E. Talanin, 2006a): (i) the recombination of intrinsic point defects at high temperatures can be neglected; (ii) background carbon and oxygen impurities are involved in the defect formation as nucleation centers; (iii) the decay of the supersaturated solid solution of point defects when the crystal is cooled from the crystallization temperature occurs in two independent ways (branches): vacancy and interstitial; (iv) the defect formation is based on primary agglomerates formed as the crystal is cooled from the crystallization temperature due to the interaction between the impurities and intrinsic point defects; (v) when the crystal is cooled at temperatures below 1423 K, depending on the thermal growth conditions, secondary grown-in microdefects are formed due to the interaction between intrinsic point defects; (vi) the secondary grown-in microdefects are formed due to the coagulation (microvoids and A-microdefects) and deformation (A-microdefects) effects; (vii) the vacancy and interstitial branches of the heterogeneous mechanism have a symmetry, which implies simultaneous processes of defect formation during the decay of

An excess concentration of intrinsic point defects (vacancies or silicon self-interstitials) arises when the crystal is cooled under certain thermal conditions (Cho et al., 2006). This process leads to the formation of secondary grown-in microdefects (А-microdefects or microvoids) (V.I. Talanin & I.E. Talanin, 2004). We have proposed the physical classification of grown-in microdefects. It is based on the differences in the physical nature of the formation of

**3. The diffusion model for formation of grown-in microdefects in dislocation-**

We propose a new diffusion model of the formation and transformation of grown-in microdefects. It is based on the experimental studies of undoped dislocation-free Si single crystals grown by the floating zone and Czochralski methods. The diffusion model combines the physical model (the heterogeneous mechanism for the formation of grown-in microdefects), the physical classification of grown-in microdefects, and mathematical

Physical model based on the assumption about the absence of recombination intrinsic point defects at high temperatures. This assumption was confirmed in several experimental works (Talanin et al., 2002a; Talanin et al., 2002b; Talanin et al., 2003). In paper (V.I. Talanin & I.E. Talanin, 2007a) we first theoretically proved the absence of recombination of intrinsic point defects at high temperatures and fast recombination at low temperatures. The experimental data and the results obtained from thermodynamic calculations have demonstrated that the process of aggregation of point defects dominates over the process of recombination of intrinsic point defects. At high temperatures, the process of recombination makes an insignificant contribution to the process of aggregation. Consequently, vacancies and intrinsic interstitial atoms coexist in thermal equilibrium. As a result, intrinsic point defects of both types are simultaneously involved in the process of aggregation. The decomposition of a supersaturated solid solution of point defects occurs upon cooling through two mechanisms, namely, the vacancy and interstitial mechanisms, with the formation of

Absence of recombination intrinsic point defects at high temperatures allows us to propose the physical model of the formation grown-in microdefects. The basic concepts of the physics model for the formation of grown-in microdefects imply the following (V.I. Talanin & I.E. Talanin, 2006a): (i) the recombination of intrinsic point defects at high temperatures can be neglected; (ii) background carbon and oxygen impurities are involved in the defect formation as nucleation centers; (iii) the decay of the supersaturated solid solution of point defects when the crystal is cooled from the crystallization temperature occurs in two independent ways (branches): vacancy and interstitial; (iv) the defect formation is based on primary agglomerates formed as the crystal is cooled from the crystallization temperature due to the interaction between the impurities and intrinsic point defects; (v) when the crystal is cooled at temperatures below 1423 K, depending on the thermal growth conditions, secondary grown-in microdefects are formed due to the interaction between intrinsic point defects; (vi) the secondary grown-in microdefects are formed due to the coagulation (microvoids and A-microdefects) and deformation (A-microdefects) effects; (vii) the vacancy and interstitial branches of the heterogeneous mechanism have a symmetry, which implies simultaneous processes of defect formation during the decay of

primary and secondary grown-in microdefects (V.I. Talanin & I.E. Talanin, 2006a).

models of the formation of primary and secondary grown-in microdefects (Fig. 1).

oxygen-vacancy and carbon- interstitial agglomerates.

**free silicon single crystals** 

supersaturated solid solution of point defects; and (viii) the consequence of this symmetry is the formation of vacancy and interstitial grown-in microdefects of the same type and, correspondingly, the growth of dislocation-free Si single crystals in the same vacancy– interstitial mode (V.I. Talanin & I.E. Talanin, 2006b). It was revealed that the growth parameter *Vg*/*G* = ξcrit describes the conditions under which the (111) face appears on the crystallization front (V.I. Talanin & I.E. Talanin, 2006a). On the basis of physical model and concepts of primary and secondary grown-in microdefects we developed of physical classification of the grown-in microdefects (V.I. Talanin & I.E. Talanin, 2006a).

Fig. 1. Diffusion model of grown-in microdefects formation

A detailed description of the heterogeneous mechanism formation of grown-in microdefects and its correspondence to the results of experimental researches are presented in the articles (V.I. Talanin & I.E. Talanin, 2004; V.I. Talanin & I.E. Talanin, 2006a).
