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Simulation of point defect distributions in silicon crystals during melt-growth
Journal of Crystal Growth 210 (2000) 49}53
Simulation of point defect distributions in silicon crystals during melt-growth K. Nakamura*, T. Saishoji, J. Tomioka Komatsu Electronic Metals Co., Ltd., Technical Division, Crystal Technology Department, 2612, Shinomiya, Hiratsuka, Kanagawa, 254-0014 Japan
Abstract Simulations of the point defect di!usion during the crystal growth process are reported for the investigation of the relationship between the distribution of the oxygen precipitation and point defect concentration in CZ silicon crystals. Distribution of the oxygen precipitation is strongly a!ected by the point defect concentrations during the growth process. There have been many proposed models for the point defect di!usion during the crystal growth process, and the correspondence between point defect concentrations and grown-in defects has been extensively investigated. In this paper, we have compared the distribution of the oxygen precipitation and the ratio of super-saturation between vacancies and self-interstitials. The experimental results agreed with the results obtained by calculation. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 61.72.Bb; 61.72.Ji; 61.72.Qq; 81.10.Fq Keywords: Silicon; Point defect; Grown-in defect; Oxygen precipitation
1. Introduction Grown-in defects in silicon crystals are formed by the aggregation of excess vacancies or selfinterstitials, and are thought to re#ect the distribution of the point defect concentration during the crystal growth process. Accordingly, there have been many papers [1}4] which the di!usion models during the growth process have used to discuss the relationship between the grown-in defects and the point defect concentrations.
* Corresponding author. Tel.: #81-463-24-8850; fax: #81463-25-1588. E-mail address: kouzo}[email protected] (K. Nakamura)
I will "rst describe two phenomena of grown-in defects that have thus far been determined to be important for the understanding of the di!usion behavior of point defects. (1) The dependence of the defect type on
0022-0248/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 6 4 5 - 4
50
K. Nakamura et al. / Journal of Crystal Growth 210 (2000) 49}53
characteristic, the vacancy type defects appear as voids [7] inside the oxidation-induced stacking fault (OSF) ring and the self-interstitial type defects appear as dislocation clusters [8] outside the OSF ring. Dornberger et al. [5] have shown that the change of defect type re#ects the change of G by the radial position and that the
(1)
where ;H is the heat of transfer, which is a cross term between the di!usion and heat #ux. Recently,
Tan [12] derived Eq. (2) QH"E.!E&#k ¹ ln(C/C%2). (2) B E. is an activation energy of migration. In this case, QH is able to become 0. We have discussed the correspondence between the point defect concentrations and the oxygen precipitation using the Tan model.
2. Experiment To compare actual "ndings with the results of calculation, we grew a 150 mm-diameter crystal (B doped, S1 0 0T orientation) with an oxygen concentration of &12.5]1017 cm~3 (ASTM F-121, 1979). The position of the OSF ring in the crystal was changed by the pulling rate during the growth process. The oxygen precipitation was evaluated by an X-ray topograph after a two-step annealing at 7803C (N )]3 h#10003C (O )]16 h. 2 2 3. Results The concentrations of point defects during the growth process were obtained by solving Eq. (3). As the boundary conditions, the point defect concentrations of the interface are assumed to be the equilibrium values.
A
B
LC D C QH V,I "+ (D +C )#+ V,I V,I V,I +¹ V,I V,I Lt k ¹2 B LC !< V,I !K (C C !C%2C%2), V I VI V I Lx
(3)
K "4p(D #D )a exp (!DG /k ¹), (4) VI V I # IV B where C denote actual concentrations of vaV,I cancies and self-interstitials, respectively. The temperature distribution of the crystal during the growth process has been evaluated by global heat transfer analysis. The thermophysical parameters have been selected using the method of parameter "tting shown in Ref. [9]. D "2.14 exp(!1.4 eV/k¹), D " 7 I 1.04]104 exp(!2.4 eV/k ¹), C%2"5.29]1022] 7 B exp(!2.6 eV/k ¹), C%2"1.06]1022 exp(!2.4 I B eV/ k ¹). The criterion for the selection is that the B
K. Nakamura et al. / Journal of Crystal Growth 210 (2000) 49}53
behavior of point defect concentrations obtained by the calculation using the parameter set corresponds well with the dependence of the defect type and the total amount of void volume on
(5)
where Co and Co%2 denote actual and equilibrium concentrations of oxygen, p denotes the interfacial energy for an SiO , X is the volume of SiO per 2 S*O2 2 one oxygen atom, and b and c are the fractions of absorbed vacancies and emitted self-interstitials per oxygen atom (b"c"0.325, for the entire release of strain). This equation describes how the increase of the ratio of supersaturation of (C /C%2) (C%2/C ) I I 7 7 decreases the critical radius of oxygen precipitates and enhances the oxygen precipitation. Therefore, we used the ratio of (C /C%2) (C%2/C ) as the criI I 7 7 terion instead of (C !C ) [2] or (C !C%2)! 7 7 I 7 (C !C%2) [4]. I I To add to the calculation of di!usion and pair annihilation of point defects, the formation and growth of vacancy clusters is also installed in this calculation model. The method to calculate the formation and growth of vacancy clusters is the same as that shown by the authors previously [4], and the model includes the e!ect of absorption of point defects by voids. Fig. 1b shows an X-ray topograph made using specimens cut along the axis of crystal grown with
51
the change of pulling rate shown in Fig. 1a and subjected to the two-step anneal. The position of the OSF ring is changed by the pulling rate. Fig. 1c shows the calculated map of (C /C%2) 7 7 (C%2/C ) when each crystal position passes the posiI I tion at 9503C. As shown, the peak position of (C /C%2) (C%2/C ) corresponds to the position of the I I 7 7 OSF ring. Because the nucleus of the OSF occurs in the temperature range of 900}10003C [14] during the growth process, formation of the OSF ring is assumed to be caused by the strong supersaturation of the point defect at 9503C. The peak of the ratio of (C /C%2) (C%2/C ) forms I I 7 7 via the following process. The vacancy concentration at the center side in the crystal is always higher than that of the periphery side due to the dependence of G on the radial position. Therefore, the vacancy cluster at the center side begins to form earlier, and the vacancy concentrations decrease with the absorption of point defects due to the vacancy clusters and become less than that at the periphery side where the vacancy clusters still have not formed. As a result, the peak of vacancy supersaturation forms just after the defect formation at the center side of crystals. This calculation has reproduced the model for OSF ring formation based on the qualitative speculation of Harada et al. [15]. Fig. 1d shows the calculated map of (C /C%2) 7 7 (C%2/C ) when each crystal position passes the posiI I tion at 7003C. As shown, the distribution of (C /C%2) (C%2/C ) at 7003C corresponds well to the I I 7 7 distribution of the oxygen precipitation by two step annealing. Based on the supposition that the nucleus precipitated by annealing at less than 10003C forms at the temperature range of 610}7603C during the growth process [16], the distribution of oxygen precipitation can be considered to correspond well with the supersaturation ratio at 7003C.
4. Summary Using the Tan model for the reduced heat of transfer (QH), we have selected the best set of thermophysical parameters of point defects. The criterion for the selection is that the behavior of point defect concentrations obtained by the calculation
52
K. Nakamura et al. / Journal of Crystal Growth 210 (2000) 49}53
Fig. 1. (a) Change of pulling rate; (b) shows an X-ray topograph made using specimens cut along the axis of crystals grown with the change of pulling rate shown in (a) and subjected to the two-step anneal. (c) and (d) show the calculated map of (C /C%2) (C%2/C ) when 7 7 I I each crystal position passes the position at 9503C and 7003C, respectively.
using the parameter set corresponds well with two kinds of defect phenomena of (1) and (2): the dependence of the defect type (1) and the total amount of void volume (2) on
[2] T. Sinno, R.A. Brown, W.v. Ammon, E. Dornberger, J. Electrochem. Soc. 145 (1998) 302. [3] R. Habu, A. Tomiura, Jpn. J. Appl. Phys. 35 (1996) 1. [4] K. Nakamura, T. Saishoji, T. Kubota et al., J. Cryst. Growth 180 (1997) 61. [5] E. Dornberger, W.v. Ammon, J. Electrochem. Soc. 143 (1996) 1648. [6] M. Hourai, T. Nagashima, H. Fujiwara, S. Umeno, S. Sadamitsu, S. Miki, T. Shigematsu, Mater. Sci. Forum 196}201 (1995) 17133. [7] M. Nishimura, S. Yoshino et al., J. Electrochem. Soc. 143 (1996) L243. [8] S. Sadamitsu, S. Umeno, Y. Koike, M. Hourai, S. Sumita, T. Shigematsu, Jpn. J. Appl. Phys. 32 (1993) 3675. [9] K. Nakamura, T. Saishoji, J. Tomioka, in: C. Claeys et al., (Eds.), High Purity Silicon, ECS, Pennington NJ, 1998, p. 41. [10] W.A. Tiller, M. Friedman, R. Shaw et al., J. Cryst. Growth 186 (1998) 113.
K. Nakamura et al. / Journal of Crystal Growth 210 (2000) 49}53 [11] S.R. de Groot, P. Mazur, Non-equilibrium Thermodynamics, Dover, New York, 1984. [12] T.Y. Tan, Appl. Phys. Lett. 73 (1998) 2678. [13] J. Vanhellemont, C. Claeys, J. Appl. Phys. 62 (1987) 3960. [14] M. Hourai, G.P. Kelly, T. Tanaka, S. Umeno, S. Ogushi, in: T. Abe, W.M. Bullis, W. Lin, P. Wagner
53
(Eds.), Defects in Silicon, ECS, Pennington NJ, 1999, p. 372. [15] K. Harada, H. Furuya, M. Kida, Jpn. J. Appl. Phys. 36 (1997) 3366. [16] N.I. Puzanov, A.M. Eidenzon, Semicond. Sci. Technol. 7 (1992) 406.
Simulation of point defect distributions in silicon crystals during melt-growth K. Nakamura*, T. Saishoji, J. Tomioka Komatsu Electronic Metals Co., Ltd., Technical Division, Crystal Technology Department, 2612, Shinomiya, Hiratsuka, Kanagawa, 254-0014 Japan
Abstract Simulations of the point defect di!usion during the crystal growth process are reported for the investigation of the relationship between the distribution of the oxygen precipitation and point defect concentration in CZ silicon crystals. Distribution of the oxygen precipitation is strongly a!ected by the point defect concentrations during the growth process. There have been many proposed models for the point defect di!usion during the crystal growth process, and the correspondence between point defect concentrations and grown-in defects has been extensively investigated. In this paper, we have compared the distribution of the oxygen precipitation and the ratio of super-saturation between vacancies and self-interstitials. The experimental results agreed with the results obtained by calculation. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 61.72.Bb; 61.72.Ji; 61.72.Qq; 81.10.Fq Keywords: Silicon; Point defect; Grown-in defect; Oxygen precipitation
1. Introduction Grown-in defects in silicon crystals are formed by the aggregation of excess vacancies or selfinterstitials, and are thought to re#ect the distribution of the point defect concentration during the crystal growth process. Accordingly, there have been many papers [1}4] which the di!usion models during the growth process have used to discuss the relationship between the grown-in defects and the point defect concentrations.
* Corresponding author. Tel.: #81-463-24-8850; fax: #81463-25-1588. E-mail address: kouzo}[email protected] (K. Nakamura)
I will "rst describe two phenomena of grown-in defects that have thus far been determined to be important for the understanding of the di!usion behavior of point defects. (1) The dependence of the defect type on
0022-0248/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 6 4 5 - 4
50
K. Nakamura et al. / Journal of Crystal Growth 210 (2000) 49}53
characteristic, the vacancy type defects appear as voids [7] inside the oxidation-induced stacking fault (OSF) ring and the self-interstitial type defects appear as dislocation clusters [8] outside the OSF ring. Dornberger et al. [5] have shown that the change of defect type re#ects the change of G by the radial position and that the
(1)
where ;H is the heat of transfer, which is a cross term between the di!usion and heat #ux. Recently,
Tan [12] derived Eq. (2) QH"E.!E&#k ¹ ln(C/C%2). (2) B E. is an activation energy of migration. In this case, QH is able to become 0. We have discussed the correspondence between the point defect concentrations and the oxygen precipitation using the Tan model.
2. Experiment To compare actual "ndings with the results of calculation, we grew a 150 mm-diameter crystal (B doped, S1 0 0T orientation) with an oxygen concentration of &12.5]1017 cm~3 (ASTM F-121, 1979). The position of the OSF ring in the crystal was changed by the pulling rate during the growth process. The oxygen precipitation was evaluated by an X-ray topograph after a two-step annealing at 7803C (N )]3 h#10003C (O )]16 h. 2 2 3. Results The concentrations of point defects during the growth process were obtained by solving Eq. (3). As the boundary conditions, the point defect concentrations of the interface are assumed to be the equilibrium values.
A
B
LC D C QH V,I "+ (D +C )#+ V,I V,I V,I +¹ V,I V,I Lt k ¹2 B LC !< V,I !K (C C !C%2C%2), V I VI V I Lx
(3)
K "4p(D #D )a exp (!DG /k ¹), (4) VI V I # IV B where C denote actual concentrations of vaV,I cancies and self-interstitials, respectively. The temperature distribution of the crystal during the growth process has been evaluated by global heat transfer analysis. The thermophysical parameters have been selected using the method of parameter "tting shown in Ref. [9]. D "2.14 exp(!1.4 eV/k¹), D " 7 I 1.04]104 exp(!2.4 eV/k ¹), C%2"5.29]1022] 7 B exp(!2.6 eV/k ¹), C%2"1.06]1022 exp(!2.4 I B eV/ k ¹). The criterion for the selection is that the B
K. Nakamura et al. / Journal of Crystal Growth 210 (2000) 49}53
behavior of point defect concentrations obtained by the calculation using the parameter set corresponds well with the dependence of the defect type and the total amount of void volume on
(5)
where Co and Co%2 denote actual and equilibrium concentrations of oxygen, p denotes the interfacial energy for an SiO , X is the volume of SiO per 2 S*O2 2 one oxygen atom, and b and c are the fractions of absorbed vacancies and emitted self-interstitials per oxygen atom (b"c"0.325, for the entire release of strain). This equation describes how the increase of the ratio of supersaturation of (C /C%2) (C%2/C ) I I 7 7 decreases the critical radius of oxygen precipitates and enhances the oxygen precipitation. Therefore, we used the ratio of (C /C%2) (C%2/C ) as the criI I 7 7 terion instead of (C !C ) [2] or (C !C%2)! 7 7 I 7 (C !C%2) [4]. I I To add to the calculation of di!usion and pair annihilation of point defects, the formation and growth of vacancy clusters is also installed in this calculation model. The method to calculate the formation and growth of vacancy clusters is the same as that shown by the authors previously [4], and the model includes the e!ect of absorption of point defects by voids. Fig. 1b shows an X-ray topograph made using specimens cut along the axis of crystal grown with
51
the change of pulling rate shown in Fig. 1a and subjected to the two-step anneal. The position of the OSF ring is changed by the pulling rate. Fig. 1c shows the calculated map of (C /C%2) 7 7 (C%2/C ) when each crystal position passes the posiI I tion at 9503C. As shown, the peak position of (C /C%2) (C%2/C ) corresponds to the position of the I I 7 7 OSF ring. Because the nucleus of the OSF occurs in the temperature range of 900}10003C [14] during the growth process, formation of the OSF ring is assumed to be caused by the strong supersaturation of the point defect at 9503C. The peak of the ratio of (C /C%2) (C%2/C ) forms I I 7 7 via the following process. The vacancy concentration at the center side in the crystal is always higher than that of the periphery side due to the dependence of G on the radial position. Therefore, the vacancy cluster at the center side begins to form earlier, and the vacancy concentrations decrease with the absorption of point defects due to the vacancy clusters and become less than that at the periphery side where the vacancy clusters still have not formed. As a result, the peak of vacancy supersaturation forms just after the defect formation at the center side of crystals. This calculation has reproduced the model for OSF ring formation based on the qualitative speculation of Harada et al. [15]. Fig. 1d shows the calculated map of (C /C%2) 7 7 (C%2/C ) when each crystal position passes the posiI I tion at 7003C. As shown, the distribution of (C /C%2) (C%2/C ) at 7003C corresponds well to the I I 7 7 distribution of the oxygen precipitation by two step annealing. Based on the supposition that the nucleus precipitated by annealing at less than 10003C forms at the temperature range of 610}7603C during the growth process [16], the distribution of oxygen precipitation can be considered to correspond well with the supersaturation ratio at 7003C.
4. Summary Using the Tan model for the reduced heat of transfer (QH), we have selected the best set of thermophysical parameters of point defects. The criterion for the selection is that the behavior of point defect concentrations obtained by the calculation
52
K. Nakamura et al. / Journal of Crystal Growth 210 (2000) 49}53
Fig. 1. (a) Change of pulling rate; (b) shows an X-ray topograph made using specimens cut along the axis of crystals grown with the change of pulling rate shown in (a) and subjected to the two-step anneal. (c) and (d) show the calculated map of (C /C%2) (C%2/C ) when 7 7 I I each crystal position passes the position at 9503C and 7003C, respectively.
using the parameter set corresponds well with two kinds of defect phenomena of (1) and (2): the dependence of the defect type (1) and the total amount of void volume (2) on
[2] T. Sinno, R.A. Brown, W.v. Ammon, E. Dornberger, J. Electrochem. Soc. 145 (1998) 302. [3] R. Habu, A. Tomiura, Jpn. J. Appl. Phys. 35 (1996) 1. [4] K. Nakamura, T. Saishoji, T. Kubota et al., J. Cryst. Growth 180 (1997) 61. [5] E. Dornberger, W.v. Ammon, J. Electrochem. Soc. 143 (1996) 1648. [6] M. Hourai, T. Nagashima, H. Fujiwara, S. Umeno, S. Sadamitsu, S. Miki, T. Shigematsu, Mater. Sci. Forum 196}201 (1995) 17133. [7] M. Nishimura, S. Yoshino et al., J. Electrochem. Soc. 143 (1996) L243. [8] S. Sadamitsu, S. Umeno, Y. Koike, M. Hourai, S. Sumita, T. Shigematsu, Jpn. J. Appl. Phys. 32 (1993) 3675. [9] K. Nakamura, T. Saishoji, J. Tomioka, in: C. Claeys et al., (Eds.), High Purity Silicon, ECS, Pennington NJ, 1998, p. 41. [10] W.A. Tiller, M. Friedman, R. Shaw et al., J. Cryst. Growth 186 (1998) 113.
K. Nakamura et al. / Journal of Crystal Growth 210 (2000) 49}53 [11] S.R. de Groot, P. Mazur, Non-equilibrium Thermodynamics, Dover, New York, 1984. [12] T.Y. Tan, Appl. Phys. Lett. 73 (1998) 2678. [13] J. Vanhellemont, C. Claeys, J. Appl. Phys. 62 (1987) 3960. [14] M. Hourai, G.P. Kelly, T. Tanaka, S. Umeno, S. Ogushi, in: T. Abe, W.M. Bullis, W. Lin, P. Wagner
53
(Eds.), Defects in Silicon, ECS, Pennington NJ, 1999, p. 372. [15] K. Harada, H. Furuya, M. Kida, Jpn. J. Appl. Phys. 36 (1997) 3366. [16] N.I. Puzanov, A.M. Eidenzon, Semicond. Sci. Technol. 7 (1992) 406.