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Kinetics of formation and growth of primary growth microdefects in the diffusion model of defect formation in dislocation-free silicon single crystals.

1. Introduction

The growth of dislocation free silicon single crystals by the Czochralski method and crucibleless zone melting is characterised by the formation of structural imperfections, referred to as growth microdefects. The microdefects represent an intermediate group between the point and linear defects and have the form of aggregates of intrinsic point defects and impurities [1].

Various technological processing methods (heat treatment, radiation) are characterised by the continuation of the growth of these microdefects together with the formation of new microdefects as a result of the continuation of the breakdown of the supersaturated solid solution of the point defects and coalescence of the microdefects. The microdefects found after processing can be regarded as post-growth microdefects.

At present, there is no unified theoretical model of the interaction of point defects and the formation of the initial defective structure of the dislocation-free silicon single crystals, although a large number of models have been proposed for describing the process of defect formation in silicon which attempt to solve the problem with various degrees of success. Unfortunately, the large majority of these models are constructed on the basis of the experimental data on the properties and characteristics of post-growth microdefects [2, 3]. However, the transfer of the results of these investigations in the case of formation of growth microdefects is not always justified and may lead to incorrect results, for example, in the calculation of the combined formation of the vacancy micropores and interstitial dislocation loops or calculation of the formation of stacking faults which do not form during the growth of silicon single crystals. In fact, only the model of dynamics of the point defects is related directly to the examination of the problems of growth microdefects. This is the common name of a large number of different approaches based on the Voronkov recombination-diffusion model [4].

In a general case, the Voronkov model assumes that the process of defect formation in dislocation-free silicon single crystals takes place in four stages: 1) the rapid recombination of intrinsic point defects in the vicinity of the solidification front; 2) the formation on the remaining intrinsic point defects in a narrow temperature range of 1423-1223 K of the vacancy micropores or interstitial dislocation loops; 3) formation of oxygen clusters in the temperature range 1223-1023 K; 4) growth of oxygen precipitates as a result of subsequent heat treatment. The currently available theoretical model of the dynamics of point defects is based on the main assumptions of the Voronkov model [5]. Both models ignore the process of interaction of the intrinsic point defects with impurities at high temperatures [4, 5].

In this article, we propose a new diffusion model of the formation of growth microdefects based on the experimental investigations of unalloyed dislocation-free single crystals of silicon grown by the Czochralski method and by the method of crucibleless zone melting.

2. Physical model

The diffusion model consists of a physical model (the mechanism of formation and transformation of growth microdefects), the physical classification of growth microdefects, and the mathematical models of the formation of primary and secondary growth microdefects.

The main assumptions of the mechanism of the formation and transformation of the growth microdefects are [6]: 1) the recombination of the intrinsic point defects at high temperatures can be ignored [7]; 2) the background impurities of carbon and oxygen take part in the process of defect formation as nucleation centres; 3) the breakdown of the supersaturated solid solution of point defects in cooling of the silicon crystals from the crystallisation temperature takes place by two independent mechanisms (branches): vacancy and interstitial; 4) the main factor of the process of defect formation of the primary agglomerates which formed during cooling of the silicon crystals from the crystallisation temperature as a result of the interaction of the 'impurity-intrinsic point defects' point defects; 5) depending on the thermal conditions of growth, the cooling stage of the crystal at temperatures below 1423 K is characterised by the formation of secondary growth microdefects as a result of the 'intrinsic point defects-intrinsic point defect' interaction; 6) the formation of the secondary growth microdefects is determined by the effect of coalescence (vacancy micropores and A-microdefects) and deformation (A-microdefects) mechanisms.

The mechanism of formation of the growth microdefects assumes that the process of defect formation in dislocation free silicon single crystals takes place in three stages: 1) formation in the vicinity of the solidification front of the impurity aggregates--primary growth microdefects; 2) the growth of the impurity precipitates during cooling of the crystal from the solidification temperature; 3) formation, in a narrow temperature range of 1423-1123 K, of the vacancy micropores or interstitial dislocation loops--the secondary growth microdefects [8].

3. Mathematical models of the formation of primary growth microdefects

At the present time, the formation of the primary growth microdefects is calculated by the model of dissociative diffusion--migration of impurities [9]. This approximation is valid in the initial stages of formation of nuclei when the dimensions of the latter are small and the Fokker-Planck continuous differential equations cannot be used. The calculations carried out on the basis of this approximation show that the boundary of the front of the reaction of formation of a complex (oxygen-vacancy and carbon-interstitial silicon atom) is situated at a distance of ~3 x [10.sup.-4] mm from the solidification front [9]. This distance represents the diffusion layer in which the concentration of the intrinsic point defects is excessively high because of the absence of recombination of these defects at high temperature. The disappearance of the excess intrinsic point defects takes place on sinks whose role is played by the uncontrolled impurities of oxygen and carbon [1]. The formation of complexes between the intrinsic point defects in the impurities in the model described in [9] is determined by the elastic interaction between them. We verify the accuracy of this model using the currently available approach in the form of the system of interlinked discrete differential equations of quasi-chemical reactions for describing the initial stages of the formation of nuclei of new phases and the identical system of the continuous differential equations of the Fokker-Planck type.

Therefore, in the group of the large number of theoretical models of the formation of post-growth microdefects it is necessary to define a model for describing the effect of heat treatment and evolution of a system of nano- and micro-sized defects of various type (oxygen precipitates, stacking faults and pores) which interact together by diffusion of point defects (intrinsic interstitial silicon atoms, vacancies and interstitial oxygen atoms) [3, 10]. The interesting feature of the model is that it can be used for modelling the combined formation and evolution of a complicated defective structure. The authors regard the post- growth microdefects as clusters of particles of different type whose formation and breakdown can be described by a reaction consisting of the random processes of annexation and separation of particles X:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where [A.sub.n] is a cluster of type A consisting of particles of type X; g (n, r, t) is the growth rate of the cluster [A.sub.n]; d(n, r, t) is the rate of breakdown of the cluster [A.sub.n]. The concentration of the clusters [A.sub.n] at the point r is determined by the function f (n, r, t). Its variation with time is described by the system of discrete kinetic differential equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

The preservation of the number of particles X is described by the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

where [n.sub.max] is the maximum number of the particles X in the cluster A.

By expanding the functions g, d, f, which depend smoothly on n, into a Taylor series to the second order inclusive, the system of the discrete equations is represented by the continual differential equation in the partial derivatives (Fokker-Planck equation) [2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

The flow of the monomers in the space of the dimensions n is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

and the kinetic coefficients A and 5 are described by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

In solving the system of the equations (4) and (6), the flows J and / are cross- linked at the point n = [n.sub.min], and the law of conservation of the particles (5) is transformed to the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Equation (6) is a diffusion-drift equation, describing the evolution of the distribution function f in the space of the dimensions n. The authors of [3, 10] show that the system of the equations (4)-(9) makes it possible to use a single model for investigating the processes of nucleation and subsequent growth of the clusters. The conventional boundary, separating the fine and large clusters, is n = [n.sub.min] which is assumed to be equal to 10-20 in the calculations. This value is the boundary between the region of the dimensions in which the thermodynamic approach to describing the physical processes (n >> [n.sub.min]) and the region where their atomic nature must be taken into account (n < [n.sub.min]) can be regarded as valid [11].

Taking these considerations into account, it is really interesting to investigate the mathematical apparatus of this theoretical study within the framework of the physical model of the formation of primary growth microdefects. In this case, to describe the kinetics of simultaneous nucleation and growth (dissolution) in the supersaturated solid solution of the impurity in silicon of the particles of the new phase of several types of it is necessary to investigate the system consisting of the atoms of oxygen, carbon, vacancies and interstitial intrinsic silicon atoms. In accordance with the diffusion model, during cooling the crystal from 1683 K the system leads to the formation of oxygen and carbon precipitates [1]. The nucleation and evolution during cooling of the crystal of a complicated system of the primary growth microdefects, which consists of the oxygen and carbon precipitates, are described by the systems of the connected differential equations (4)-(9) for each type of defects. The relationship between these systems is realised through the laws of conservation of point defects which determine the actual values of the concentration of the defects in the crystal and influenced the rate of growth and dissolution of the clusters of both types. For the case of a thin plane-parallel crystal sheet of a large diameter when the conditions in the plane parallel to the surface of the crystal can be regarded as uniform and we examine the diffusion only along the normal to the surface (the coordinate axis z), the mass balance of the point defects in the crystal is described by the system of diffusion equations for the interesting interstitial silicon atoms, and the atoms of oxygen, carbon and vacancies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

in the equations (10) it is taken into account that the oxygen precipitates act as both sinks for the oxygen atoms and vacancies and the sources of interstitial silicon atoms. Consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

At the same time, the carbon precipitates are the sinks for the carbon atoms and the interstitial silicon atoms, and also the sources of vacancies. Consequently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

In a general case, the proportionality multipliers [[gamma].sub.v],[[gamma].sub.i], [[gamma].sup.*.sub.v], [[gamma].sup.*.sub.i] can depend on the quantities [n.sub.o], [n.sub.c] and are determined by the thermodynamic equilibrium conditions [12, 13]. In addition, in the equations (10), on the basis of the assumptions of the diffusion model, the recombination of the pairs the intrinsic interstitial silicon atoms in the vacancies is not taken into account.

The appropriate system of the inter-linked Fokker-Planck equations can be transformed to the dimensions less form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

where [tau] = t/[t.sub.o] is the dimensions less time. The time constants in (13) are given by the expressions: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], where the critical rate of growth of the precipitates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] The normalised dimensions of the precipitates are determined in the equations (13) as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], are the normalisation critical sizes of the precipitates. The quantities [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] represent the numbers of the particles in the vicinity of the appropriate precipitates with the critical dimensions. The size distribution functions of the precipitates in (13) are normalised with respect to initial concentration of the appropriate nucleation centres:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

The fluxes of the particles in the right-hand part of (13) are described by the expressions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

in which the following notations are used for the normalised kinetic coefficients:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

The normalised rates of growth and dissolution of the precipitates in the equations (16) and (17) have the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

The critical size of the precipitates can be determined in accordance with [3, 13]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the supersaturation of respectively the atoms of oxygen, carbon, interstitial intrinsic silicon atoms and vacancies; [sigma] is the density of the surface interfacial energy between the precipitate and the matrix; [mu] is the shear modulus of silicon; [delta] and [epsilon] are the linear and volume deformation of mismatch between the precipitate and the matrix; [[gamma].sub.i] and [[gamma].sub.v] at the fractions of the intrinsic interstitial silicon atoms and the vacancies, related to a single impurity atom connected with the 3 ecipitates; [V.sub.p] is the volume of the precipitate molecule; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

The number of the impurity atoms in the compressed precipitates with the radii [r.sub.o] and [r.sub.c] is determined as follows [14, 15]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

where [V.sub.p] is the volume of the precipitate, x [less than or equal to] 2, [[gamma].sub.i] [less than or equal to] 1/2, [[gamma].sub.v], [less than or equal to] 1/2..

4. Experimental results and discussion

The following values were used in the calculations: [V.sub.p] = 4.302 x [10.sup.-2] [nm.sup.3] (Si[O.sub.2]); [V.sub.p] = 2.04 x [10.sup.2] [nm.sup.3] (SiC); [sigma] = 310 erg/[cm.sup.2] (Si[O.sub.2]); [sigma] = 1000 erg/ [cm.sup.2] (SiC)- [mu] = 6.41 x [10.sup.10] Pa; [sigma] = 0.3; 8 = 0.15 [3,10]; [[gamma].sub.i] = 0.4; [[gamma].sub.v] = 0.1; x = 1.5; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] = 0.5431 nm; [[delta].sub.Sic] = 0.4359nm; [C.sup.eq.sub.o] = 8 x [10.sup.16] [cm.sup.-3]; [C.sup.eq.sub.c] 1 x [10.sup.16] [cm.sup.-3];

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In analysis of the evolution of the primary growth microdefects in the process of calling the crystal during the growth the important parameters are the characteristic constants in the space of the dimensions: the critical size of the appropriate growth microdefects, and the associated characteristic time constants which define the scale of the measurements with respect to time of the size distribution function of the microdefects. The increase of the values of supersaturation of the point defects (oxygen and carbon atoms, intrinsic interstitial silicon atoms and vacancies) results in a decrease of the appropriate critical size of the precipitates and accelerates the growth of the precipitates. The acceleration of the precipitation processes during the increase of the degree of supersaturation of the point defects is also caused by a decrease of the characteristic time. A reversed tendency is detected in cooling the crystal from the solidification temperature.

An important property of the characteristic time is the inverse proportionality to the products of the characteristics of the point defects (the diffusion coefficient and the equilibrium concentration):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

Since the product for the oxygen atoms is considerably higher than the identical product for the carbon atoms, the rate of evolution of the size distribution function of the carbon precipitates will be higher than the appropriate for the oxygen precipitates. This means that the pattern of development of the microdefects structure of the dislocation-free single crystals of silicon during cooling of the crystal after growth is determined primarily by the rate of growth of the oxygen precipitates.

The algorithm of the solution of the problem of modelling the simultaneous growth and dissolution of the oxygen and carbon precipitates as a result of the interaction of the point defects during cooling of the crystal from the solidification temperature is based on the application of the monotonic explicit difference scheme of the first although the accuracy for the Fokker-Planck equations (13). Figures 1 and 2 shows the dependence of the critical radius of the oxygen and carbon precipitates respectively. In the vicinity of the solidification front (at T = 1682 K) the size of the critical nucleus of the oxygen precipitate is 0.81 nm, and the size of the critical nucleus of the carbon precipitates is approximately 1.1 nm.

The maximum values [n.sup.cr.sub.o] = [n.sup.cr,0.sub.o] and [n.sup.cr.sub.c] = [n.sup.cr,0.sub.c] are recorded in the initial state at T = 1682 K and increase with decreasing temperature. The increase of the critical radius of the precipitates during cooling of the crystal results in a large decrease of the growth rate of the crystals and, consequently, a large decrease of the kinetics of precipitation of these crystals.

The simulation of the defect formation kinetics in cooling the growing crystal in accordance with the exponential law in the temperature range 1682-1403 K is shown in Figure 3 and 4. In these computing experiments it was assumed that the concentration of the nucleation centres for the oxygen and carbon precipitates is ~[10.sup.12] [cm.sup.-3]. These values correspond to the experimental data obtained by transmission electron microscopy [1]. Figure 3 shows the size distribution function of the spherical precipitates of oxygen, and Figure 4 shows the same for the carbon precipitates.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

The absorption of the vacancies by the growing oxygen precipitates results in the emission of silicon atoms into interstitials. The intrinsic interstitial silicon atoms in turn interact with the growing carbon precipitates which during their growth represent vacancies for the growing oxygen precipitates. As a result of this interaction, firstly, the growth of the precipitates is not so rapid as a result of the slower increase of the degree of supersaturation of the intrinsic point defects in the volume of the growing crystal and, secondly, the critical radius of formation of the carbon precipitates increases at a lower rate, and this results in more rapid growth of the carbon precipitates.

The high rate of evolution of the distribution function with respect to the dimensions of the carbon precipitates may be determined by the higher mobility of the interstitial silicon atoms in comparison with the vacancies in the high- temperature range. It may be assumed that the mutual of formation and growth of the oxygen and carbon precipitates the reduces the rate of evolution of the size distribution function of the oxygen precipitates, regardless of the smaller critical size of this precipitate at the initial moment of time, as a result of the effect of the carbon impurity.

The computing experiments carried out in this work are in agreement with the experimental results published in [16]. In the latter study using the methods of low-energy electron spectroscopy and electronic Auger spectroscopy it was shown that the resultant carbon precipitates prevent the growth of the oxygen precipitates causing at the same time the formation of new oxygen precipitates. Similar results were obtained in [17, 18] where the results obtained by transmission electron microscopy showed that cooling of a crystal with the diameter 150 mm in the conditions V/G >> [[xi].sub.crit] in the temperature range 1682-1403 K is accompanied by the formation of oxygen and carbon precipitates (V is the growth rate of the crystal, G is the axial temperature gradient). These precipitates are precursors and centres of the formation of vacancy micropores during cooling of the crystal below 1403 K [19]. In this connection, it is also important to mention the study [20] in which transmission electron microscopy confirmed the formation of plate-shaped and amorphous carbon precipitates distributed in the planes {111} and {100}.

These results of the approximate calculations of the differential equations in the partial derivatives of the Fokker-Planck type are in good correlation with the results of analytical solution of the equations of a successive model of dissociative diffusion in the approximation of strong complexing [9]. In particular, it should be mentioned that these mathematical models together with experimental results obtained in the investigations of the quenched crystals show that the nucleation processes take place at a very high rate in the vicinity of the solidification front.

5. Conclusions

In conclusion, it should be mentioned that the process of formation and growth of the precipitates (primary growth microdefects) during cooling of the crystal after growth is the controlling stage in the formation of the growth defective structure of the dislocation-free single crystals of silicon. This stage in the temperature range 1682-1403 K is characterised by the formation and growth of the oxygen and carbon precipitates ((/ + V)-microdefects, D(C)-microdefects).

The defective structure of the dislocation free single crystals of silicon was calculated in the stage of formation and growth of the oxygen and carbon precipitates on the basis of the approximate solution of the differential equations in the partial derivatives of the Fokker-Planck type. It is shown that the precipitation process starts in the vicinity of the solidification front and is determined by the disappearance of the excess intrinsic point defects at the sinks whose role is played by the impurities of oxygen and carbon.

References

[1.] V. I. Talanin, Modelling and properties of the defective structure of dislocation-free silicon single crystals, GU ZIGMU, Zaporozh'e, 2007.

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[5.] M. S. Kulkarni, V. V. Voronkov, and R. Falster, J. Electrochem. Soc., 151, No. 5, G663 (2004).

[6.] V. I. Talanin and I. E. Talanin, New Research on Semiconductors (New York, Nova Science Publishers, Inc., 2006), p. 31.

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[10.] S. I. Olikhovskii, et al., Metallofiz. Noveishiie Tekhnol., 29, No. 5, 649 (2007).

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[12.] V. V. Voronkov and R. Falster, J. Appl. Phys., 86, No. 11, 2100 (1999).

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[14.] J. Vanhellemont and C. Claeys, J. Appl. Phys., 62, No. 9, 3960 (1987).

[15.] V. V. Voronkov and R. Falster, J. Appl. Phys., 91, No. 9, 5802 (2002).

[16.] T. Ueki, M. Itsumi, and T. Takeda, Jap. J. Appl. Phys., 38, No. 10, 5695 (1999).

[17.] M. Itsumi, M. Maeda, S. Ohfuji et al., Jap. J. Appl. Phys., 38, No. 10, 5720 (1999).

[18.] H. Yamanaka, Jap. J. Appl. Phys., 33, No. 6, 3319 (1994).

[19.] M. Itsumi, J. Cryst. Growth, 237-239, No.3, 1773 (2002).

[20.] M. Albrecht, S. B. Aldabergenova, Sh. B. Baiganatova et al., Cryst. Res. Technol., 35, No. 6-7, 899 (2000).

V.I. Talanin, I.E. Talanin, A.I. Mazurskii and M.L. Maksimchuk Classic Private University, Zhukovskii street, 706, 69012 Zaporozh'e, Ukraine
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Title Annotation:CRYSTAL LATTICE DEFECTS
Author:Talanin, V.I.; Talanin, I.E.; Mazurskii, A.I.; Maksimchuk, M.L.
Publication:Physics of Metals and Advanced Technologies
Article Type:Report
Geographic Code:4EXUR
Date:Jan 1, 2010
Words:4050
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