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Effect of the crystal–melt interface on the grown-in defects in silicon CZ growth
ARTICLE IN PRESS
Journal of Crystal Growth 299 (2007) 152–157 www.elsevier.com/locate/jcrysgro
Effect of the crystal–melt interface on the grown-in defects in silicon CZ growth Bok-Cheol Sim, Yo-Han Jung, Jai-Eun Lee, Hong-Woo Lee Crystal Growth Technology Team, LG Siltron, 283 Imsoo-dong, Gumi, Gyeong-buk 730-724, Republic of Korea Received 14 August 2006; received in revised form 27 November 2006; accepted 27 November 2006 Communicated by T. Hibiya Available online 17 January 2007
Abstract Single crystals of 206 mm diameter are grown by magnetic Czochralski method in different growing conditions. Effect of the shape of the crystal–melt interface on the defect behavior in the crystal is experimentally investigated. In order to obtain the various shapes of the interfaces, the melt convection is changed with various heating conditions, which are different crucible positions at a fixed heater, and various heaters. The crystal–melt interface becomes more concave (convex to the crystal) with increasing natural convection and the critical pulling speed for defect-free crystal increases. The experimental results are numerically analyzed. r 2006 Published by Elsevier B.V. PACS: 81.05.Cy; 81.10.Fy Keywords: A1. Convection; A1. Defects; A1. Interface shape; A2. Czochralski method; B2. Semiconducting silicon
1. Introduction The growing technology of silicon single crystals is very important in microelectronics industry and most of the silicon crystals used today are grown by the Czochralski (CZ) method. Since the design rule of the semiconductor device decreases to tens of nanometer, defect-free wafer, which has neither vacancy-rich nor self-interstitial-rich defects, becomes more necessary. Therefore, the control of point defect behavior is much more important during crystal growth. It is well known that the shape of the crystal–melt interface and the temperature gradient near the interface play an important role to control the point defect behavior in the crystal. Convection in the melt influences the crystal–melt interface shape and temperature distribution near the interface during crystal growth. In a commercial system, melt convection in a large amount of silicon melt is complex, strong and unsteady because larger melt makes the natural convection much stronger. Thus, in a large Corresponding author. Tel.:+ 82 54 470 6021; fax:+ 82 54 470 6283.
E-mail address: [email protected] (B.-C. Sim). 0022-0248/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.jcrysgro.2006.11.217
amount of the melt, it is difficult to control the melt convection. In the present study, 206 mm single crystals are grown by magnetic CZ method in different growing conditions. Effect of the shape of the crystal–melt interface on the defect behavior in the crystal is experimentally investigated, and the experimental results are numerically analyzed. In order to obtain the various shapes of the interfaces, the melt convection is changed with various heating conditions, which are different crucible positions at a fixed heater, and various heaters. 2. Experiments Commercial furnace with the cusp-magnetic field is used to grow 206 mm diameter Si single crystals. The investigated system is sketched in Fig. 1. The melt level, M=L, is defined as the distance from the heater top to the melt free surface as shown in Fig. 1, which corresponds to different crucible positions at the fixed heater. The crucible has an inner diameter of 600 mm and is made of quartz. The initial charge size of the silicon melt is 140 kg. The crystal (seed) is rotated with 11 rpm, and the
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
153 M/ L = 124
10
M / L = 154 M / L = 174
9
Height of the interface [mm]
8 7 6 5 4 3 2 1 Fig. 1. The Czochralski system.
0 0
crucible rotation is in 0–0.1 rpm. Argon flow rate and furnace pressure are fixed. The cusp-magnetic field is applied in the melt. Crystal growth rate or pulling speed, V , is varied to observe grown-in defect behavior as crystal length increases. Vertical samples are cut from crystals along ingot axial direction and investigated using minority carrier life time wafer mapping (MCLT-mapping) and X-ray topography (XRT) after 2-step annealing ð800 C=4 h r þ 1000 C= 16230 hÞ. The MCLT-map and XRT enable us to observe the crystal quality and get information such as the critical V , oxygen-induced stacking fault (OSF) distribution pattern, the shape of the crystal–melt interface, and the radial V =G profile, where G is the axial temperature gradient in the crystal at the crystal–melt interface, and the critical V , V , indicates the pulling speed at which Pv/Pi boundary is formed in the crystal. Pi is interstitial pure and Pv is vacancy pure. Unfortunately, we could not recognize V and OSF distribution pattern from the MCLT-map and XRT because the oxygen concentration in the crystal was very low and the resolutions of the MCLT-map and XRT were bad. Thus, flow pattern defect (FPD) densities and distribution patterns are measured from the slug samples, where the slugs are sampled at a distance of 150 mm in the vertical direction of the crystal. FPD density and distribution pattern are measured by an optical microscope after Secco etching for 30 min with the slug. In the paper, V indicates the critical V at which FPD disappears. The shape of the crystal–melt interface is measured by growth striation from XRT near the position where FPD disappears. 3. Results and discussion 3.1. Shapes of the crystal–melt interfaces with various M=L Three different crystals are grown with various M=L: (i) M=L ¼ 124 mm, (ii) M=L ¼ 154 mm and (iii) M=L ¼
25
50 Radial position [mm]
75
100
Fig. 2. Shapes of the crystal–melt interfaces with various M=L. The interface is more concave with decreasing M=L.
174 mm at a fixed heater. Fig. 2 shows the shapes of the crystal–melt interfaces with various M=L. The interface is more concave (more convex toward the crystal) with decreasing M=L. FPD density and FPD distribution pattern are shown in Fig. 3 with various M=L in the crystal whose V is changed. The grown-in defect increases with increasing V . The higher the FPD density, the larger the diameter of FPD area. V are in the range, 0.671–0.691, 0.628–0.676 and 0:56520:628 mm= min with, respectively, M=L ¼ 124, 154 and 174 mm. It is evident that the interface becomes more concave and V increases as M=L decreases. There have been a large number of experimental and numerical studies of the shape of the crystal–melt interface and the grown-in defect in CZ method. Nakamura et al. [1] reported that the convex crystal–melt interface increases the axial temperature gradient, G, in the crystal at the crystal–melt interface, and thus the growth rate increases by V =G theory [2]. They suggested the dependence of the critical V =G ratio on the shape of the crystal–melt interface as follows. For crystal with a convex interface, self-interstitials become dominant in the region of crystal center. Therefore, V increases and hence the critical V =G increases for crystals with a convex interface. Crystals with a concave interface are the opposites of these: vacancies become dominant in the region of crystal center. However, the results in Figs. 2 and 3 contradict their theory. In our experiments, more concave interface has higher V as shown in Figs. 2 and 3. Recently, Watanabe et al. [3] reported that point defects in the concave interface would diffuse outward from the crystal center to the outside according to the concentration difference, and the
ARTICLE IN PRESS
Averaged FPD density [ea /cm2]
400 M/L = 124 M/L = 154 350 M/L = 174 300 250 200 150 100 1.11 50 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8
Diameter of FPD Area [mm]
B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
154
180 160 140 120 100 80 60 40 20 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8
V [mm/min]
V [mm/min]
Fig. 3. Axial profiles of (a) averaged FPD density and (b) diameter of FPD area with various M=L. V increases with decreasing M=L.
a
b -1.926E - 4
c -1.668E - 4
-1.437E - 4
Fig. 4. Streamlines and isotherms corresponding to Fig. 3 with (a) M=L ¼ 124 mm; (b) M=L ¼ 154 mm; and (c) M=L ¼ 174 mm, where the isotherms are in steps of 1.667 K from the melting temperature. Convection is more vigorous with decreasing M=L.
driving force was large in more concave interface. Also, they suggested that no outward diffusion could occur at the flat interface, and point defects in the convex interface moved toward the crystal center. As a result, the concave shape of the crystal–melt interface can make defect-free crystals at high V . Kim et al. [4] showed experimentally that compared with magnetic CZ, the crystal–melt interface in electro-magnetic CZ could become more concave and V could be higher with more concave interface. Our experimental results are in agreement with the theory in [3] and the other experimental results [4]. The flow patterns corresponding to Figs. 2 and 3 are analyzed by numerical simulation. The numerical simulations are performed by finite element code Femag 2D using the geometry of the real CZ furnace, and the simulation conditions are the same as those for the experimental conditions. The validity of the code for CZ
system has been confirmed in other studies [5,6]. Convective flows with different M=L are shown in Fig. 4. The flow fields show two toroidal cells, which is a characteristic of CZ crystal growth. The inner flow cell is the forced convection due to the seed (crystal) rotation and the outer cell is the natural convection which is buoyant [7] and thermocapillary(Marangoni) flows [8,9] with the low crucible rotation. M=L changes the heating condition and hence natural convection. As expected, the minimum value of the streamfunction in the silicon melt decreases with decreasing M=L. Convection is more vigorous with decreasing M=L. As M=L decreases, natural convection is stronger, the bulk temperature of the melt near the interface is higher in Fig. 4, and hence the crystal–melt interface becomes more concave as shown in Fig. 5. The numerical results in Fig. 5 are in good qualitative but not in good quantitative agreement with the experiments in
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
Fig. 2. Effect of the natural convection (M=L) on oxygen concentration in CZ system can be found in the other paper [10].
growth: one is to change the cooling rate of the crystal in order to control G at the crystal–melt interface, and the other is to change the melt convection in order to control Gl at the interface as shown in Eq. (1), where G l is the temperature gradient in the melt at the crystal–melt interface. At the crystal–melt interface, the heat balance equation is as follows:
3.2. Shapes of the crystal–melt interfaces with various heaters On the basis of V =G theory [2] and Eq. (1), there are two kinds of methods available to control V during the crystal
M/L = 174
KG c ¼ K l G lc þ V L, V V ¼ Constant, ¼ Gc K l G l c =K þ V L=K
14 12 10
ð2Þ ð3Þ
where subscript c means the critical value for defect-free crystal. Eq. (3) is rearranged in simple form as follows:
8
V ¼ Constant. Glc
6
2
0
25
50 Radial position [mm]
75
100
Fig. 5. Shapes of the crystal–melt interfaces corresponding to Fig. 4 computed by numerical studies.
b
a
100
100 Heater - F Heater - U
90
90
80
80
70
70
60
60
Length [%]
Length [%]
(4)
The decrease of M=L means both the increase of cooling rate of the crystal, because the crystal moves closer to the fixed cooling pipe at the furnace top, and the increase of natural convection. Thus, the decrease of M=L induces the increases of both G and G l . At the fixed M=L and other growing parameters, the heater is only changed to control G l . Fig. 6 shows the heating proportion and the temperature distribution of the heater, where the temperature distribution of the heater is from the
4
0
(1)
where K and K l are the conductivities of the crystal and the melt, respectively, and L is the latent heat of fusion. It is well known that the critical value of V =G for the defectfree crystal is almost constant [11]. Assuming constant properties, Eq. (1) can be changed into the critical equation for defect-free crystal:
M/L = 154
16
Height of the interface [mm]
KG ¼ K l Gl þ VL,
M/L = 124
18
155
50 40
50 40
30
30
20
20
10
10
0
0
10
20
30
40
50
Heating Proportion [%]
60
0 1350 1400 1450 1500 1550 1600 1650 Temperature [C]
Fig. 6. (a) Heating proportion and (b) temperature distribution in two types of heaters. The location of the maximum heating position in Heater-F is lower than that of Heater-U.
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
156
steady numerical result. The location of the maximum heating position in Heater-F is lower than one of Heater-U as shown in Fig. 6. Compared with Heater-U, Heater-F is heating the lower position of the crucible at the fixed crucible, which is similar to the decrease of M=L. As expected, the interface with Heater-F is more concave than the interface of Heater-U as shown in Fig. 7. Fig. 8 shows the distribution of the grown-in defect with various heaters. V are in the ranges, 0.671–0.691 and 0:62420:668 mm= min with, respectively, Heater-F and Heater-U. It is evident that V of Heater-F is higher than that of Heater-U because of a more concave interface. The melt flow patterns are shown in Fig. 9. Convection
with Heater-F is stronger, the bulk melt temperature near the crystal–melt interface is hotter, and the interface is more concave compared with the case of Heater-U. In the case of Heater-F, Gl is higher than G l of Heater-U because the isotherms with Heater-F are denser near the crystal–melt interface than those from Heater-U as shown in Fig. 9. G l is higher, the interface is more concave, and V is higher in Heater-F. Thus, this is relatively included in the second method, which controls G l in the melt at the interface. The interface shape computed by numerical study in Fig. 10 are in good qualitative but not in good quantitative agreement with the experiments in Fig. 7.
a
b
10
-1.926E - 4
-1.792E - 4
9
Height of the interface [mm]
8 7 6 5 4 3 M /L = 124 with Heater - F
2
M/L = 124 with Heater - U
1 0 0
25
50 Radial position [mm]
75
100
Fig. 7. Shapes of the crystal–melt interfaces with various heaters. The crystal–melt interface in Heater-F is more concave.
a
Fig. 9. Streamlines and isotherms corresponding to Fig. 8 with (a) HeaterF and (b) Heater-U, where the isotherms are in steps of 1.667 K from the melting temperature. Convection in Heater-F is more vigorous.
b Diameter of FPD Area [mm]
Averaged FPD density [ea/cm2]
150 M/L = 124 with Heater - F M/L = 124 with Heater - U
100
50 5 0 0.6
1.11
0.65 0.7 V [mm /min]
0.75
180 160 140 120 100 80 60 40 20 0 0.6
0.65 0.7 V [mm /min]
0.75
Fig. 8. Axial profiles of (a) averaged FPD density and (b) diameter of FPD area with various heaters. V in Heater-F is higher than that of Heater-U.
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
Convection is more vigorous with decreasing M=L. As M=L decreases, natural convection is stronger, the bulk temperature of the melt near the crystal–melt interface is higher, the crystal–melt interface becomes more concave, and V increases. Two types of heaters are used to study the influence of G l with fixed other parameters. Convection with Heater-F is stronger, the bulk melt temperature near the crystal–melt interface is hotter, and the interface is more concave compared with the case of Heater-U. In the case of Heater-F, Gl is higher than G l of Heater-U, the interface is more concave, and the V is higher in Heater-F. The crystal–melt interface becomes more concave (convex to the crystal) with increasing natural convection, and the critical pulling speed for defect-free crystal increases. The numerical results are in good qualitative but not in good quantitative agreement with the experiments.
18 16
Height of the interface [mm]
14 12 10 8 6 4
M/L = 124 with Heater - F M/L = 124 with Heater - U
2 0
0
25
50 Radial position [mm]
157
75
100
Fig. 10. Shapes of the crystal–melt interfaces corresponding to Fig. 9 computed by numerical studies.
4. Conclusions Single crystals of 206 mm diameter are grown by magnetic Czochralski method with different growing conditions. Effect of the shape of the crystal–melt interface on the defect behavior in the crystal is experimentally investigated, and the experimental results are numerically analyzed. In order to obtain the various shapes of the interfaces, the melt convection is changed with various heating conditions, which are different crucible positions at a fixed heater, and various heaters.
References [1] K. Nakamura, S. Maeda, S. Togawa, T. Saishoji, J. Tomioka, ECS Proceedings 17 (2000) 31. [2] V. Voronkov, J. Crystal Growth 59 (1982) 625. [3] M. Watanabe, D. Vizman, J. Friedrich, G. Muller, J. Crystal Growth 292 (2006) 252. [4] K.-H. Kim, B.-C. Sim, I.-S. Choi, H.-W. Lee, J. Crystal Growth, in press. [5] N. Van den Bogaert, F. Dupret, J. Crystal Growth 171 (1997) 65. [6] N. Van den Bogaert, F. Dupret, J. Crystal Growth 171 (1997) 77. [7] D. Schwabe, J. Crystal Growth 237–239 (2002) 1849. [8] D. Schwabe, A. Zebib, B.-C. Sim, J. Fluid Mech. 491 (2003) 239. [9] B.-C. Sim, A. Zebib, D. Schwabe, J. Fluid Mech. 491 (2003) 259. [10] B.-C. Sim, I.-K. Lee, K.-H. Kim, H.-W. Lee, J. Crystal Growth 275 (2005) 455. [11] W. Ammon, E. Dornberger, H. Oelkrug, H. Weidner, J. Crystal Growth 151 (1995) 273.
Journal of Crystal Growth 299 (2007) 152–157 www.elsevier.com/locate/jcrysgro
Effect of the crystal–melt interface on the grown-in defects in silicon CZ growth Bok-Cheol Sim, Yo-Han Jung, Jai-Eun Lee, Hong-Woo Lee Crystal Growth Technology Team, LG Siltron, 283 Imsoo-dong, Gumi, Gyeong-buk 730-724, Republic of Korea Received 14 August 2006; received in revised form 27 November 2006; accepted 27 November 2006 Communicated by T. Hibiya Available online 17 January 2007
Abstract Single crystals of 206 mm diameter are grown by magnetic Czochralski method in different growing conditions. Effect of the shape of the crystal–melt interface on the defect behavior in the crystal is experimentally investigated. In order to obtain the various shapes of the interfaces, the melt convection is changed with various heating conditions, which are different crucible positions at a fixed heater, and various heaters. The crystal–melt interface becomes more concave (convex to the crystal) with increasing natural convection and the critical pulling speed for defect-free crystal increases. The experimental results are numerically analyzed. r 2006 Published by Elsevier B.V. PACS: 81.05.Cy; 81.10.Fy Keywords: A1. Convection; A1. Defects; A1. Interface shape; A2. Czochralski method; B2. Semiconducting silicon
1. Introduction The growing technology of silicon single crystals is very important in microelectronics industry and most of the silicon crystals used today are grown by the Czochralski (CZ) method. Since the design rule of the semiconductor device decreases to tens of nanometer, defect-free wafer, which has neither vacancy-rich nor self-interstitial-rich defects, becomes more necessary. Therefore, the control of point defect behavior is much more important during crystal growth. It is well known that the shape of the crystal–melt interface and the temperature gradient near the interface play an important role to control the point defect behavior in the crystal. Convection in the melt influences the crystal–melt interface shape and temperature distribution near the interface during crystal growth. In a commercial system, melt convection in a large amount of silicon melt is complex, strong and unsteady because larger melt makes the natural convection much stronger. Thus, in a large Corresponding author. Tel.:+ 82 54 470 6021; fax:+ 82 54 470 6283.
E-mail address: [email protected] (B.-C. Sim). 0022-0248/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.jcrysgro.2006.11.217
amount of the melt, it is difficult to control the melt convection. In the present study, 206 mm single crystals are grown by magnetic CZ method in different growing conditions. Effect of the shape of the crystal–melt interface on the defect behavior in the crystal is experimentally investigated, and the experimental results are numerically analyzed. In order to obtain the various shapes of the interfaces, the melt convection is changed with various heating conditions, which are different crucible positions at a fixed heater, and various heaters. 2. Experiments Commercial furnace with the cusp-magnetic field is used to grow 206 mm diameter Si single crystals. The investigated system is sketched in Fig. 1. The melt level, M=L, is defined as the distance from the heater top to the melt free surface as shown in Fig. 1, which corresponds to different crucible positions at the fixed heater. The crucible has an inner diameter of 600 mm and is made of quartz. The initial charge size of the silicon melt is 140 kg. The crystal (seed) is rotated with 11 rpm, and the
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
153 M/ L = 124
10
M / L = 154 M / L = 174
9
Height of the interface [mm]
8 7 6 5 4 3 2 1 Fig. 1. The Czochralski system.
0 0
crucible rotation is in 0–0.1 rpm. Argon flow rate and furnace pressure are fixed. The cusp-magnetic field is applied in the melt. Crystal growth rate or pulling speed, V , is varied to observe grown-in defect behavior as crystal length increases. Vertical samples are cut from crystals along ingot axial direction and investigated using minority carrier life time wafer mapping (MCLT-mapping) and X-ray topography (XRT) after 2-step annealing ð800 C=4 h r þ 1000 C= 16230 hÞ. The MCLT-map and XRT enable us to observe the crystal quality and get information such as the critical V , oxygen-induced stacking fault (OSF) distribution pattern, the shape of the crystal–melt interface, and the radial V =G profile, where G is the axial temperature gradient in the crystal at the crystal–melt interface, and the critical V , V , indicates the pulling speed at which Pv/Pi boundary is formed in the crystal. Pi is interstitial pure and Pv is vacancy pure. Unfortunately, we could not recognize V and OSF distribution pattern from the MCLT-map and XRT because the oxygen concentration in the crystal was very low and the resolutions of the MCLT-map and XRT were bad. Thus, flow pattern defect (FPD) densities and distribution patterns are measured from the slug samples, where the slugs are sampled at a distance of 150 mm in the vertical direction of the crystal. FPD density and distribution pattern are measured by an optical microscope after Secco etching for 30 min with the slug. In the paper, V indicates the critical V at which FPD disappears. The shape of the crystal–melt interface is measured by growth striation from XRT near the position where FPD disappears. 3. Results and discussion 3.1. Shapes of the crystal–melt interfaces with various M=L Three different crystals are grown with various M=L: (i) M=L ¼ 124 mm, (ii) M=L ¼ 154 mm and (iii) M=L ¼
25
50 Radial position [mm]
75
100
Fig. 2. Shapes of the crystal–melt interfaces with various M=L. The interface is more concave with decreasing M=L.
174 mm at a fixed heater. Fig. 2 shows the shapes of the crystal–melt interfaces with various M=L. The interface is more concave (more convex toward the crystal) with decreasing M=L. FPD density and FPD distribution pattern are shown in Fig. 3 with various M=L in the crystal whose V is changed. The grown-in defect increases with increasing V . The higher the FPD density, the larger the diameter of FPD area. V are in the range, 0.671–0.691, 0.628–0.676 and 0:56520:628 mm= min with, respectively, M=L ¼ 124, 154 and 174 mm. It is evident that the interface becomes more concave and V increases as M=L decreases. There have been a large number of experimental and numerical studies of the shape of the crystal–melt interface and the grown-in defect in CZ method. Nakamura et al. [1] reported that the convex crystal–melt interface increases the axial temperature gradient, G, in the crystal at the crystal–melt interface, and thus the growth rate increases by V =G theory [2]. They suggested the dependence of the critical V =G ratio on the shape of the crystal–melt interface as follows. For crystal with a convex interface, self-interstitials become dominant in the region of crystal center. Therefore, V increases and hence the critical V =G increases for crystals with a convex interface. Crystals with a concave interface are the opposites of these: vacancies become dominant in the region of crystal center. However, the results in Figs. 2 and 3 contradict their theory. In our experiments, more concave interface has higher V as shown in Figs. 2 and 3. Recently, Watanabe et al. [3] reported that point defects in the concave interface would diffuse outward from the crystal center to the outside according to the concentration difference, and the
ARTICLE IN PRESS
Averaged FPD density [ea /cm2]
400 M/L = 124 M/L = 154 350 M/L = 174 300 250 200 150 100 1.11 50 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8
Diameter of FPD Area [mm]
B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
154
180 160 140 120 100 80 60 40 20 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8
V [mm/min]
V [mm/min]
Fig. 3. Axial profiles of (a) averaged FPD density and (b) diameter of FPD area with various M=L. V increases with decreasing M=L.
a
b -1.926E - 4
c -1.668E - 4
-1.437E - 4
Fig. 4. Streamlines and isotherms corresponding to Fig. 3 with (a) M=L ¼ 124 mm; (b) M=L ¼ 154 mm; and (c) M=L ¼ 174 mm, where the isotherms are in steps of 1.667 K from the melting temperature. Convection is more vigorous with decreasing M=L.
driving force was large in more concave interface. Also, they suggested that no outward diffusion could occur at the flat interface, and point defects in the convex interface moved toward the crystal center. As a result, the concave shape of the crystal–melt interface can make defect-free crystals at high V . Kim et al. [4] showed experimentally that compared with magnetic CZ, the crystal–melt interface in electro-magnetic CZ could become more concave and V could be higher with more concave interface. Our experimental results are in agreement with the theory in [3] and the other experimental results [4]. The flow patterns corresponding to Figs. 2 and 3 are analyzed by numerical simulation. The numerical simulations are performed by finite element code Femag 2D using the geometry of the real CZ furnace, and the simulation conditions are the same as those for the experimental conditions. The validity of the code for CZ
system has been confirmed in other studies [5,6]. Convective flows with different M=L are shown in Fig. 4. The flow fields show two toroidal cells, which is a characteristic of CZ crystal growth. The inner flow cell is the forced convection due to the seed (crystal) rotation and the outer cell is the natural convection which is buoyant [7] and thermocapillary(Marangoni) flows [8,9] with the low crucible rotation. M=L changes the heating condition and hence natural convection. As expected, the minimum value of the streamfunction in the silicon melt decreases with decreasing M=L. Convection is more vigorous with decreasing M=L. As M=L decreases, natural convection is stronger, the bulk temperature of the melt near the interface is higher in Fig. 4, and hence the crystal–melt interface becomes more concave as shown in Fig. 5. The numerical results in Fig. 5 are in good qualitative but not in good quantitative agreement with the experiments in
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
Fig. 2. Effect of the natural convection (M=L) on oxygen concentration in CZ system can be found in the other paper [10].
growth: one is to change the cooling rate of the crystal in order to control G at the crystal–melt interface, and the other is to change the melt convection in order to control Gl at the interface as shown in Eq. (1), where G l is the temperature gradient in the melt at the crystal–melt interface. At the crystal–melt interface, the heat balance equation is as follows:
3.2. Shapes of the crystal–melt interfaces with various heaters On the basis of V =G theory [2] and Eq. (1), there are two kinds of methods available to control V during the crystal
M/L = 174
KG c ¼ K l G lc þ V L, V V ¼ Constant, ¼ Gc K l G l c =K þ V L=K
14 12 10
ð2Þ ð3Þ
where subscript c means the critical value for defect-free crystal. Eq. (3) is rearranged in simple form as follows:
8
V ¼ Constant. Glc
6
2
0
25
50 Radial position [mm]
75
100
Fig. 5. Shapes of the crystal–melt interfaces corresponding to Fig. 4 computed by numerical studies.
b
a
100
100 Heater - F Heater - U
90
90
80
80
70
70
60
60
Length [%]
Length [%]
(4)
The decrease of M=L means both the increase of cooling rate of the crystal, because the crystal moves closer to the fixed cooling pipe at the furnace top, and the increase of natural convection. Thus, the decrease of M=L induces the increases of both G and G l . At the fixed M=L and other growing parameters, the heater is only changed to control G l . Fig. 6 shows the heating proportion and the temperature distribution of the heater, where the temperature distribution of the heater is from the
4
0
(1)
where K and K l are the conductivities of the crystal and the melt, respectively, and L is the latent heat of fusion. It is well known that the critical value of V =G for the defectfree crystal is almost constant [11]. Assuming constant properties, Eq. (1) can be changed into the critical equation for defect-free crystal:
M/L = 154
16
Height of the interface [mm]
KG ¼ K l Gl þ VL,
M/L = 124
18
155
50 40
50 40
30
30
20
20
10
10
0
0
10
20
30
40
50
Heating Proportion [%]
60
0 1350 1400 1450 1500 1550 1600 1650 Temperature [C]
Fig. 6. (a) Heating proportion and (b) temperature distribution in two types of heaters. The location of the maximum heating position in Heater-F is lower than that of Heater-U.
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156
steady numerical result. The location of the maximum heating position in Heater-F is lower than one of Heater-U as shown in Fig. 6. Compared with Heater-U, Heater-F is heating the lower position of the crucible at the fixed crucible, which is similar to the decrease of M=L. As expected, the interface with Heater-F is more concave than the interface of Heater-U as shown in Fig. 7. Fig. 8 shows the distribution of the grown-in defect with various heaters. V are in the ranges, 0.671–0.691 and 0:62420:668 mm= min with, respectively, Heater-F and Heater-U. It is evident that V of Heater-F is higher than that of Heater-U because of a more concave interface. The melt flow patterns are shown in Fig. 9. Convection
with Heater-F is stronger, the bulk melt temperature near the crystal–melt interface is hotter, and the interface is more concave compared with the case of Heater-U. In the case of Heater-F, Gl is higher than G l of Heater-U because the isotherms with Heater-F are denser near the crystal–melt interface than those from Heater-U as shown in Fig. 9. G l is higher, the interface is more concave, and V is higher in Heater-F. Thus, this is relatively included in the second method, which controls G l in the melt at the interface. The interface shape computed by numerical study in Fig. 10 are in good qualitative but not in good quantitative agreement with the experiments in Fig. 7.
a
b
10
-1.926E - 4
-1.792E - 4
9
Height of the interface [mm]
8 7 6 5 4 3 M /L = 124 with Heater - F
2
M/L = 124 with Heater - U
1 0 0
25
50 Radial position [mm]
75
100
Fig. 7. Shapes of the crystal–melt interfaces with various heaters. The crystal–melt interface in Heater-F is more concave.
a
Fig. 9. Streamlines and isotherms corresponding to Fig. 8 with (a) HeaterF and (b) Heater-U, where the isotherms are in steps of 1.667 K from the melting temperature. Convection in Heater-F is more vigorous.
b Diameter of FPD Area [mm]
Averaged FPD density [ea/cm2]
150 M/L = 124 with Heater - F M/L = 124 with Heater - U
100
50 5 0 0.6
1.11
0.65 0.7 V [mm /min]
0.75
180 160 140 120 100 80 60 40 20 0 0.6
0.65 0.7 V [mm /min]
0.75
Fig. 8. Axial profiles of (a) averaged FPD density and (b) diameter of FPD area with various heaters. V in Heater-F is higher than that of Heater-U.
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 299 (2007) 152–157
Convection is more vigorous with decreasing M=L. As M=L decreases, natural convection is stronger, the bulk temperature of the melt near the crystal–melt interface is higher, the crystal–melt interface becomes more concave, and V increases. Two types of heaters are used to study the influence of G l with fixed other parameters. Convection with Heater-F is stronger, the bulk melt temperature near the crystal–melt interface is hotter, and the interface is more concave compared with the case of Heater-U. In the case of Heater-F, Gl is higher than G l of Heater-U, the interface is more concave, and the V is higher in Heater-F. The crystal–melt interface becomes more concave (convex to the crystal) with increasing natural convection, and the critical pulling speed for defect-free crystal increases. The numerical results are in good qualitative but not in good quantitative agreement with the experiments.
18 16
Height of the interface [mm]
14 12 10 8 6 4
M/L = 124 with Heater - F M/L = 124 with Heater - U
2 0
0
25
50 Radial position [mm]
157
75
100
Fig. 10. Shapes of the crystal–melt interfaces corresponding to Fig. 9 computed by numerical studies.
4. Conclusions Single crystals of 206 mm diameter are grown by magnetic Czochralski method with different growing conditions. Effect of the shape of the crystal–melt interface on the defect behavior in the crystal is experimentally investigated, and the experimental results are numerically analyzed. In order to obtain the various shapes of the interfaces, the melt convection is changed with various heating conditions, which are different crucible positions at a fixed heater, and various heaters.
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