**1. Introduction**

618 Advances in Crystallization Processes

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Synthesis, Structure and Properties of Undecatungstozincate Containing 3A

Dislocation-free silicon single crystals are the basic material of microelectronics and nanoelectronics. Physical properties of semiconductor silicon are determined by the structural perfection of the crystals grown by the Czochralski and float-zone processes (Huff, 2002). In such crystals during their growth are formed grown-in microdefects.

Grown-in microdefects degrade the electronic properties of microdevices fabricated on silicon wafers. Optimizing the number and size of grown-in microdefects is crucial to improving processing yield of microelectronic devices. Many of the advances in integratedcircuit manufacturing achieved in recent years would not have been possible without parallel advances in silicon-crystal quality and defect engineering (Yang et al., 2009). The problem of defect formation in dislocation-free silicon single crystals during their growth is a fundamental problem of physics and chemistry of silicon. In particular it is the key to solving the problem engineering applications of silicon crystals. This is connected with the transformation grown-in microdefects during the technological treatment of silicon monocrystals.

Formation of grown-in microdefects occurs as a result of the interaction of point defects during crystal cooling. The distribution of grown-in microdefects in a growing crystal is influenced by its temperature field and the boundary conditions defined by its surfaces. Until recently it was assumed that the formation of grown-in microdefects is due to condensation of intrinsic point defects (Voronkov et. al., 2011). Recombination-diffusion model assumes fast recombination of intrinsic point defects at the initial moment of cooling the grown crystal. Fast recombination determines the type of dominant intrinsic point defects in the crystal. In this model was first used mathematical tool which allows you to associate the defect structure of crystal with distribution in the crystal thermal fields during the growth (Prostomolotov et al., 2011). It has been suggested that the fast recombination of intrinsic point defects near the crystallization front as a function of the growth parameter Vg/G (where Vg is the rate of crystal growth; G is the axial temperature gradient) leads to the formation of microvoids or interstitial dislocation loops (Voronkov, 2008). It is assumed that in the case Vg/G < ξcrit formed only interstitial A-microdefects as a result of aggregation of intrinsic interstitial silicon atoms. It is assumed that in the case Vg/G > ξcrit formed only

The Diffusion Model of Grown-In Microdefects Formation

**2. Classification of grown-in microdefects** 

grown-in microdefects.

range from 1682 to 1423 K (V.I. Talanin & I.E. Talanin, 2010a).

During Crystallization of Dislocation-Free Silicon Single Crystals 621

is based (V.I. Talanin & I.E. Talanin, 2006a; V.I. Talanin & I.E. Talanin, 2007a).With the help of the diffusion model of formation grown-in microdefects was calculated process of hightemperature precipitation (V.I. Talanin & I.E. Talanin, 2010a). The processes of formation and growth of precipitates during cooling of the crystal is a controlling stage in the formation of the grown-in defect structure of dislocation-free silicon single crystals. At this stage, the formation and growth of oxygen and carbon precipitates occur in the temperature

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 dislocation loops (V.I. Talanin & I.E. Talanin, 2010b). The model of point defect dynamics can be considered as component of the diffusion model for formation

The aim of this paper is to present a diffusion model of formation grown-in microdefects in general and to discuss the possibility of its use as a tool for building the defect structure of

Currently, there are three classifications of grown-in microdefects: experimental

Experimental classification of grown-in microdefects is based on the use of methods of selective etching, X-ray topography and transmission electron microscopy (Kock, 1970; Petroff & Kock, 1975; Foll & Kolbesen, 1975; Veselovskaya et al., 1977; Sitnikova et al., 1984; Sitnikova at al., 1985). A.J.R. de Kock entered the name of A-microdefects and Bmicrodefects, whereas Е.G. Sheikhet entered the name C-microdefects, D-microdefects. We are entered the name (I+V)-microdefects (V.I.Talanin et al., 2002a, 2002b). These research allowed to establish the physical nature of A-microdefects, B-microdefects, C-microdefects, D-microdefects and (I+V)-microdefects. Experimental results indicated the identity of the processes of defect formation in crystals of FZ-Si and CZ-Si (Kock et al., 1979; V.I. Talanin & I.E. Talanin, 2003). This means that the classifications of grown-in microdefects in both types

Technological classification is used for large-scale crystals. The larger the diameter of the growing crystal, the lower growth rate, at which the same type of grown-in microdefects is formed. This occurs by reducing the axial temperature gradient in the crystal (Ammon et al., 1999). This leads to the appearance of a new type of grown-in microdefects (microvoids) and dislocation-free crystal growth in a narrow range of growth rates (Voronkov & Falster, 1998). In large crystals of interstitial dislocation loops and microvoids are considered as major grown-in microdefects in dislocation-free silicon crystals (Kulkarni et al., 2004).

Analysis of the experimental results of investigations of grown-in microdefects indicates that there are only three types of grown-in microdefects: precipitates of impurities ((I+V) microdefects, D(C)-microdefects, B-microdefects), dislocation loops (A-microdefects) and microvoids (V.I. Talanin et. al., 2011b). We established that the basic elements of defect formation are primary oxygen-vacancy and carbon-interstitial agglomerates, which are formed at impurity centers near the crystallization front (V.I. Talanin & I.E. Talanin, 2006a).

dislocation-free silicon single crystal and device structures based on them.

classification, technological classification and physical classification.

of crystals should also be identical (V.I. Talanin & I.E. Talanin, 2004).

microvoids as a result of aggregation of vacancies (Goethem et al., 2008; Kulkarni, 2008a). In this physical model, the interaction between the impurities and intrinsic point defects is not considered (Kulkarni et. al., 2004).

Recent versions of this model have suggested that part of the vacancies (v) in the temperature range 1683 ... 1373 K, due to the interaction with oxygen (O) and nitrogen (N) impurities, are bound into complexes of the vO, vO2, and vN types (Kulkarni 2007; 2008b). After the formation of microvoids, the aforementioned complexes grow and take up vacancies. This model has ignored the growth of the complexes by means of the injection of intrinsic interstitial silicon atoms and the interaction of an impurity with intrinsic interstitial silicon atoms (Kulkarni 2007; 2008b).

In the general case recombination-diffusion model assumes that the process of defect formation in dislocation-free silicon single crystals occurs in four stages: (i) fast recombination of intrinsic point defects near the crystallization front; (ii) the formation in the narrow temperature range 1423...1223 K depending on the value of Vg/G microvoids or interstitial dislocation loops; (iii) the formation of oxygen clusters in the temperature range 1223...1023 K; (iv) growth of precipitates as a result of subsequent heat treatments.

Recombination-diffusion model is the physical basis for models of the dynamics of point defects. The mathematical model of point defect dynamics in silicon quantitatively explains the homogeneous mechanism of formation of microvoids and dislocation loops. It should be noted that, in the general case, the model of point defect dynamics includes three approximations: rigorous, simplified, and discrete–continuum approaches (Sinno, 1999; Dornberger et. al., 2001; Wang & Brown, 2001; Kulkarni et. al., 2004; Kulkarni, 2005; Prostomolotov & Verezub, 2009). The rigorous model requires the solution to integrodifferential equations for point defect concentration fields, and the distribution of grown-in microdefects in this model is a function of the coordinates, the time, and the time of evolution of the size distribution of microdefects. A high consumption of time and cost for the performance of calculations required the development of a simplified model in which the average defect radius is approximated by the square root of the average defect area. This approximation is taken into account in the additional variable, which is proportional to the total area of the defect surface. The simplified model is effective for calculating the twodimensional distribution of grown-in microdefects. Both models use the classical nucleation theory and suggest the calculation of the formation of stable nuclei and the kinetics of diffusion-limited growth of defects. The discrete–continuum approximation suggests a complex approach: the solution to discrete equations for the smallest defects and the solution to the Fokker–Planck equation for large-sized defects.

Recently, we proposed a new model for the formation grown-in microdefects. The physical model of the formation of grown-in microdefects assumes that the defect formation in dislocation-free Si single crystals upon cooling occurs in three stages: (i) the formation of impurity aggregates near the crystallization front, (ii) the formation and growth of impurity precipitates upon cooling from the crystallization temperature, and (iii) the formation of microvoids or dislocation loops (depending on the growth parameter *Vg*/*G*) *–* in a narrow temperature range of 1423…1223 K (V.I. Talanin & I.E. Talanin, 2006a; V.I. Talanin & I.E. Talanin, 2010b). This model on the experimentally and theoretically established fact the absence of recombination of intrinsic point defects near the crystallization front of the crystal

microvoids as a result of aggregation of vacancies (Goethem et al., 2008; Kulkarni, 2008a). In this physical model, the interaction between the impurities and intrinsic point defects is not

Recent versions of this model have suggested that part of the vacancies (v) in the temperature range 1683 ... 1373 K, due to the interaction with oxygen (O) and nitrogen (N) impurities, are bound into complexes of the vO, vO2, and vN types (Kulkarni 2007; 2008b). After the formation of microvoids, the aforementioned complexes grow and take up vacancies. This model has ignored the growth of the complexes by means of the injection of intrinsic interstitial silicon atoms and the interaction of an impurity with intrinsic interstitial

In the general case recombination-diffusion model assumes that the process of defect formation in dislocation-free silicon single crystals occurs in four stages: (i) fast recombination of intrinsic point defects near the crystallization front; (ii) the formation in the narrow temperature range 1423...1223 K depending on the value of Vg/G microvoids or interstitial dislocation loops; (iii) the formation of oxygen clusters in the temperature range

Recombination-diffusion model is the physical basis for models of the dynamics of point defects. The mathematical model of point defect dynamics in silicon quantitatively explains the homogeneous mechanism of formation of microvoids and dislocation loops. It should be noted that, in the general case, the model of point defect dynamics includes three approximations: rigorous, simplified, and discrete–continuum approaches (Sinno, 1999; Dornberger et. al., 2001; Wang & Brown, 2001; Kulkarni et. al., 2004; Kulkarni, 2005; Prostomolotov & Verezub, 2009). The rigorous model requires the solution to integrodifferential equations for point defect concentration fields, and the distribution of grown-in microdefects in this model is a function of the coordinates, the time, and the time of evolution of the size distribution of microdefects. A high consumption of time and cost for the performance of calculations required the development of a simplified model in which the average defect radius is approximated by the square root of the average defect area. This approximation is taken into account in the additional variable, which is proportional to the total area of the defect surface. The simplified model is effective for calculating the twodimensional distribution of grown-in microdefects. Both models use the classical nucleation theory and suggest the calculation of the formation of stable nuclei and the kinetics of diffusion-limited growth of defects. The discrete–continuum approximation suggests a complex approach: the solution to discrete equations for the smallest defects and the

Recently, we proposed a new model for the formation grown-in microdefects. The physical model of the formation of grown-in microdefects assumes that the defect formation in dislocation-free Si single crystals upon cooling occurs in three stages: (i) the formation of impurity aggregates near the crystallization front, (ii) the formation and growth of impurity precipitates upon cooling from the crystallization temperature, and (iii) the formation of microvoids or dislocation loops (depending on the growth parameter *Vg*/*G*) *–* in a narrow temperature range of 1423…1223 K (V.I. Talanin & I.E. Talanin, 2006a; V.I. Talanin & I.E. Talanin, 2010b). This model on the experimentally and theoretically established fact the absence of recombination of intrinsic point defects near the crystallization front of the crystal

1223...1023 K; (iv) growth of precipitates as a result of subsequent heat treatments.

solution to the Fokker–Planck equation for large-sized defects.

considered (Kulkarni et. al., 2004).

silicon atoms (Kulkarni 2007; 2008b).

is based (V.I. Talanin & I.E. Talanin, 2006a; V.I. Talanin & I.E. Talanin, 2007a).With the help of the diffusion model of formation grown-in microdefects was calculated process of hightemperature precipitation (V.I. Talanin & I.E. Talanin, 2010a). The processes of formation and growth of precipitates during cooling of the crystal is a controlling stage in the formation of the grown-in defect structure of dislocation-free silicon single crystals. At this stage, the formation and growth of oxygen and carbon precipitates occur in the temperature range from 1682 to 1423 K (V.I. Talanin & I.E. Talanin, 2010a).

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 dislocation loops (V.I. Talanin & I.E. Talanin, 2010b). The model of point defect dynamics can be considered as component of the diffusion model for formation grown-in microdefects.

The aim of this paper is to present a diffusion model of formation grown-in microdefects in general and to discuss the possibility of its use as a tool for building the defect structure of dislocation-free silicon single crystal and device structures based on them.
