**4.1 Model of formation complex "impurity + intrinsic point defect"**

The solution is sought within the model of dissociative diffusion–migration of impurities (Bulyarskii & Fistul', 1997). In this case, the difference from the decomposition phenomenon

The Diffusion Model of Grown-In Microdefects Formation

intrinsic point defects appears.

differential equations.

and vacancies respectively.

During Crystallization of Dislocation-Free Silicon Single Crystals 625

the model of dissociative diffusion (Talanin et al., 2007b; 2008). This approximation is valid at the initial stages of the formation of nuclei, when their sizes are small and the use of Fokker–Planck continuity differential equations is impossible. The calculations performed in the framework of this approximation have demonstrated that the edge of the reaction front of the formation of a complex is located at a distance of ~3·10–4 mm from the crystallization front. This spacing represents a diffusion layer in which an excessive concentration of

We have considered the modern approach based on solving systems of coupled discrete differential equations of quasi-chemical reactions for the description of the initial stages of the formation of nuclei of new phases and a similar system of Fokker–Planck continuity

In order to describe the kinetics of the simultaneous nucleation and growth (dissolution) of a new phase particles of several types in a supersaturated solid solution of an impurity in silicon was considered a system consisting of oxygen and carbon atoms, vacancies, and intrinsic interstitial silicon atoms. The interaction in this system during cooling of the crystal from 1683 K results in the formation of oxygen and carbon precipitates. In order to perform the computational experiments and to interpret their results was conducted a dimensional analysis of the kinetic equations and the conservation laws with the use of characteristic time constants and critical sizes of defects. This is made it possible to perform a comparative analysis of the joint evolution of oxygen and carbon precipitates and to optimize the

For example, for the case of a thin plane-parallel crystal plate of a large diameter, when the conditions in the plane parallel to the surface of the crystal can be considered to be uniform and the diffusion can be treated only along the normal to the surface (the z coordinate axis), the mass balance of point defects in the crystal is described by the system of diffusion equations for intrinsic interstitial silicon atoms, oxygen atoms, carbon atoms, and vacancies:

*C CC <sup>D</sup> t t z C CC <sup>D</sup> t t z*

*SiO o oo <sup>o</sup>*

*SiC c cc <sup>c</sup>*

*i*

*i ii i*

*CC C <sup>C</sup> <sup>D</sup> t tt z C CC C <sup>D</sup> t tt z*

 

*SiO SiC v vv v <sup>v</sup>*

where ,,, *CCCC ociv* are the concentrations of oxygen, carbon, self-interstitials and vacancies respectively; ,,, *DDDD ociv* are the diffusion coefficients of oxygen, carbon, self-interstitials

In the system of equations (1), we took into account that the oxygen precipitates serves as sinks for oxygen atoms and vacancies and as sources of interstitial silicon atoms. At the same time, the carbon precipitates, in turn, also serve as sinks for carbon atoms and interstitial silicon atoms and as sources for vacancies. Kinetic model of decomposition of

2

2

*SiO SiC*

(1)

2

computational algorithm for the numerical solution of the equations.

is that during diffusion (as a technological process), a diffusant is supplied to the sample from an external source, whereas in the case of decomposition it is produced by an internal source (lattice sites).

Vas'kin & Uskov are considered the problem of successive diffusion of a component *A* into a sample singly doped with a component *B*, taking into account the complex formation at the initial and boundary conditions (Vas'kin & Uskov, 1968). We are conducted similar consideration for our conditions (Talanin et al., 2007b; 2008). Under physical-model conditions (heterogeneous mechanism of grown-in microdefect formation), we assume that the component *A* is the background impurity (oxygen *O* or carbon *C*) and the component *B*  is intrinsic point defects (vacancies *V* or interstitials *I*). For the vacancy and interstitial mechanisms, we consider, respectively, the oxygen+vacancy (*O*+*V*) and carbon+interstitial (*C* +*I*) interactions. The calculations performed in the framework of approximation of strong complex formation have demonstrated that the edge of the reaction front of the formation of a complex (i.e., the "oxygen+vacancy" and "carbon + self-interstitials" complex) is located at a distance of ~ 3·10–4 mm from the crystallization front (Talanin et al., 2007b). We have shown that complex formation occurs near the crystallization front. Detailed calculations are presented in the articles (Talanin et al., 2007b, 2008).

#### **4.2 Model for the formation of precipitates**

Let us consider a system of a growing undoped dislocation-free silicon single crystal. The concentrations of all point defects at the crystallization front are assumed to be equilibrium, and both the vacancies and the intrinsic interstitial silicon atoms are present in comparable concentrations. During cooling of the crystal after passing through the diffusion zone, an excessive (nonequilibrium) concentration of intrinsic point defects appears. Excess intrinsic point defects disappear on sinks whose role in this process is played by uncontrollable (background) impurities of oxygen and carbon (V.I. Talanin & I.E. Talanin, 2006a). In real silicon crystals, the concentrations of carbon and oxygen impurities are higher than the concentrations of the intrinsic point defects. The formation of complexes between the intrinsic point defects and impurities is governed, on the one hand, by the fact that both the intrinsic point defects and the impurities are sources of internal stresses in the lattice (elastic interaction) and, on the other hand, by the Coulomb interaction between them (provided the defects and the impurities are present in the charged state). The mathematical model under consideration allows for the elastic interaction and the absence of the recombination of intrinsic point defects in the high-temperature range (V.I. Talanin & I.E. Talanin, 2007a). The concentrations of intrinsic point defects *C rt i v*, , in the growing crystal satisfy the diffusion

equation , ,, , *i v i v i v ie ve <sup>С</sup> DCC <sup>t</sup>* where *r* is the coordinate and *t* is the time. In the vicinity of the sinks (oxygen and carbon atoms), the concentration of intrinsic point defects *Cie ve* , is kept equilibrium, whereas the diffusion coefficients *Di v*, and the concentrations *Cie ve* , of intrinsic point defects decrease exponentially with decreasing temperature. Under these conditions, the formation of microvoids and interstitial dislocation loops is possible only at significant supersaturations of intrinsic point defects, which take place at a temperature *TT K <sup>m</sup>* 300 (where *Tm* is the crystallization temperature). For the formation of precipitates in the high-temperature range *T* ~ 1683…1403 K has been calculated using

is that during diffusion (as a technological process), a diffusant is supplied to the sample from an external source, whereas in the case of decomposition it is produced by an internal

Vas'kin & Uskov are considered the problem of successive diffusion of a component *A* into a sample singly doped with a component *B*, taking into account the complex formation at the initial and boundary conditions (Vas'kin & Uskov, 1968). We are conducted similar consideration for our conditions (Talanin et al., 2007b; 2008). Under physical-model conditions (heterogeneous mechanism of grown-in microdefect formation), we assume that the component *A* is the background impurity (oxygen *O* or carbon *C*) and the component *B*  is intrinsic point defects (vacancies *V* or interstitials *I*). For the vacancy and interstitial mechanisms, we consider, respectively, the oxygen+vacancy (*O*+*V*) and carbon+interstitial (*C* +*I*) interactions. The calculations performed in the framework of approximation of strong complex formation have demonstrated that the edge of the reaction front of the formation of a complex (i.e., the "oxygen+vacancy" and "carbon + self-interstitials" complex) is located at a distance of ~ 3·10–4 mm from the crystallization front (Talanin et al., 2007b). We have shown that complex formation occurs near the crystallization front. Detailed calculations are

Let us consider a system of a growing undoped dislocation-free silicon single crystal. The concentrations of all point defects at the crystallization front are assumed to be equilibrium, and both the vacancies and the intrinsic interstitial silicon atoms are present in comparable concentrations. During cooling of the crystal after passing through the diffusion zone, an excessive (nonequilibrium) concentration of intrinsic point defects appears. Excess intrinsic point defects disappear on sinks whose role in this process is played by uncontrollable (background) impurities of oxygen and carbon (V.I. Talanin & I.E. Talanin, 2006a). In real silicon crystals, the concentrations of carbon and oxygen impurities are higher than the concentrations of the intrinsic point defects. The formation of complexes between the intrinsic point defects and impurities is governed, on the one hand, by the fact that both the intrinsic point defects and the impurities are sources of internal stresses in the lattice (elastic interaction) and, on the other hand, by the Coulomb interaction between them (provided the defects and the impurities are present in the charged state). The mathematical model under consideration allows for the elastic interaction and the absence of the recombination of intrinsic point defects in the high-temperature range (V.I. Talanin & I.E. Talanin, 2007a). The concentrations of intrinsic point defects *C rt i v*, , in the growing crystal satisfy the diffusion

where *r* is the coordinate and *t* is the time. In the

vicinity of the sinks (oxygen and carbon atoms), the concentration of intrinsic point defects *Cie ve* , is kept equilibrium, whereas the diffusion coefficients *Di v*, and the concentrations *Cie ve* , of intrinsic point defects decrease exponentially with decreasing temperature. Under these conditions, the formation of microvoids and interstitial dislocation loops is possible only at significant supersaturations of intrinsic point defects, which take place at a temperature *TT K <sup>m</sup>* 300 (where *Tm* is the crystallization temperature). For the formation of precipitates in the high-temperature range *T* ~ 1683…1403 K has been calculated using

source (lattice sites).

presented in the articles (Talanin et al., 2007b, 2008).

**4.2 Model for the formation of precipitates** 

equation , ,, , *i v*

*<sup>С</sup> DCC <sup>t</sup>* 

*i v i v ie ve*

the model of dissociative diffusion (Talanin et al., 2007b; 2008). This approximation is valid at the initial stages of the formation of nuclei, when their sizes are small and the use of Fokker–Planck continuity differential equations is impossible. The calculations performed in the framework of this approximation have demonstrated that the edge of the reaction front of the formation of a complex is located at a distance of ~3·10–4 mm from the crystallization front. This spacing represents a diffusion layer in which an excessive concentration of intrinsic point defects appears.

We have considered the modern approach based on solving systems of coupled discrete differential equations of quasi-chemical reactions for the description of the initial stages of the formation of nuclei of new phases and a similar system of Fokker–Planck continuity differential equations.

In order to describe the kinetics of the simultaneous nucleation and growth (dissolution) of a new phase particles of several types in a supersaturated solid solution of an impurity in silicon was considered a system consisting of oxygen and carbon atoms, vacancies, and intrinsic interstitial silicon atoms. The interaction in this system during cooling of the crystal from 1683 K results in the formation of oxygen and carbon precipitates. In order to perform the computational experiments and to interpret their results was conducted a dimensional analysis of the kinetic equations and the conservation laws with the use of characteristic time constants and critical sizes of defects. This is made it possible to perform a comparative analysis of the joint evolution of oxygen and carbon precipitates and to optimize the computational algorithm for the numerical solution of the equations.

For example, for the case of a thin plane-parallel crystal plate of a large diameter, when the conditions in the plane parallel to the surface of the crystal can be considered to be uniform and the diffusion can be treated only along the normal to the surface (the z coordinate axis), the mass balance of point defects in the crystal is described by the system of diffusion equations for intrinsic interstitial silicon atoms, oxygen atoms, carbon atoms, and vacancies:

$$\begin{aligned} \frac{\partial \mathcal{C}\_o}{\partial t} &= D\_o \frac{\partial^2 \mathcal{C}\_o}{\partial z^2} - \frac{\partial \mathcal{C}\_o^{SiO\_2}}{\partial t} \\ \frac{\partial \mathcal{C}\_c}{\partial t} &= D\_c \frac{\partial^2 \mathcal{C}\_c}{\partial z^2} - \frac{\partial \mathcal{C}\_c^{SiC}}{\partial t} \\ \frac{\partial \mathcal{C}\_i}{\partial t} &= D\_i \frac{\partial^2 \mathcal{C}\_i}{\partial z^2} + \frac{\partial \mathcal{C}\_i^{SiO\_2}}{\partial t} - \frac{\partial \mathcal{C}\_i^{SiC}}{\partial t} \\ \frac{\partial \mathcal{C}\_v}{\partial t} &= D\_v \frac{\partial^2 \mathcal{C}\_v}{\partial z^2} - \frac{\partial \mathcal{C}\_v^{SiO\_2}}{\partial t} + \frac{\partial \mathcal{C}\_v^{SiC}}{\partial t} \end{aligned} \tag{1}$$

where ,,, *CCCC ociv* are the concentrations of oxygen, carbon, self-interstitials and vacancies respectively; ,,, *DDDD ociv* are the diffusion coefficients of oxygen, carbon, self-interstitials and vacancies respectively.

In the system of equations (1), we took into account that the oxygen precipitates serves as sinks for oxygen atoms and vacancies and as sources of interstitial silicon atoms. At the same time, the carbon precipitates, in turn, also serve as sinks for carbon atoms and interstitial silicon atoms and as sources for vacancies. Kinetic model of decomposition of

The Diffusion Model of Grown-In Microdefects Formation

not separated in time and proceed in parallel.

growth conditions, these are microvoids or dislocation loops).

decrease in the average size of the precipitates.

During Crystallization of Dislocation-Free Silicon Single Crystals 627

Talanin & I.E. Talanin, 2010a) has demonstrated that the phase transition occurs according to the mechanism of nucleation and growth of a new phase so that these two processes are

The condition providing changeover to the stage of the coalescence is written in the form *Rt R t cr* , which is satisfied for large-sized crystals at the temperature T ≈ 1423 K. Taking into account the computational errors, this temperature for large-sized crystals corresponds to the initial point of the range of the formation of microvoids (at *Vg* = 0.6 mm/min). In this range, all impurities are bound and there arises a supersaturation with respect to vacancies, which is removed as a result of the formation of microvoids. With a change in the thermal conditions of the growth (for example, at *Vg* = 0.3 mm/min), there arises a supersaturation with respect to interstitial silicon atoms, which leads to the formation of dislocation loops. In this case, the condition *Rt R t cr* is satisfied at *T* ≈ 1418 K. Consequently, the stage of the coalescence in large-sized silicon single crystals begins at temperatures close to the temperatures of the formation of clusters of intrinsic point defects (depending on the thermal

The absorption of vacancies by growing oxygen precipitates results in the emission of silicon atoms in interstitial sites. In turn, the intrinsic interstitial silicon atoms interact with growing carbon precipitates, which, in the course of their growth, supply vacancies for growing oxygen precipitates. This interplay between the processes leads to an accelerated changeover of the subsystems of oxygen and carbon precipitates to the stage of the coalescence as compared to the independent evolution of these two subsystems. The change in the thermal conditions for the growth of small-sized FZ-Si single crystals (high growth rates and axial temperature gradients) leads to the fact that the stage of the coalescence begins far in advance (at *TT K <sup>m</sup>* 20 ). The results of theoretical calculations have demonstrated that a decrease in the concentrations of oxygen and carbon in small-sized single crystals leads to a further decrease in the time of occurrence of the growth stage of precipitates. The change in thermal conditions of crystal growth (in particular, an increase in the growth rate and in the axial temperature gradient in the crystal) substantially affects the stage of the growth of precipitates. In turn, the decrease in the time of occurrence of the growth stage of precipitates is associated, to a lesser extent, with the decrease in the concentration of impurities in crystals. Eventually, these factors are responsible for the

The kinetic model of growth and coalescence of oxygen and carbon precipitates in combination with the kinetic models describing their formation represents a unified model

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

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

of the process of precipitation in dislocation-free silicon single crystals.

solid solutions of oxygen and carbon impurities not only allows one to simulate the processes of precipitation during cooling of the as-grown silicon crystal to a temperature of 300 K but also adequately describes the available experimental data on the oxygen and carbon precipitation (V.I. Talanin & I.E. Talanin, 2011a).

The algorithm used for solving the problem of simulation of the simultaneous growth and dissolution of the oxygen and carbon precipitates due to the interaction of point defects during cooling of the crystal from the crystallization temperature is based on the monotonic explicit difference scheme of the first-order accuracy as applied to the Fokker–Planck equations.

Detailed calculations are presented in the articles (V.I. Talanin & I.E. Talanin, 2010a). These calculations demonstrate that intrinsic point defects (vacancies and intrinsic interstitial silicon atoms) exert a significant influence on the dynamics of mass exchange and mass transfer of point defects between the oxygen and carbon precipitates. The absorption of vacancies by the growing oxygen precipitates leads to the emission of silicon atoms into interstitial positions. The intrinsic interstitial silicon atoms, in turn, interact with the growing carbon precipitates, which, in the process of growth, supply vacancies for growing oxygen precipitates. This interaction leads to such a situation that, first, the growth of the precipitates is suppressed more weakly because of the slower increase in the supersaturation of the intrinsic point defects in the bulk of the growing crystal and, second, the critical radius of the formation of carbon precipitates increases more slowly, which favors a more rapid growth of the carbon precipitates. The higher rate of the evolution of the size distribution function for carbon precipitates can be associated with the higher mobility of interstitial silicon atoms as compared to vacancies in the high-temperature range. It can be assumed that the mutual formation and growth of oxygen and carbon precipitates result in a lower rate of the evolution of the size distribution function of the oxygen precipitates, regardless of their smaller critical size at the initial instant of time, owing to the effect of the carbon impurity.

#### **4.3 Model of growth and coalescence of precipitates**

In the classical theory of nucleation and growth of new-phase particles, the process of precipitation in a crystal is treated as a first-order phase transition and the kinetics of this process is divided into three stages: the formation of new-phase nuclei, the growth of clusters, and the coalescence stage. At the second stage of the precipitation process, clusters grow without a change in their number. At the third stage of the precipitation process, when the particles of the new phase are sufficiently large, the supersaturation is relatively low, new particles are not formed and the decisive role is played by the coalescence, which is accompanied by the dissolution of small-sized particles and the growth of large-sized particles. The condition providing for changeover to the coalescence stage is the ratio ( ) ( )/ ( ) 1 *ut Rt R t cr* , where *R t cr* is the critical radius of the precipitate.

Detailed calculations stages of the growth and the coalescence are presented in the article (V.I. Talanin & I.E. Talanin, 2011a). The analysis was carried out under the assumption that precipitates grow at a fixed number of nucleation centers according to the diffusion mechanism of growth. The model corresponds to the precipitation uniform in the volume. An analysis of the results obtained and the data taken from (Talanin et al., 2007b; V.I.

solid solutions of oxygen and carbon impurities not only allows one to simulate the processes of precipitation during cooling of the as-grown silicon crystal to a temperature of 300 K but also adequately describes the available experimental data on the oxygen and

The algorithm used for solving the problem of simulation of the simultaneous growth and dissolution of the oxygen and carbon precipitates due to the interaction of point defects during cooling of the crystal from the crystallization temperature is based on the monotonic explicit difference scheme of the first-order accuracy as applied to the Fokker–Planck

Detailed calculations are presented in the articles (V.I. Talanin & I.E. Talanin, 2010a). These calculations demonstrate that intrinsic point defects (vacancies and intrinsic interstitial silicon atoms) exert a significant influence on the dynamics of mass exchange and mass transfer of point defects between the oxygen and carbon precipitates. The absorption of vacancies by the growing oxygen precipitates leads to the emission of silicon atoms into interstitial positions. The intrinsic interstitial silicon atoms, in turn, interact with the growing carbon precipitates, which, in the process of growth, supply vacancies for growing oxygen precipitates. This interaction leads to such a situation that, first, the growth of the precipitates is suppressed more weakly because of the slower increase in the supersaturation of the intrinsic point defects in the bulk of the growing crystal and, second, the critical radius of the formation of carbon precipitates increases more slowly, which favors a more rapid growth of the carbon precipitates. The higher rate of the evolution of the size distribution function for carbon precipitates can be associated with the higher mobility of interstitial silicon atoms as compared to vacancies in the high-temperature range. It can be assumed that the mutual formation and growth of oxygen and carbon precipitates result in a lower rate of the evolution of the size distribution function of the oxygen precipitates, regardless of their smaller critical size at the initial instant of time, owing to the effect of the

In the classical theory of nucleation and growth of new-phase particles, the process of precipitation in a crystal is treated as a first-order phase transition and the kinetics of this process is divided into three stages: the formation of new-phase nuclei, the growth of clusters, and the coalescence stage. At the second stage of the precipitation process, clusters grow without a change in their number. At the third stage of the precipitation process, when the particles of the new phase are sufficiently large, the supersaturation is relatively low, new particles are not formed and the decisive role is played by the coalescence, which is accompanied by the dissolution of small-sized particles and the growth of large-sized particles. The condition providing for changeover to the coalescence stage is the ratio

Detailed calculations stages of the growth and the coalescence are presented in the article (V.I. Talanin & I.E. Talanin, 2011a). The analysis was carried out under the assumption that precipitates grow at a fixed number of nucleation centers according to the diffusion mechanism of growth. The model corresponds to the precipitation uniform in the volume. An analysis of the results obtained and the data taken from (Talanin et al., 2007b; V.I.

( ) ( )/ ( ) 1 *ut Rt R t cr* , where *R t cr* is the critical radius of the precipitate.

carbon precipitation (V.I. Talanin & I.E. Talanin, 2011a).

**4.3 Model of growth and coalescence of precipitates** 

equations.

carbon impurity.

Talanin & I.E. Talanin, 2010a) has demonstrated that the phase transition occurs according to the mechanism of nucleation and growth of a new phase so that these two processes are not separated in time and proceed in parallel.

The condition providing changeover to the stage of the coalescence is written in the form *Rt R t cr* , which is satisfied for large-sized crystals at the temperature T ≈ 1423 K. Taking into account the computational errors, this temperature for large-sized crystals corresponds to the initial point of the range of the formation of microvoids (at *Vg* = 0.6 mm/min). In this range, all impurities are bound and there arises a supersaturation with respect to vacancies, which is removed as a result of the formation of microvoids. With a change in the thermal conditions of the growth (for example, at *Vg* = 0.3 mm/min), there arises a supersaturation with respect to interstitial silicon atoms, which leads to the formation of dislocation loops. In this case, the condition *Rt R t cr* is satisfied at *T* ≈ 1418 K. Consequently, the stage of the coalescence in large-sized silicon single crystals begins at temperatures close to the temperatures of the formation of clusters of intrinsic point defects (depending on the thermal growth conditions, these are microvoids or dislocation loops).

The absorption of vacancies by growing oxygen precipitates results in the emission of silicon atoms in interstitial sites. In turn, the intrinsic interstitial silicon atoms interact with growing carbon precipitates, which, in the course of their growth, supply vacancies for growing oxygen precipitates. This interplay between the processes leads to an accelerated changeover of the subsystems of oxygen and carbon precipitates to the stage of the coalescence as compared to the independent evolution of these two subsystems. The change in the thermal conditions for the growth of small-sized FZ-Si single crystals (high growth rates and axial temperature gradients) leads to the fact that the stage of the coalescence begins far in advance (at *TT K <sup>m</sup>* 20 ). The results of theoretical calculations have demonstrated that a decrease in the concentrations of oxygen and carbon in small-sized single crystals leads to a further decrease in the time of occurrence of the growth stage of precipitates. The change in thermal conditions of crystal growth (in particular, an increase in the growth rate and in the axial temperature gradient in the crystal) substantially affects the stage of the growth of precipitates. In turn, the decrease in the time of occurrence of the growth stage of precipitates is associated, to a lesser extent, with the decrease in the concentration of impurities in crystals. Eventually, these factors are responsible for the decrease in the average size of the precipitates.

The kinetic model of growth and coalescence of oxygen and carbon precipitates in combination with the kinetic models describing their formation represents a unified model of the process of precipitation in dislocation-free silicon single crystals.
