**6. Construction of the defect structure of dislocation-free silicon single crystal and device structures on their base**

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

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

The Diffusion Model of Grown-In Microdefects Formation

concentrations (V.I. Talanin & I.E. Talanin, 2010b).

Fig. 3. Form of calculation of the defect structure

et. al., 2011b).

The software complex performs imitation of a real experiment and with the maximum precision reproduces the thermal characteristics of silicon single crystal growing. The software complex allows to determine the thermal conditions of crystal growth, to predict and control the defect structure of the crystal. Calculations formations of grown-in microdefects are in good agreement with the experimental results of research (V.I. Talanin

During Crystallization of Dislocation-Free Silicon Single Crystals 635

The first area of the calculation is characterized by high rates of crystal growth, when the above V-shaped distribution of precipitates formed only microvoids and precipitates. The second area of calculation is characterized by the average growth rate of the crystal, when a ring of precipitate in the plane perpendicular to the direction of growth crystal is formed. In this case inside the ring are formed precipitates and microvoids, outside the ring are formed precipitates and interstitial dislocation loops. The third area of calculation is characterized by low rates of crystal growth, when the below V-shaped distribution of precipitates are formed precipitates and interstitial dislocation loops. Calculation of the precipitates is carried out within the classical theory of nucleation, growth and coalescence of precipitates by means of the analytical and approximate calculations. Critical radius of precipitates, the distribution of precipitates in size, change in the average size of precipitates during the cooling of the crystal, and other parameters of the precipitation of carbon and oxygen are determined (Fig. 3). Mathematical models and calculation parameters are given in (V.I. Talanin et al., 2008; V.I. Talanin & I.E. Talanin, 2010a; 2011a). When calculating the vacancy microvoids initially are tested of conditions their formation, since microvoids are not formed at a cooling rate of the crystal *V K cool* 40 /min (Nakamura et al., 2002) and in crystals with a diameter less than 70 mm (V.I. Talanin et al., 2010c). Calculation of the microvoids and interstitial dislocation loops to determines for each of these types of defects such parameters as the critical radii and the

At the stage of determining the type of the structure defect of software system works as follows. Initially is the choice of method of growing dislocation-free silicon single crystals (Czochralski method or the floating-zone method) and then is the choice of certain diameter of the crystal. The ratio of *V G gcrit* / theoretically and experimentally determined in a certain range of values (0.06 mm2/K·min ≤ ≤ 0.3 mm2/K·min). Therefore, we choose the certain value for the calculation. For a given diameter in a certain range of values are selected: the value of the axial temperature gradient in the center of the crystal ( *Ga* ) and the value of the minimum (*V*min ) and maximum (*V*max ) crystal growth rate. These values are determined from the analysis of experimental and theoretical data for different diameters of the crystal. Then produced choice of the axial temperature gradient at the edge of the crystal ( *Ge* ) in the range *<sup>e</sup> a G <sup>G</sup>* = 1.0 ... 2.5. The reliability and accuracy of the computational experiments can be experimentally verified by means of selective etching crystal of the plane which passes through the center of the crystal and parallel to the direction of growth. In case of deviations can by using of the selection of parameters , , *G Ga e* achieve full compliance with theoretical and experimental data. In addition, this technique avoids the difficulties of experimental determination of the *Ga* and *Ge* , especially for large diameter crystals (V.I. Talanin et. al., 2011b). At the last stage of determining the type of defect structure on the resulting dependence of the critical growth rate ( ) ( ) *V r Gr gcrit* is imposed value of real crystal growth rate *V const* (Fig. 2). This procedure allows determining the type of defect

structure of a real crystal. Depending on the position of the line *V const* relative to the curve () () *V r Gr gcrit* may be three areas of the defect structure. Calculation of these areas produced the block of calculation and graphs (Fig. 3).

Fig. 2. Shape analysis of the defect structure

At the stage of determining the type of the structure defect of software system works as follows. Initially is the choice of method of growing dislocation-free silicon single crystals (Czochralski method or the floating-zone method) and then is the choice of certain diameter

selected: the value of the axial temperature gradient in the center of the crystal ( *Ga* ) and the value of the minimum (*V*min ) and maximum (*V*max ) crystal growth rate. These values are determined from the analysis of experimental and theoretical data for different diameters of the crystal. Then produced choice of the axial temperature gradient at the edge of the crystal

experiments can be experimentally verified by means of selective etching crystal of the plane which passes through the center of the crystal and parallel to the direction of growth. In case

with theoretical and experimental data. In addition, this technique avoids the difficulties of experimental determination of the *Ga* and *Ge* , especially for large diameter crystals (V.I. Talanin et. al., 2011b). At the last stage of determining the type of defect structure on the

crystal growth rate *V const* (Fig. 2). This procedure allows determining the type of defect structure of a real crystal. Depending on the position of the line *V const* relative to the

for the calculation. For a given diameter in a certain range of values are

*<sup>G</sup>* = 1.0 ... 2.5. The reliability and accuracy of the computational

may be three areas of the defect structure. Calculation of these areas

theoretically and experimentally determined in a

achieve full compliance

is imposed value of real

≤ 0.3 mm2/K·min). Therefore, we choose the

of the crystal. The ratio of *V G gcrit* /

( *Ge* ) in the range *<sup>e</sup>*

curve () () *V r Gr gcrit*

certain value

certain range of values (0.06 mm2/K·min ≤

*G*

*a*

produced the block of calculation and graphs (Fig. 3).

Fig. 2. Shape analysis of the defect structure

of deviations can by using of the selection of parameters , , *G Ga e*

resulting dependence of the critical growth rate ( ) ( ) *V r Gr gcrit*

The first area of the calculation is characterized by high rates of crystal growth, when the above V-shaped distribution of precipitates formed only microvoids and precipitates. The second area of calculation is characterized by the average growth rate of the crystal, when a ring of precipitate in the plane perpendicular to the direction of growth crystal is formed. In this case inside the ring are formed precipitates and microvoids, outside the ring are formed precipitates and interstitial dislocation loops. The third area of calculation is characterized by low rates of crystal growth, when the below V-shaped distribution of precipitates are formed precipitates and interstitial dislocation loops. Calculation of the precipitates is carried out within the classical theory of nucleation, growth and coalescence of precipitates by means of the analytical and approximate calculations. Critical radius of precipitates, the distribution of precipitates in size, change in the average size of precipitates during the cooling of the crystal, and other parameters of the precipitation of carbon and oxygen are determined (Fig. 3). Mathematical models and calculation parameters are given in (V.I. Talanin et al., 2008; V.I. Talanin & I.E. Talanin, 2010a; 2011a). When calculating the vacancy microvoids initially are tested of conditions their formation, since microvoids are not formed at a cooling rate of the crystal *V K cool* 40 /min (Nakamura et al., 2002) and in crystals with a diameter less than 70 mm (V.I. Talanin et al., 2010c). Calculation of the microvoids and interstitial dislocation loops to determines for each of these types of defects such parameters as the critical radii and the concentrations (V.I. Talanin & I.E. Talanin, 2010b).

Fig. 3. Form of calculation of the defect structure

The software complex performs imitation of a real experiment and with the maximum precision reproduces the thermal characteristics of silicon single crystal growing. The software complex allows to determine the thermal conditions of crystal growth, to predict and control the defect structure of the crystal. Calculations formations of grown-in microdefects are in good agreement with the experimental results of research (V.I. Talanin et. al., 2011b).

The Diffusion Model of Grown-In Microdefects Formation

crystal structure can be defined as postgrowth microdefects.

the initial wafer.

**7. Conclusion** 

**8. References** 

the expensive experimental researches.

398, ISSN 0022-0248.

defect structure of silicon can be used for other materials.

During Crystallization of Dislocation-Free Silicon Single Crystals 637

Production technologies of devices based on silicon technology are connected with various impacts on the material. Heat treatments, ionizing radiations and mechanical effects have a critical impact on the initial defect structure of dislocation-free silicon single crystals. Technological impacts are lead to the transformation of grown-in microdefects. In the general case technological impacts are leads to: (i) the growth of initial grown-in microdefects; (ii) formation on grown-in microdefects of new defects (e.g., stacking faults); (iii) the formation of new defects. In contrast to grown-in microdefects these defects of

At the present time for the description of the formation of postgrowth microdefects are used a variety of models. Overview of the main models has been presented in (Sadamitsu et. al., 1993). The purpose of all models is to consider the formation and growth of defects as a result of technological impacts (e.g., by heating the crystal). A common deficiency of these models is the lack of consideration of the influence of grown-in microdefects, which are in

Diffusion model of formation grown-in microdefects makes it possible to calculate of the defect structure of the initial silicon wafers. So from our point of view of theoretical analysis of the formation of postgrowth microdefects must be based on the diffusion model. It should be noted that taking into account all components of the general solution may be difficult. However, in some cases (e.g., heat treatment of silicon with a certain type of grown-in microdefects) can be solved in the near future. Building a general model for the formation of postgrowth microdefects will help optimize technological processes for production of devices.

The diffusion model of the formation grown-in microdefects provides the unity and adequacy of physical and mathematical modeling. This model simulates of the defect structure of dislocation-free silicon single crystals of any diameters. The model of point defects dynamics can be considered as component of the diffusion model for formation grown-in microdefects. The diffusion model allowed to create software for personal computer. With the help of software can be conducted analytical researches which replace

Further development and modification of the software will lead to the development of information system of formation grown-in microdefects in dislocation-free silicon single crystals. The combinations of an information system with software for control the crystal growth will allow construct the defect structure of crystals during their growth. In turn, application a diffusion model of formation grown-in microdefects in the calculation of the formation of postgrowth microdefects allow to calculate the defect structure of silicon-based devices. In this case, it is possible to adequately construction the defect structure of silicon devices. We believe that the proposed in an article algorithm for the engineering of the

Ammon von, W.; Dornberger, E. & Hansson P.O. (1999). Bulk properties of very large

diameter silicon single crystal. *Journal Crystal Growth*, Vol. 198-199, No. 1-4, pp. 390-

In this case, we can construct the defect structure of devices during their manufacture.

The program complex is the first experience of a virtual experimental device for research the real structure of dislocation-free silicon single crystals. Currently, it can be used for the analysis and calculation of the defect structure of undoped single-crystal silicon. Depending on the thermal conditions of crystal growth can determine crucial parameters such as size and concentration of grown-in microdefects. Approach to the analysis and calculation of formation grown-in microdefects has an important advantage in simplicity, accessibility and sufficient adequacy of mathematical modeling in comparison with other methods. For its implementation does not require supercomputers, and can effectively use the experimental data, experience and intuition of physicists, materials scientists and technologists for the analytical calculation and design of the defect structure. The program complex is easy to implement on a personal computer in technology and research practices.

Disadvantages software system is determined deficiencies of the diffusion model of formation grown-in microdefects. These include: (i) one-dimensional model; (ii) failure to account for the width of the V-shaped distribution of precipitates; (iii) uncertainty in determining of thermal conditions of growth; (iv) the error of approximate numerical methods. Elimination of these deficiencies will increase the accuracy of the calculations.

The influence of other impurities (e.g., dopants, nitrogen, hydrogen, iron, and others) on the formation of the defect structure of silicon can be taken into account by using two approaches: a rigorous approach and a simplified approach. A rigorous approach requires accurate accounting of all the components in equations (1) mass balance of point defects in the crystal. In this case, the solution of the corresponding system of coupled equations of the Fokker-Planck equation can be considerably more difficult.

A simplified approach assumes the separation of impurities in the two groups. The first group contains impurities, which interact with vacancies. The second group contains impurities, which interact with self-interstitial atoms. This interaction for the first group of impurities is similar to the interaction of oxygen atoms with vacancies. This interaction for the second group of impurities is similar to carbon atoms interaction with self-interstitial atoms.. Therefore, in equations (1) the sum of the concentrations of the components of the first group of impurities is equivalent to the impurity concentration of oxygen, and oxygen diffusion coefficient is the sum of the diffusion coefficients of the system components. Are defined analogously the components of interaction in the second group of impurities. Then the system of equations reduced to the system of equations (1), followed by the task of determining the chemical nature of precipitates.

The task of construction the defect structure of dislocation-free silicon single crystal is an inverse problem the analysis and calculation of grown-in microdefects. In this case, you must first specify the type, size and concentration of grown-in microdefects. Parameters of grown-in microdefects are selected based on the requirements for defect structure of devices and integrated circuits. In the next stage are calculated parameters of crystal growth (the growth rate for a certain diameter of the crystal and temperature gradients), which provide presence of given defects of structure. We get that the defect structure determines thermal conditions of crystal growth. To automate the process of crystal growth need to carry out the development of software products based on the diffusion model in combination with known software products for modelling crystal growth.

Production technologies of devices based on silicon technology are connected with various impacts on the material. Heat treatments, ionizing radiations and mechanical effects have a critical impact on the initial defect structure of dislocation-free silicon single crystals. Technological impacts are lead to the transformation of grown-in microdefects. In the general case technological impacts are leads to: (i) the growth of initial grown-in microdefects; (ii) formation on grown-in microdefects of new defects (e.g., stacking faults); (iii) the formation of new defects. In contrast to grown-in microdefects these defects of crystal structure can be defined as postgrowth microdefects.

At the present time for the description of the formation of postgrowth microdefects are used a variety of models. Overview of the main models has been presented in (Sadamitsu et. al., 1993). The purpose of all models is to consider the formation and growth of defects as a result of technological impacts (e.g., by heating the crystal). A common deficiency of these models is the lack of consideration of the influence of grown-in microdefects, which are in the initial wafer.

Diffusion model of formation grown-in microdefects makes it possible to calculate of the defect structure of the initial silicon wafers. So from our point of view of theoretical analysis of the formation of postgrowth microdefects must be based on the diffusion model. It should be noted that taking into account all components of the general solution may be difficult. However, in some cases (e.g., heat treatment of silicon with a certain type of grown-in microdefects) can be solved in the near future. Building a general model for the formation of postgrowth microdefects will help optimize technological processes for production of devices. In this case, we can construct the defect structure of devices during their manufacture.
