**5.3 Model of the vacancy coalescence**

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 point defects in the form of microvoids.

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 with impurity atoms.

The Diffusion Model of Grown-In Microdefects Formation

loops (Kolesnikova et al., 2007).

follows

During Crystallization of Dislocation-Free Silicon Single Crystals 631

as initial model (Chaldyshev et al., 2002; Kolesnikova & Romanov, 2004; Chaldyshev et al., 2005; Kolesnikova et al., 2007). According to these representations, as far as the precipitate grows, its elastic field induces the formation of a circular interstitial dislocation loop of mismatch. This process contributes to the decrease in total strain energy of the system. A growing precipitate displaces the matrix material in the crystal volume. Interstitial atoms form an interstitial dislocation loop near to the precipitate. At the same time, a mismatch dislocation loop is formed on the very precipitate (Kolesnikova et al., 2007). At the same time, the critical sizes of precipitates, at which formation of dislocations is energy favorable, have the same order as the critical size of dislocation

In the volume of silicon the precipitate produces a stress field caused by mismatch between the lattice parameters of precipitate *a*<sup>1</sup> and the surrounding matrix *a*<sup>2</sup> (Kolesnikova et al., 2007). Then, the intrinsic deformation of the precipitate is defined as described bellow

> 1 2 1 *a a a*

 

In general, the precipitate intrinsic deformation in the matrix volume can be expressed as

*xx xy xz*

 

matrix lattices; the other terms are shear components;

Fourier transform (Kolesnikova & Romanov, 2004).

Elastic fields of precipitate (stresses

where *J* is the shear modulus;

precipitate *pr* 

 

2007).

 

radius of precipitate *Rpr* increases as a cubic law (Kolesnikova et al., 2007):

*zx zy zz*

where the diagonal terms constitute a dilatation mismatch between the precipitate and

are calculated taking into account their own deformation (3) and region of localization of the

known scheme by using the elastic modules, Green's function of an elastic medium or its

Consider the simplest model of a spherical precipitate with equiaxed own deformation

takes effect mechanism for resetting the elastic energy of the precipitate. This mechanism leads to the formation of circular interstitial dislocation loop. Energy criterion of this mechanism is the condition *initial final E E* , here , *initial final E E* is the elastic energy of the system with the precipitate before and after relaxation, respectively (Kolesnikova et al.,

*ii ij* , 0 ;, , , , *<sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>x</sup> <sup>y</sup> <sup>z</sup>* . The elastic strain energy of spheroidal defect with increasing

32 2 3 45 1 *E JR pr pr* 

(4)

is the Poisson's ratio. From a certain critical radius *Rcrit*

*ij* and deformation *ij*

. The calculation of elastic fields of the precipitate is carried out by well-

*xy yy yz pr*

(2)

(3)

*pr* is the Kronecker symbol.

) and field of full displacements

The growth parameter / *V G <sup>g</sup>* describes the fundamental reasons related to the systematic nonuniform impurity distribution during crystal growth from a melt. Based on an analysis of the experimental results, one can suggest that the parameter / *V G <sup>g</sup>* controls the growth because it describes the condition for the emergence of the (111) face at the crystallization front. Therefore, the impurity depletion inside the ring of D*-*microdefects upon crystal cooling at *Т* < 1423 K is caused by two things: the impurity bonding during the formation of primary grown-in microdefects ((I+V)-microdefects) and the impurity drift to the (111) face, which is equivalent to the annular distribution of primary D*-*type grown-in microdefects. In this case, excess vacancies arise within the ring of D*-*microdefects to form a supersaturated solid solution with its subsequent decay and the formation of vacancy microvoids. In contrast, excess silicon self-interstitials arise beyond the D*-*ring to form a supersaturated solid solution with its subsequent decay and the formation of interstitial dislocation loops (А*-*microdefects) (V.I. Talanin & I.E. Talanin, 2010c).

The experimental classification of grown-in microdefects employs the terms such as Amicrodefects, B-microdefects, D(C)-microdefects, (I+V)-microdefects and microvoids (V.I. Talanin & I.E. Talanin, 2006a). It was found that A-microdefects constitute interstitial-type dislocation loops, and B-microdefects, D(C)-microdefects, (I+V)-microdefects constitute precipitates of background oxygen and carbon impurities at different stages of their evolution (V.I. Talanin & I.E. Talanin, 2006a; V.I. Talanin & I.E. Talanin, 2011a)). At present, it is difficult to apply the experimental classification, since it is necessary to interpret the terms of every type of the grown-in microdefects for each publication. At the same time, from the physical point of view there are only three types of grown-in microdefects, i.e. impurity precipitates, dislocation loops and microvoids. Besides, when considering the formation of defects in silicon after processing (post-growth microdefects) the terms such as precipitates, dislocation loops and microvoids are also employed. Therefore, in order to harmonize a defect structure, we propose to switch to the physical classification of grown-in microdefects (V.I. Talanin & I.E. Talanin, 2004).

#### **5.4 Kinetic model for the formation and growth of dislocation loops**

Kinetics of high-temperature precipitation involves three stages: (i) the nucleation of a new phase, (ii) the growth stage and (iii) the stage coalescence. Precipitates originate from elastic interaction between point defects. They are, initially, present in coherent, elastic and deformable state, when lattice distortions close to the precipitate-matrix boundary are not large, and one atom of the precipitate corresponds to one atom of the matrix (Goldstein et al., 2011). Elastic deformations and any mechanical stress connected with them cause a transfer of excessive (deficient) substance from the precipitate or vice versa. Storage of elastic strain energy during the precipitate growth results in a loss of coherence by matrix. In this case it is impossible to establish one-to-one correspondence between atoms at different sides of the boundary. It results in structural relaxation of precipitates which occurs due to formation and movement of dislocation loops.

To simulate a stress state of the precipitate and the matrix surrounding it, it is sufficient to observe the precipitate which is simple spherical in shape. There can be found analytical solutions in respect of spherical precipitates (Kolesnikova & Romanov, 2004). Let us take the theoretical and experimental researches of stress relaxation at volume quantum dots

The growth parameter / *V G <sup>g</sup>* describes the fundamental reasons related to the systematic nonuniform impurity distribution during crystal growth from a melt. Based on an analysis of the experimental results, one can suggest that the parameter / *V G <sup>g</sup>* controls the growth because it describes the condition for the emergence of the (111) face at the crystallization front. Therefore, the impurity depletion inside the ring of D*-*microdefects upon crystal cooling at *Т* < 1423 K is caused by two things: the impurity bonding during the formation of primary grown-in microdefects ((I+V)-microdefects) and the impurity drift to the (111) face, which is equivalent to the annular distribution of primary D*-*type grown-in microdefects. In this case, excess vacancies arise within the ring of D*-*microdefects to form a supersaturated solid solution with its subsequent decay and the formation of vacancy microvoids. In contrast, excess silicon self-interstitials arise beyond the D*-*ring to form a supersaturated solid solution with its subsequent decay and the formation of interstitial dislocation loops

The experimental classification of grown-in microdefects employs the terms such as Amicrodefects, B-microdefects, D(C)-microdefects, (I+V)-microdefects and microvoids (V.I. Talanin & I.E. Talanin, 2006a). It was found that A-microdefects constitute interstitial-type dislocation loops, and B-microdefects, D(C)-microdefects, (I+V)-microdefects constitute precipitates of background oxygen and carbon impurities at different stages of their evolution (V.I. Talanin & I.E. Talanin, 2006a; V.I. Talanin & I.E. Talanin, 2011a)). At present, it is difficult to apply the experimental classification, since it is necessary to interpret the terms of every type of the grown-in microdefects for each publication. At the same time, from the physical point of view there are only three types of grown-in microdefects, i.e. impurity precipitates, dislocation loops and microvoids. Besides, when considering the formation of defects in silicon after processing (post-growth microdefects) the terms such as precipitates, dislocation loops and microvoids are also employed. Therefore, in order to harmonize a defect structure, we propose to switch to the physical classification of grown-in

Kinetics of high-temperature precipitation involves three stages: (i) the nucleation of a new phase, (ii) the growth stage and (iii) the stage coalescence. Precipitates originate from elastic interaction between point defects. They are, initially, present in coherent, elastic and deformable state, when lattice distortions close to the precipitate-matrix boundary are not large, and one atom of the precipitate corresponds to one atom of the matrix (Goldstein et al., 2011). Elastic deformations and any mechanical stress connected with them cause a transfer of excessive (deficient) substance from the precipitate or vice versa. Storage of elastic strain energy during the precipitate growth results in a loss of coherence by matrix. In this case it is impossible to establish one-to-one correspondence between atoms at different sides of the boundary. It results in structural relaxation of precipitates which

To simulate a stress state of the precipitate and the matrix surrounding it, it is sufficient to observe the precipitate which is simple spherical in shape. There can be found analytical solutions in respect of spherical precipitates (Kolesnikova & Romanov, 2004). Let us take the theoretical and experimental researches of stress relaxation at volume quantum dots

(А*-*microdefects) (V.I. Talanin & I.E. Talanin, 2010c).

microdefects (V.I. Talanin & I.E. Talanin, 2004).

**5.4 Kinetic model for the formation and growth of dislocation loops** 

occurs due to formation and movement of dislocation loops.

as initial model (Chaldyshev et al., 2002; Kolesnikova & Romanov, 2004; Chaldyshev et al., 2005; Kolesnikova et al., 2007). According to these representations, as far as the precipitate grows, its elastic field induces the formation of a circular interstitial dislocation loop of mismatch. This process contributes to the decrease in total strain energy of the system. A growing precipitate displaces the matrix material in the crystal volume. Interstitial atoms form an interstitial dislocation loop near to the precipitate. At the same time, a mismatch dislocation loop is formed on the very precipitate (Kolesnikova et al., 2007). At the same time, the critical sizes of precipitates, at which formation of dislocations is energy favorable, have the same order as the critical size of dislocation loops (Kolesnikova et al., 2007).

In the volume of silicon the precipitate produces a stress field caused by mismatch between the lattice parameters of precipitate *a*<sup>1</sup> and the surrounding matrix *a*<sup>2</sup> (Kolesnikova et al., 2007). Then, the intrinsic deformation of the precipitate is defined as described bellow

$$
\varepsilon = \frac{a\_1 - a\_2}{a\_1} \tag{2}
$$

In general, the precipitate intrinsic deformation in the matrix volume can be expressed as follows

$$
\boldsymbol{\varepsilon}^\* = \begin{pmatrix}
\boldsymbol{\varepsilon}\_{xx} & \boldsymbol{\varepsilon}\_{xy} & \boldsymbol{\varepsilon}\_{xz} \\
\boldsymbol{\varepsilon}\_{xy} & \boldsymbol{\varepsilon}\_{yy} & \boldsymbol{\varepsilon}\_{yz} \\
\boldsymbol{\varepsilon}\_{zx} & \boldsymbol{\varepsilon}\_{zy} & \boldsymbol{\varepsilon}\_{zz}
\end{pmatrix} \boldsymbol{\delta} \begin{pmatrix} \boldsymbol{\mathcal{Q}}\_{pr} \\
\end{pmatrix} \tag{3}
$$

where the diagonal terms constitute a dilatation mismatch between the precipitate and matrix lattices; the other terms are shear components; *pr* is the Kronecker symbol. Elastic fields of precipitate (stresses *ij* and deformation *ij* ) and field of full displacements are calculated taking into account their own deformation (3) and region of localization of the precipitate *pr* . The calculation of elastic fields of the precipitate is carried out by wellknown scheme by using the elastic modules, Green's function of an elastic medium or its Fourier transform (Kolesnikova & Romanov, 2004).

Consider the simplest model of a spherical precipitate with equiaxed own deformation *ii ij* , 0 ;, , , , *<sup>i</sup> <sup>j</sup> <sup>i</sup> <sup>j</sup> <sup>x</sup> <sup>y</sup> <sup>z</sup>* . The elastic strain energy of spheroidal defect with increasing radius of precipitate *Rpr* increases as a cubic law (Kolesnikova et al., 2007):

$$E\_{pr} = \frac{32 \cdot \pi}{45 \cdot (1 - \nu)} \cdot J \cdot \varepsilon^2 \cdot R\_{pr}^3 \tag{4}$$

where *J* is the shear modulus; is the Poisson's ratio. From a certain critical radius *Rcrit* takes effect mechanism for resetting the elastic energy of the precipitate. This mechanism leads to the formation of circular interstitial dislocation loop. Energy criterion of this mechanism is the condition *initial final E E* , here , *initial final E E* is the elastic energy of the system with the precipitate before and after relaxation, respectively (Kolesnikova et al., 2007).

The Diffusion Model of Grown-In Microdefects Formation

crystal is determined from the dependence:

where *M*( )*t* is the concentration of precipitates.

If the parameter of crystal growth / *V G <sup>g</sup> crit*

**crystal and device structures on their base** 

real experiment by the software (V.I. Talanin et. al., 2011b).

the type of defect structure and (ii) the unit of calculation and graphs.

dislocation loops.

During Crystallization of Dislocation-Free Silicon Single Crystals 633

where *D t*( ) is the diffusion coefficient of intrinsic interstitial silicon atoms; *t* is the time cooling the crystal; *j* is the proportionality factor. The value of the cooling time of the

temperature (melting) of silicon; *UVG <sup>g</sup>* is the cooling rate of the crystal. The loop

( ) ( ) 1 () 2 *crit M t N t*

Initially, the precipitates act as stoppers for the dislocation loops restraining their distribution and generation. Then, the precipitates facilitate the formation of dislocation loops due to the action of Bardeen-Herring or Frank-Read sources (Gyseva et al., 1986). These processes lead to the formation and growth of complex dislocation loops. Formation and development of dislocation loops caused by the high-temperature precipitation of background impurities (oxygen and carbon). Growth and coalescence of dislocation loops are generally maintained due to the generation of growing precipitates instead of silicon

relaxation precipitate adsorbs vacancies. In this case is suppressed the formation of

Experimental studies require large material and time costs, while theoretical studies are carried out for single crystals with selected fixed parameters of their growth. It is necessary to develop a new method for studying the defect structure of silicon without these drawbacks. In the diffusion model of formation grown-in microdefects all the parameters of precipitates, microvoids and dislocation loops are determined through the thermal conditions of growth. Therefore, definition the type of defect structure and calculation of the formation of microdefects is conducted depending on the values of crystal growth rate, temperature gradients and cooling rate of the crystal. On this basis, we have developed a new method for studying the defect structure of silicon. This method allows to simulate a

Electronic equivalent of an object for direct test on the computer are programs that converted the mathematical models and algorithms to the available computer language (C++). The program is written high-level language programming in C++ compiler Borland C++ Builder. Program complex consists of two consecutive parts: (i) the unit determination

*Dt t R*

concentration depends on the crystal cooling time (Burton & Speight, 1985):

self-interstitials, and as well to the dissolution of small dislocation loops.

interstitial silicon atoms. If the parameter of crystal growth / *V G <sup>g</sup> crit*

**6. Construction of the defect structure of dislocation-free silicon single** 

2

2

*T Ut* , where *Tm* is the crystallization

, for stress relaxation precipitate generates

, for stress

(8)

( ) *<sup>m</sup> m <sup>T</sup> T t*

In respect of a spherical precipitate with equiaxial intrinsic deformation, the calculation of elastic fields of the precipitate is substantially simplified. Let us assume that the intrinsic elastic strain energy of the precipitate before and after the formation of a dislocation loop of mismatch remains constant *initial final E E pr pr* . Then the criterion of nucleation loop of misfit dislocation can be represented by the condition 0 *E E D prD* , where *ED* is the energy of loop of misfit dislocation; *EprD* is the energy of interaction of precipitate with the dislocation loop (Kolesnikova et al., 2007).

To estimate believe that loop of misfit dislocation is the equatorial location on the spheroidal precipitate *R R D pr* the self-energy prismatic loop (Kolesnikova et al., 2007)

$$E\_{loop} = \frac{J \cdot b^2 \cdot R\_D}{2 \cdot (1 - \nu)} \cdot \left( \ln \frac{2 \cdot R\_D}{f} - 2 \right) \tag{5}$$

where *f* is the radius of the core loop; *b* is the magnitude of the Burgers vector. The critical radius of precipitate for the formation of dislocation loop is determined from the expression (Kolesnikova et al., 2007)

$$R\_{crit} = \frac{3b}{8\pi (1+\nu)\varepsilon} \left( \ln \frac{1.08aR\_{crit}}{b} \right) \tag{6}$$

where is a constant contribution of the dislocation core. Expression (6) is approximate and can only be used to determine the value critical radius *Rcrit* .

This paper (Bonafos et al., 1998) theoretically considers the increase kinetics for dislocation loops at the stages of loop growth and coalescence. It is assumed that, in general, the growth is either controlled by energy barrier when atom is captured by the loop, or by activation energy of interstitial atom diffusion. In conditions of cooling the crystal after being grown, we presume that the diffusion processes play a core role. The model (Burton & Speight, 1985) is further used in the calculations for evolution in size-dependant distribution of loops and for evolution in loop density.

The dislocation loops with a radius of *R R crit* become bigger in size at the coalescence stage, while small dislocation loops with a radius of *R R crit* will dissolve (Bonafos et al., 1998; Burton & Speight, 1985). The growth of dislocation loops during cooling after the growth of single crystal silicon occurs as due to dissolution of small loops with sizes less than critical, and as a result supersaturation for intrinsic interstitial silicon atoms. In this case, the crystal growth ratio is *<sup>g</sup> crit V <sup>G</sup>* . When oversaturation of vacancies ( *<sup>g</sup> crit V <sup>G</sup>* ) occurs, the interstitial dislocation loops start to dissolve. Increase in the radius of interstitial dislocation loop can be defined by the formula depending on the crystal cooling time (Burton & Speight, 1985):

$$R(t) = \sqrt{R\_{crit}^2 + j \cdot D(t) \cdot t} \tag{7}$$

In respect of a spherical precipitate with equiaxial intrinsic deformation, the calculation of elastic fields of the precipitate is substantially simplified. Let us assume that the intrinsic elastic strain energy of the precipitate before and after the formation of a dislocation loop of mismatch remains constant *initial final E E pr pr* . Then the criterion of nucleation loop of misfit dislocation can be represented by the condition 0 *E E D prD* , where *ED* is the energy of loop of misfit dislocation; *EprD* is the energy of interaction of precipitate with the

To estimate believe that loop of misfit dislocation is the equatorial location on the spheroidal

where *f* is the radius of the core loop; *b* is the magnitude of the Burgers vector. The critical radius of precipitate for the formation of dislocation loop is determined from the expression

<sup>3</sup> 1.08 ln

This paper (Bonafos et al., 1998) theoretically considers the increase kinetics for dislocation loops at the stages of loop growth and coalescence. It is assumed that, in general, the growth is either controlled by energy barrier when atom is captured by the loop, or by activation energy of interstitial atom diffusion. In conditions of cooling the crystal after being grown, we presume that the diffusion processes play a core role. The model (Burton & Speight, 1985) is further used in the calculations for evolution in size-dependant distribution of loops

The dislocation loops with a radius of *R R crit* become bigger in size at the coalescence stage, while small dislocation loops with a radius of *R R crit* will dissolve (Bonafos et al., 1998; Burton & Speight, 1985). The growth of dislocation loops during cooling after the growth of single crystal silicon occurs as due to dissolution of small loops with sizes less than critical, and as a result supersaturation for intrinsic interstitial silicon atoms. In this

occurs, the interstitial dislocation loops start to dissolve. Increase in the radius of interstitial dislocation loop can be defined by the formula depending on the crystal cooling time

is a constant contribution of the dislocation core. Expression (6) is approximate

<sup>2</sup> <sup>2</sup> ln 2

*D D*

(5)

(6)

. When oversaturation of vacancies ( *<sup>g</sup> crit*

<sup>2</sup> ( ) ( ) *Rt R j Dt t crit* (7)

*V <sup>G</sup>* )

*crit*

*b* 

 

*<sup>b</sup> <sup>R</sup> <sup>R</sup>*

8 1

*Jb R R <sup>E</sup> f*

2 1

*loop*

*crit*

and can only be used to determine the value critical radius *Rcrit* .

*V <sup>G</sup>* 

precipitate *R R D pr* the self-energy prismatic loop (Kolesnikova et al., 2007)

dislocation loop (Kolesnikova et al., 2007).

(Kolesnikova et al., 2007)

and for evolution in loop density.

case, the crystal growth ratio is *<sup>g</sup> crit*

(Burton & Speight, 1985):

where

where *D t*( ) is the diffusion coefficient of intrinsic interstitial silicon atoms; *t* is the time cooling the crystal; *j* is the proportionality factor. The value of the cooling time of the crystal is determined from the dependence: 2 ( ) *<sup>m</sup> m <sup>T</sup> T t T Ut* , where *Tm* is the crystallization temperature (melting) of silicon; *UVG <sup>g</sup>* is the cooling rate of the crystal. The loop concentration depends on the crystal cooling time (Burton & Speight, 1985):

$$N(t) = \frac{M(t)}{1 + D(t) \cdot t / 2 \cdot R\_{crit}^2} \tag{8}$$

where *M*( )*t* is the concentration of precipitates.

Initially, the precipitates act as stoppers for the dislocation loops restraining their distribution and generation. Then, the precipitates facilitate the formation of dislocation loops due to the action of Bardeen-Herring or Frank-Read sources (Gyseva et al., 1986). These processes lead to the formation and growth of complex dislocation loops. Formation and development of dislocation loops caused by the high-temperature precipitation of background impurities (oxygen and carbon). Growth and coalescence of dislocation loops are generally maintained due to the generation of growing precipitates instead of silicon self-interstitials, and as well to the dissolution of small dislocation loops.

If the parameter of crystal growth / *V G <sup>g</sup> crit* , for stress relaxation precipitate generates interstitial silicon atoms. If the parameter of crystal growth / *V G <sup>g</sup> crit* , for stress relaxation precipitate adsorbs vacancies. In this case is suppressed the formation of dislocation loops.
