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Boron segregation control in silicon crystal ingots grown in Czochralski process
ARTICLE IN PRESS
Journal of Crystal Growth 290 (2006) 665–669 www.elsevier.com/locate/jcrysgro
Boron segregation control in silicon crystal ingots grown in Czochralski process Bok-Cheol Sim, Kwang-Hun Kim, Hong-Woo Lee Crystal Growth Technology Team, LG Siltron, 283 Imsoo-dong, Gumi, Gyeong-buk 730-724, Korea Received 10 August 2005; received in revised form 2 February 2006; accepted 6 February 2006 Communicated by T. Hibiya
Abstract Boron-doped silicon single crystals of 207 mm diameter with various growing conditions are grown from a large amount of the melt in the cusp-magnetic Czochralski method, and the effects of growing parameters on dopant concentrations in the crystals are experimentally investigated. Equilibrium distribution coefficient of boron calculated by BPS model is 0.716. With the crystal rotation ðoÞ of 13 rpm and the crucible rotation of 0:5 rpm, the effective distribution coefficient (ke ) is 0.751 in zero magnetic strength and increases up to 0.78 in the magnetic strength of 640 G. For oX7 rpm, there is no significant influence of o on ke . With op3 rpm, ke is almost unity. The experimental results are compared with theory. r 2006 Elsevier B.V. All rights reserved. PACS: 81.05.Cy; 81.10.Fq Keywords: A1. Distribution coefficient; A1. Dopant concentration; A2. Czochralski method; B2. Semiconducting silicon
1. Introduction In order to obtain silicon single crystal of a required resistivity in Czochralski (CZ) method, dopant is added to the polysilicon charged in the crucible. The dopant is incorporated into the single crystal from the silicon melt. Dopant concentration and its distribution in the crystal are important for predictable device performance. Therefore, there have been a large number of experiments of dopant segregation process in the silicon melt during the CZ crystal growth. Dopant distribution in the crystals depends on the effective distribution coefficient, ke , at the solid–liquid interface, and convection in the melt plays a major role in determining ke of the dopant. Since Burton, Prim and Slichter (BPS) [1] developed a model for segregation in steady convective flows which is based on boundary layer in the melt, most of the studies have focused on the melt 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 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2006.02.003
convection to investigate dopant segregation during crystal growth. Carruthers et al. [2] reported ke of antimony, Sb, in stirring silicon melt in ‘‘CZ’’ system. Effect of magnetic field on ke was investigated by Hirata and Inoue [3] and Thomas et al. [4] for low ke dopants. They found that the magnetic field stabilized the melt convection and ke increased with increasing magnetic strength. These facts were verified in theoretical studies with a simplified model by Hurle and Series [5]. Various types of magnetic fields were discussed by Series and Hurls [6]. Ostrogorsky [7] developed more complex model to describe ke , which included the simple melt velocities in the solute boundary layers. ke can vary from ke ¼ k0 (for convection dominant segregation) to ke ¼ 1 (for convection-free segregation), where k0 is the equilibrium distribution coefficient of the dopant. For diffusion system without buoyant convection in zero gravity, Witt et al. [8] reported the dramatic increase of the dopant concentration in the axial direction and non-uniform radial distribution in the crystals, compared with results on earth.
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666
Ravishankar et al. [9] and Huff et al. [10] investigated ke of Boron in small amount of melt. They found that ke in ‘‘CZ’’ system was different from ke in floating zone system [9], and ke did not change with crystal growth rate [9,10]. The brief review of segregation was performed by Hurle and Rudolph [11] and Wang et al. [12]. Convection in the melt influences dopant distribution at the crystal-melt interface during crystal growth. In a commercial system, melt charge size should increase for large crystal size and mass products. Melt convection in a large amount of the silicon melt is complex and strong because the larger melt makes the natural convection [13–15] much stronger. Thus, in a large amount of the melt, it is difficult to control convection and hence dopant distribution. In the present study, 207 mm diameter silicon single crystals with various growing conditions are grown from a large amount of the melt in the cusp-magnetic Czochralski method, and the effects of growing parameters on dopant concentrations in the boron-doped silicon single crystal ingots are experimentally investigated. The experimental results are compared with theoretical results.
where g is gravity acceleration, b is the volume expansion coefficient, DT is the temperature difference, l is the characteristic length or melt height, n is kinematic viscosity, a is thermal diffusivity, o is the crystal or crucible rotation rate, and r is the crystal or crucible radius. In the experiments, Ra is in the range 4.7–8:2 107 with DT ¼ 20 K and l ¼ 216–260 mm. Re due to crystal rotation is 4:05 103 –8:1 104 with o ¼ 1–20 rpm, and Re for the crucible rotation is 0–2:38 105 . Thermophysical properties used in calculations are found in Won et al. [16]. Boron concentrations in the crystal are measured from the individual slices. The dopant, boron, concentrations and its distributions in the axial and radial directions are obtained from four-point probe resistivity measurements with the aid of the Irvin curve [17]. Since usual ‘‘CZ’’ Si wafer has almost completely axisymmetric structure, the four-point evaluation method for resistivity-radial gradient measurement is assumed valid. The measurement error is less than 1%. The resistivity-radial gradient, RRG, is expressed as follows:
2. Experiments
RRG½% ¼
Commercial furnace with a cusp-magnetic field is used to grow 207 mm diameter silicon single crystals doped with boron. The crucible has an inner diameter of 600 mm and is made of quartz. The initial charge sizes of the silicon melts are in the range 110–140 kg in various experiments, and the initial melt heights in the quartz crucibles are 216–260 mm, which depends on the initial charge size of the melt. The single crystals of more than 110 cm in length are grown in the h1 0 0i direction with 8–12 O cm. The crystal (seed) is rotated in the range 1–20 rpm, and the crucible rotation is in the range 0–7 rmp in opposite to crystal rotation direction. Argon flow-rate and furnace pressure are fixed. The cusp-magnetic field with the zero-Gauss plane at 110 mm below the free surface is applied to the melt. The magnetic strength indicated here represents the intensity of the horizontal component of the magnetic field orthogonal to crucible side wall at the zero-Gauss plane. The pulling speed of the crystal is changed to keep the crystal diameter constant. The pulling speeds varied in the range 0:65 0:05 mm= min. Two main flows exist in the melt; one is natural convection, which is the buoyant and thermocapillary forces, and the other is forced convection due to crystal and crucible rotations. Usually, buoyant force is much stronger than thermocapillary force, and thermocapillary force can be neglected. The buoyant and forced convection is characterized by the Rayleigh, Ra, and Reynolds, Re, numbers as follows: Ra ¼
gbDTl 3 , na
or2 , Re ¼ n
Center [Res] Average [Res] at the edge 4 points 100, Center [Res]
(3) where Center [Res] is the resistivity at the crystal center, and the 4 edge [Res] are measured at 6 mm from the crystal edges as shown in Fig. 1 and are averaged.
(1)
(2)
Fig. 1. Resistivity-measurement points for RRG at the crystal, where the 4 edge [Res] are measured at 6 mm from the crystal edges and are averaged.
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 290 (2006) 665–669
3. Results and discussion
0.15 0.1
ke is determined by fitting the experimental data to the normal freezing equation, C ¼ C 0 ke ð1 f Þke 1 ,
0.05 0
where C is the dopant concentration in the solid for a given fraction, f, solidified, and C 0 is the initial dopant concentration in the melt. The equation, Eq. (4), is rearranged in logarithmic form as follows:
-0.05
lnðC=C 0 Þ ¼ lnðke Þ þ ðke 1Þ lnð1 f Þ.
(5)
The log–log plot, which is lnðC=C 0 Þ vs. lnð1 f Þ, of Eq. (5) results in a straight line, and ke can be calculated from the slope in the log–log plot. The slope is determined by a least square fit of the experimental data. Fig. 2 shows a typical log–log curve of the boron concentration in the two crystals grown with the same growing parameters. The growing condition for the crystals is the crystal rotation ðoÞ of 20 rpm, the crucible rotation (C=R) of 7 rpm and the initial melt size of 120 kg without the magnetic field. ke at each crystal is 0.732 and 0.744. It is determined that ke is 0:738 0:006 in Fig. 2. The square of the correlation coefficient, R2 , is 0.997 close to unity, which demonstrates a good linear fit to the data. ke ¼ 0:738 0:006 for boron in the silicon melt is in reasonable agreement with other works [9,10], where ke ¼ 0:7 and ke ¼ 0:78 0:03 are reported irrespective of the crystal growth rates. Fig. 3 shows ke with o ¼ 13 rpm, C=R ¼ 0:5 rpm, the initial melt size of 120 kg and no magnetic field. For reproducibility, three different ingots are grown with the same above growing parameters. ke for three ingots are 0.75, 0.755 and 0.76, and the deviation of ke in three ingots 0.15 0.1 Ke=0.738
0
ln (Cs/C0)
-0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 0
-0.5
-1
-1.5
ln (1-f) Fig. 2. Macroscopic boron concentration in the axial direction with o ¼ 20 rpm and C=R ¼ 7 rpm, where two symbols represent experimental results in two crystal ingots grown with the same growing condition.
ln (Cs/C0)
(4)
0.05
667
Ke =0.751
-0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 0
-0.5
-1
-1.5
ln (1-f) Fig. 3. Macroscopic boron concentration in the axial direction with o ¼ 13 rpm and C=R ¼ 0:5 rpm. The square of the correlation coefficient in a least square fit is 0.999.
with the same growth conditions is less than 1.5%. ke determined by a least square fit of the total data is 0.751 as shown in Fig. 3. BPS model relates ke to k0 as follows: ke ¼
k0 , k0 þ ð1 k0 Þ expðvd=DÞ
(6)
where v is the crystal growth rate, d is the diffusion-layer thickness, and D is diffusion coefficient. With the experimental ke of 0.738 for o ¼ 20 rpm, k0 is calculated by Eq. (6), where v ¼ 1:083 103 cm=s and d=D ¼ 120 19 s=cm for o ¼ 20 rpm from Kodera [18] and Tiller [19]. k0 calculated by Eq. (6) is in the range, 0.708–0.716. Also, with o ¼ 13 rpm and ke ¼ 0:751, k0 calculated is 0.715–0.724, where d=D ¼ 150 19 s=cm for o ¼ 13 rpm is used due to do0:5 [18,19]. Almost same value of k0 ¼ 0:716 is found from two experimental ke even though C=R is ignored in Eq. (6). It is concluded that k0 for boron is 0:716 0:08 in silicon ‘‘CZ’’ system. Effect of the magnetic force on ke is shown in Fig. 4 with o ¼ 13 rpm, C=R ¼ 0:3 rpm and different magnetic strengths. ke increases with increasing magnetic strength in Fig. 4. Compared with ke of zero magnetic strength, ke with the magnetic strength of 640 G is 5.4% higher. The magnetic field makes melt convection stable and hence ke higher. Unfortunately, 640 G was the maximum magnetic strength in our experiments. It is well known that ke increases dramatically with increasing magnetic strength in a low ke dopant such as P, Sb, Ga and Al [3,4]. Fig. 5 shows the effect of o on ke with C=R ¼ 0:3 rpm, the magnetic strength of 394 G and oX7 rpm. The difference between two ke is within the measurement error, with oX7 rpm, there is no significant influence of o on ke as shown in Fig. 5. For op7 rpm, the effect of o on ke with
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 290 (2006) 665–669
0.15
0.15 Mag. strength=394 Gauss Mag. strength=640 Gauss Least Square Fit
0.1 0.05
0.1
0
5
-0.05 ln (Cs/C0)
-0.05 ln (Cs/C0)
6
0.05
Ke=0.764
0
7
-0.1 -0.15
Ke=0.7 8
Ke=0.791
-0.1 -0.15
-0.25
-0.25
-0.3
-0.3
-0.35
-0.35
-0.4
Ke=0.981
1 0
-0.5
-1
-0.4 -1
Fig. 4. Effect of the magnetic strength on ke with o ¼ 13 rpm. The magnetic strength lower than 640 G has a little influence on ke .
S/R=13rpm S/R=7rpm Least Square Fit
0.05 0
Ke=0.762
ln (Cs/C0)
-0.05
Fig. 6. Effect of o ðp7 rpmÞ on ke with the magnetic strength of 640 G. With op3 rpm, ke is almost unity.
Table 1 Comparison of experimental ke with theoretical ke [20], where magnetic strength is the value at the melt-crystal interface
0.15 0.1
-1.5
ln (1-f)
-1.5
ln (1-f)
2
Ke=0.814
0 -0.5
3
Ke=0.813
-0.2
-0.2
0
4
ω [rpm]
668
-0.1 Ke=0.764
-0.15
o ðrpmÞ
C=R (rpm)
Magnetic strength (G)
1 3 5 7 13 13
0 0 0.3 0.3 0.3 0.5
383 383 383 383 383 0
Exp. ke 0.981 0.981 0.814 0.791 0.78 0.751
Theo. ke 0.943 0.835 0.803 0.782 0.758 0.750
-0.2 -0.25 -0.3 -0.35 -0.4 0
-1
-0.5
-1.5
ln (1-f) Fig. 5. Effect of o (X7 rpm) on ke with the magnetic strength of 394 G. o ðX7 rpmÞ has no influence on ke .
the magnetic strength of 640 G is shown in Fig. 6, where C=R ¼ 0 rpm at op3 rpm and C=R ¼ 0:3 rpm at oX5 rpm. o changes along with the solidified fraction in the experiment, and C 0 is the same in Figs. 4–6. With op3 rpm, ke is 0.981 close to unity in Fig. 6. ke increases with decreasing o as shown in Fig. 6. ke can be calculated from the slope or the intercept in the log–log plot of Eq. (5). If ke is close to unity, lnðC=C 0 Þ should be 0 at f ¼ 0 as shown in Eq. (5). However, the result near f ¼ 0 in Fig. 6 shows lnðC=C 0 Þ ¼ lnðke Þ ¼
0:2215, which means ke ¼ 0:801. ke obtained by the intercept in the log–log plot of Fig. 6 is not in agreement with the value obtained by the slope. The reason may be due to the shoulder process of the crystal. It is known that the shoulder process during crystal growth is in a transient state and the segregation phenomenon of the shoulder process is very different from that of the crystal body growth [3]. In Eq. (5), C 0 has no influence on the slope of the plot. Hence, we calculate ke from the slope. As o increases, the segregation process becomes an interface phenomenon even more, which is isolated from the bulk convection in the melt, and ke is close to k0 . Table 1 shows the comparison of experimental ke with theoretical ke [20], where the theoretical ke is calculated by Eq. (16) of the analytical paper [20] with n ¼ 3:38 107 m2 =s, D ¼ 2:4 108 m2 =s [18], and k0 ¼ 0:716. The silicon density of 2530 kg=m3 , magnetic strength of 383 G and electric conductivity of 1:25 106 ðomÞ1 are used to calculate theoretical ke . In the case of 640 G at the zeroGauss plane, the magnetic strength at the melt-crystal interface is 383 G.
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 290 (2006) 665–669
0
-0.5
RRG [%]
-1
-1.5
-2
-2.5 0
-0.5
-1
-1.5
ln (1-f) Fig. 7. RRG corresponding to Fig. 6. Even with very low o, RRG is in 2.2%.
The experimental results are in good agreement with the theoretical results [20], except at the highest value of ke with op3 rpm. The errors between the experimental and theoretical ke with oX5 rpm are less than 2.8% as shown in Table 1. Note that theory [20] has many assumptions such as neglecting natural convection, axisymmetric flow, and so on. Natural convection in the melt becomes dominant as o decreases, and natural convection becomes unstable, non-axisymmetric flow at a certain critical condition. The large error between the experimental and theoretical results at op3 rpm may be due to natural convection. The experimental ke with op3 rpm is higher than the theoretical result. It can be concluded that natural convection increases ke and should be considered to get exact ke in future theoretical study. Fig. 7 shows RRG corresponding to Fig. 6. It is well known that radial distribution of the dopant is influenced by o, and it is difficult to obtain uniform RRG in nostirring flow [8]. However, in our experimental data, RRG is 2.2% even with very low o as shown in Fig. 7, which shows a good uniformity. 4. Conclusions Boron-doped silicon single crystals of 207 mm diameter with various growing conditions are grown from a large amount of the melt in the cusp-magnetic Czochralski method, and the effects of growing parameters on dopant concentrations in the crystals are experimentally investigated.
669
With the experimental ke of 0.738 at o ¼ 20 rpm and no magnetic field, k0 calculated by BPS equation is in the range, 0.708–0.716. Also, k0 calculated with ke ¼ 0:751, o ¼ 13 rpm and no magnetic field is 0.715–0.724. It is concluded that k0 for boron is 0:716 0:08 in silicon ‘‘CZ’’ system. With o ¼ 13 rpm and C=R ¼ 0:5 rpm, ke in zero magnetic strength is 0.751 and increases up to 0.78 in the magnetic strength of 640 G. Magnetic field makes melt convection stable and hence ke higher. ke increases with increasing magnetic strength. For oX7 rpm, there is no significant influence of o on ke at the crystal-melt interface. With op3 rpm and the magnetic strength of 640 G, ke is 0.981 close to unity. The experimental results for oX5 rpm are in good agreement with the theoretical results, where the error between the experimental and theoretical ke is less than 2.8%. The large error between the experimental and theoretical results at op3 rpm is due to natural convection. The experimental ke with op3 rpm is higher than the theoretical result. It can be concluded that natural convection increases ke and should be considered to get exact ke in future theoretical study. Even with very low o, RRG is in 2.2%.
References [1] J. Burton, R. Prim, W. Slichter, J. Chem. Phys. 21 (1953) 1987. [2] J. Carruthers, A. Witt, R. Reusser, in: H. Huff, E. Sirtl (Eds.), Semiconductor Silicon 1977, The Electrochem. Soc., Princeton, p. 61. [3] H. Hirata, N. Inoue, Japan J. Appl. Phys. 24 (1985) 1399. [4] R. Thomas, H. Hobgood, P. Ravishankar, T. Braggins, Solid State Technol. (1990) 163. [5] D. Hurle, R. Series, J. Crystal Growth 73 (1985) 1. [6] R. Series, D. Hurle, J. Crystal Growth 113 (1991) 305. [7] A. Ostrogorsky, J. Crystal Growth 121 (1992) 587. [8] A. Witt, H. Gatos, M. Lichtensteiger, C. Herman, J. Electrochem. Soc. 125 (1978) 1832. [9] P. Ravishankar, L. Hunt, R. Francis, J. Electrochem. Soc. 131 (1984) 872. [10] H. Huff, T. Digges, O. Cecil, J. Appl. Phys. 42 (1971) 1235. [11] D. Hurle, P. Rudolph, J. Crystal Growth 264 (2004) 550. [12] C. Wang, D. Carlson, S. Motakef, M. Wiegel, M. Wargo, J. Crystal Growth 264 (2004) 565. [13] D. Schwabe, J. Crystal Growth 237–239 (2002) 1849. [14] D. Schwabe, A. Zebib, B.-C. Sim, J. Fluid Mech. 491 (2003) 239. [15] B.-C. Sim, A. Zebib, D. Schwabe, J. Fluid Mech. 491 (2003) 259. [16] Y. Won, K. Kakimoto, H. Ozoe, J. Crystal Growth 233 (2001) 622. [17] J. Irvin, Bell Syst. Tech. J. 41 (1962) 387. [18] H. Kodera, Japan J. Appl. Phys. 2 (1963) 212. [19] W. Tiller, The Science of Crystallization: Macroscopic Phenomena and Defect Generation, Cambridge University Press, Cambridge, 1991, p. 112. [20] R. Cartwright, D. Hurle, R. Series, J. Szekely, J. Crystal Growth 82 (1987) 327.
Journal of Crystal Growth 290 (2006) 665–669 www.elsevier.com/locate/jcrysgro
Boron segregation control in silicon crystal ingots grown in Czochralski process Bok-Cheol Sim, Kwang-Hun Kim, Hong-Woo Lee Crystal Growth Technology Team, LG Siltron, 283 Imsoo-dong, Gumi, Gyeong-buk 730-724, Korea Received 10 August 2005; received in revised form 2 February 2006; accepted 6 February 2006 Communicated by T. Hibiya
Abstract Boron-doped silicon single crystals of 207 mm diameter with various growing conditions are grown from a large amount of the melt in the cusp-magnetic Czochralski method, and the effects of growing parameters on dopant concentrations in the crystals are experimentally investigated. Equilibrium distribution coefficient of boron calculated by BPS model is 0.716. With the crystal rotation ðoÞ of 13 rpm and the crucible rotation of 0:5 rpm, the effective distribution coefficient (ke ) is 0.751 in zero magnetic strength and increases up to 0.78 in the magnetic strength of 640 G. For oX7 rpm, there is no significant influence of o on ke . With op3 rpm, ke is almost unity. The experimental results are compared with theory. r 2006 Elsevier B.V. All rights reserved. PACS: 81.05.Cy; 81.10.Fq Keywords: A1. Distribution coefficient; A1. Dopant concentration; A2. Czochralski method; B2. Semiconducting silicon
1. Introduction In order to obtain silicon single crystal of a required resistivity in Czochralski (CZ) method, dopant is added to the polysilicon charged in the crucible. The dopant is incorporated into the single crystal from the silicon melt. Dopant concentration and its distribution in the crystal are important for predictable device performance. Therefore, there have been a large number of experiments of dopant segregation process in the silicon melt during the CZ crystal growth. Dopant distribution in the crystals depends on the effective distribution coefficient, ke , at the solid–liquid interface, and convection in the melt plays a major role in determining ke of the dopant. Since Burton, Prim and Slichter (BPS) [1] developed a model for segregation in steady convective flows which is based on boundary layer in the melt, most of the studies have focused on the melt 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 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2006.02.003
convection to investigate dopant segregation during crystal growth. Carruthers et al. [2] reported ke of antimony, Sb, in stirring silicon melt in ‘‘CZ’’ system. Effect of magnetic field on ke was investigated by Hirata and Inoue [3] and Thomas et al. [4] for low ke dopants. They found that the magnetic field stabilized the melt convection and ke increased with increasing magnetic strength. These facts were verified in theoretical studies with a simplified model by Hurle and Series [5]. Various types of magnetic fields were discussed by Series and Hurls [6]. Ostrogorsky [7] developed more complex model to describe ke , which included the simple melt velocities in the solute boundary layers. ke can vary from ke ¼ k0 (for convection dominant segregation) to ke ¼ 1 (for convection-free segregation), where k0 is the equilibrium distribution coefficient of the dopant. For diffusion system without buoyant convection in zero gravity, Witt et al. [8] reported the dramatic increase of the dopant concentration in the axial direction and non-uniform radial distribution in the crystals, compared with results on earth.
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 290 (2006) 665–669
666
Ravishankar et al. [9] and Huff et al. [10] investigated ke of Boron in small amount of melt. They found that ke in ‘‘CZ’’ system was different from ke in floating zone system [9], and ke did not change with crystal growth rate [9,10]. The brief review of segregation was performed by Hurle and Rudolph [11] and Wang et al. [12]. Convection in the melt influences dopant distribution at the crystal-melt interface during crystal growth. In a commercial system, melt charge size should increase for large crystal size and mass products. Melt convection in a large amount of the silicon melt is complex and strong because the larger melt makes the natural convection [13–15] much stronger. Thus, in a large amount of the melt, it is difficult to control convection and hence dopant distribution. In the present study, 207 mm diameter silicon single crystals with various growing conditions are grown from a large amount of the melt in the cusp-magnetic Czochralski method, and the effects of growing parameters on dopant concentrations in the boron-doped silicon single crystal ingots are experimentally investigated. The experimental results are compared with theoretical results.
where g is gravity acceleration, b is the volume expansion coefficient, DT is the temperature difference, l is the characteristic length or melt height, n is kinematic viscosity, a is thermal diffusivity, o is the crystal or crucible rotation rate, and r is the crystal or crucible radius. In the experiments, Ra is in the range 4.7–8:2 107 with DT ¼ 20 K and l ¼ 216–260 mm. Re due to crystal rotation is 4:05 103 –8:1 104 with o ¼ 1–20 rpm, and Re for the crucible rotation is 0–2:38 105 . Thermophysical properties used in calculations are found in Won et al. [16]. Boron concentrations in the crystal are measured from the individual slices. The dopant, boron, concentrations and its distributions in the axial and radial directions are obtained from four-point probe resistivity measurements with the aid of the Irvin curve [17]. Since usual ‘‘CZ’’ Si wafer has almost completely axisymmetric structure, the four-point evaluation method for resistivity-radial gradient measurement is assumed valid. The measurement error is less than 1%. The resistivity-radial gradient, RRG, is expressed as follows:
2. Experiments
RRG½% ¼
Commercial furnace with a cusp-magnetic field is used to grow 207 mm diameter silicon single crystals doped with boron. The crucible has an inner diameter of 600 mm and is made of quartz. The initial charge sizes of the silicon melts are in the range 110–140 kg in various experiments, and the initial melt heights in the quartz crucibles are 216–260 mm, which depends on the initial charge size of the melt. The single crystals of more than 110 cm in length are grown in the h1 0 0i direction with 8–12 O cm. The crystal (seed) is rotated in the range 1–20 rpm, and the crucible rotation is in the range 0–7 rmp in opposite to crystal rotation direction. Argon flow-rate and furnace pressure are fixed. The cusp-magnetic field with the zero-Gauss plane at 110 mm below the free surface is applied to the melt. The magnetic strength indicated here represents the intensity of the horizontal component of the magnetic field orthogonal to crucible side wall at the zero-Gauss plane. The pulling speed of the crystal is changed to keep the crystal diameter constant. The pulling speeds varied in the range 0:65 0:05 mm= min. Two main flows exist in the melt; one is natural convection, which is the buoyant and thermocapillary forces, and the other is forced convection due to crystal and crucible rotations. Usually, buoyant force is much stronger than thermocapillary force, and thermocapillary force can be neglected. The buoyant and forced convection is characterized by the Rayleigh, Ra, and Reynolds, Re, numbers as follows: Ra ¼
gbDTl 3 , na
or2 , Re ¼ n
Center [Res] Average [Res] at the edge 4 points 100, Center [Res]
(3) where Center [Res] is the resistivity at the crystal center, and the 4 edge [Res] are measured at 6 mm from the crystal edges as shown in Fig. 1 and are averaged.
(1)
(2)
Fig. 1. Resistivity-measurement points for RRG at the crystal, where the 4 edge [Res] are measured at 6 mm from the crystal edges and are averaged.
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 290 (2006) 665–669
3. Results and discussion
0.15 0.1
ke is determined by fitting the experimental data to the normal freezing equation, C ¼ C 0 ke ð1 f Þke 1 ,
0.05 0
where C is the dopant concentration in the solid for a given fraction, f, solidified, and C 0 is the initial dopant concentration in the melt. The equation, Eq. (4), is rearranged in logarithmic form as follows:
-0.05
lnðC=C 0 Þ ¼ lnðke Þ þ ðke 1Þ lnð1 f Þ.
(5)
The log–log plot, which is lnðC=C 0 Þ vs. lnð1 f Þ, of Eq. (5) results in a straight line, and ke can be calculated from the slope in the log–log plot. The slope is determined by a least square fit of the experimental data. Fig. 2 shows a typical log–log curve of the boron concentration in the two crystals grown with the same growing parameters. The growing condition for the crystals is the crystal rotation ðoÞ of 20 rpm, the crucible rotation (C=R) of 7 rpm and the initial melt size of 120 kg without the magnetic field. ke at each crystal is 0.732 and 0.744. It is determined that ke is 0:738 0:006 in Fig. 2. The square of the correlation coefficient, R2 , is 0.997 close to unity, which demonstrates a good linear fit to the data. ke ¼ 0:738 0:006 for boron in the silicon melt is in reasonable agreement with other works [9,10], where ke ¼ 0:7 and ke ¼ 0:78 0:03 are reported irrespective of the crystal growth rates. Fig. 3 shows ke with o ¼ 13 rpm, C=R ¼ 0:5 rpm, the initial melt size of 120 kg and no magnetic field. For reproducibility, three different ingots are grown with the same above growing parameters. ke for three ingots are 0.75, 0.755 and 0.76, and the deviation of ke in three ingots 0.15 0.1 Ke=0.738
0
ln (Cs/C0)
-0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 0
-0.5
-1
-1.5
ln (1-f) Fig. 2. Macroscopic boron concentration in the axial direction with o ¼ 20 rpm and C=R ¼ 7 rpm, where two symbols represent experimental results in two crystal ingots grown with the same growing condition.
ln (Cs/C0)
(4)
0.05
667
Ke =0.751
-0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 0
-0.5
-1
-1.5
ln (1-f) Fig. 3. Macroscopic boron concentration in the axial direction with o ¼ 13 rpm and C=R ¼ 0:5 rpm. The square of the correlation coefficient in a least square fit is 0.999.
with the same growth conditions is less than 1.5%. ke determined by a least square fit of the total data is 0.751 as shown in Fig. 3. BPS model relates ke to k0 as follows: ke ¼
k0 , k0 þ ð1 k0 Þ expðvd=DÞ
(6)
where v is the crystal growth rate, d is the diffusion-layer thickness, and D is diffusion coefficient. With the experimental ke of 0.738 for o ¼ 20 rpm, k0 is calculated by Eq. (6), where v ¼ 1:083 103 cm=s and d=D ¼ 120 19 s=cm for o ¼ 20 rpm from Kodera [18] and Tiller [19]. k0 calculated by Eq. (6) is in the range, 0.708–0.716. Also, with o ¼ 13 rpm and ke ¼ 0:751, k0 calculated is 0.715–0.724, where d=D ¼ 150 19 s=cm for o ¼ 13 rpm is used due to do0:5 [18,19]. Almost same value of k0 ¼ 0:716 is found from two experimental ke even though C=R is ignored in Eq. (6). It is concluded that k0 for boron is 0:716 0:08 in silicon ‘‘CZ’’ system. Effect of the magnetic force on ke is shown in Fig. 4 with o ¼ 13 rpm, C=R ¼ 0:3 rpm and different magnetic strengths. ke increases with increasing magnetic strength in Fig. 4. Compared with ke of zero magnetic strength, ke with the magnetic strength of 640 G is 5.4% higher. The magnetic field makes melt convection stable and hence ke higher. Unfortunately, 640 G was the maximum magnetic strength in our experiments. It is well known that ke increases dramatically with increasing magnetic strength in a low ke dopant such as P, Sb, Ga and Al [3,4]. Fig. 5 shows the effect of o on ke with C=R ¼ 0:3 rpm, the magnetic strength of 394 G and oX7 rpm. The difference between two ke is within the measurement error, with oX7 rpm, there is no significant influence of o on ke as shown in Fig. 5. For op7 rpm, the effect of o on ke with
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 290 (2006) 665–669
0.15
0.15 Mag. strength=394 Gauss Mag. strength=640 Gauss Least Square Fit
0.1 0.05
0.1
0
5
-0.05 ln (Cs/C0)
-0.05 ln (Cs/C0)
6
0.05
Ke=0.764
0
7
-0.1 -0.15
Ke=0.7 8
Ke=0.791
-0.1 -0.15
-0.25
-0.25
-0.3
-0.3
-0.35
-0.35
-0.4
Ke=0.981
1 0
-0.5
-1
-0.4 -1
Fig. 4. Effect of the magnetic strength on ke with o ¼ 13 rpm. The magnetic strength lower than 640 G has a little influence on ke .
S/R=13rpm S/R=7rpm Least Square Fit
0.05 0
Ke=0.762
ln (Cs/C0)
-0.05
Fig. 6. Effect of o ðp7 rpmÞ on ke with the magnetic strength of 640 G. With op3 rpm, ke is almost unity.
Table 1 Comparison of experimental ke with theoretical ke [20], where magnetic strength is the value at the melt-crystal interface
0.15 0.1
-1.5
ln (1-f)
-1.5
ln (1-f)
2
Ke=0.814
0 -0.5
3
Ke=0.813
-0.2
-0.2
0
4
ω [rpm]
668
-0.1 Ke=0.764
-0.15
o ðrpmÞ
C=R (rpm)
Magnetic strength (G)
1 3 5 7 13 13
0 0 0.3 0.3 0.3 0.5
383 383 383 383 383 0
Exp. ke 0.981 0.981 0.814 0.791 0.78 0.751
Theo. ke 0.943 0.835 0.803 0.782 0.758 0.750
-0.2 -0.25 -0.3 -0.35 -0.4 0
-1
-0.5
-1.5
ln (1-f) Fig. 5. Effect of o (X7 rpm) on ke with the magnetic strength of 394 G. o ðX7 rpmÞ has no influence on ke .
the magnetic strength of 640 G is shown in Fig. 6, where C=R ¼ 0 rpm at op3 rpm and C=R ¼ 0:3 rpm at oX5 rpm. o changes along with the solidified fraction in the experiment, and C 0 is the same in Figs. 4–6. With op3 rpm, ke is 0.981 close to unity in Fig. 6. ke increases with decreasing o as shown in Fig. 6. ke can be calculated from the slope or the intercept in the log–log plot of Eq. (5). If ke is close to unity, lnðC=C 0 Þ should be 0 at f ¼ 0 as shown in Eq. (5). However, the result near f ¼ 0 in Fig. 6 shows lnðC=C 0 Þ ¼ lnðke Þ ¼
0:2215, which means ke ¼ 0:801. ke obtained by the intercept in the log–log plot of Fig. 6 is not in agreement with the value obtained by the slope. The reason may be due to the shoulder process of the crystal. It is known that the shoulder process during crystal growth is in a transient state and the segregation phenomenon of the shoulder process is very different from that of the crystal body growth [3]. In Eq. (5), C 0 has no influence on the slope of the plot. Hence, we calculate ke from the slope. As o increases, the segregation process becomes an interface phenomenon even more, which is isolated from the bulk convection in the melt, and ke is close to k0 . Table 1 shows the comparison of experimental ke with theoretical ke [20], where the theoretical ke is calculated by Eq. (16) of the analytical paper [20] with n ¼ 3:38 107 m2 =s, D ¼ 2:4 108 m2 =s [18], and k0 ¼ 0:716. The silicon density of 2530 kg=m3 , magnetic strength of 383 G and electric conductivity of 1:25 106 ðomÞ1 are used to calculate theoretical ke . In the case of 640 G at the zeroGauss plane, the magnetic strength at the melt-crystal interface is 383 G.
ARTICLE IN PRESS B.-C. Sim et al. / Journal of Crystal Growth 290 (2006) 665–669
0
-0.5
RRG [%]
-1
-1.5
-2
-2.5 0
-0.5
-1
-1.5
ln (1-f) Fig. 7. RRG corresponding to Fig. 6. Even with very low o, RRG is in 2.2%.
The experimental results are in good agreement with the theoretical results [20], except at the highest value of ke with op3 rpm. The errors between the experimental and theoretical ke with oX5 rpm are less than 2.8% as shown in Table 1. Note that theory [20] has many assumptions such as neglecting natural convection, axisymmetric flow, and so on. Natural convection in the melt becomes dominant as o decreases, and natural convection becomes unstable, non-axisymmetric flow at a certain critical condition. The large error between the experimental and theoretical results at op3 rpm may be due to natural convection. The experimental ke with op3 rpm is higher than the theoretical result. It can be concluded that natural convection increases ke and should be considered to get exact ke in future theoretical study. Fig. 7 shows RRG corresponding to Fig. 6. It is well known that radial distribution of the dopant is influenced by o, and it is difficult to obtain uniform RRG in nostirring flow [8]. However, in our experimental data, RRG is 2.2% even with very low o as shown in Fig. 7, which shows a good uniformity. 4. Conclusions Boron-doped silicon single crystals of 207 mm diameter with various growing conditions are grown from a large amount of the melt in the cusp-magnetic Czochralski method, and the effects of growing parameters on dopant concentrations in the crystals are experimentally investigated.
669
With the experimental ke of 0.738 at o ¼ 20 rpm and no magnetic field, k0 calculated by BPS equation is in the range, 0.708–0.716. Also, k0 calculated with ke ¼ 0:751, o ¼ 13 rpm and no magnetic field is 0.715–0.724. It is concluded that k0 for boron is 0:716 0:08 in silicon ‘‘CZ’’ system. With o ¼ 13 rpm and C=R ¼ 0:5 rpm, ke in zero magnetic strength is 0.751 and increases up to 0.78 in the magnetic strength of 640 G. Magnetic field makes melt convection stable and hence ke higher. ke increases with increasing magnetic strength. For oX7 rpm, there is no significant influence of o on ke at the crystal-melt interface. With op3 rpm and the magnetic strength of 640 G, ke is 0.981 close to unity. The experimental results for oX5 rpm are in good agreement with the theoretical results, where the error between the experimental and theoretical ke is less than 2.8%. The large error between the experimental and theoretical results at op3 rpm is due to natural convection. The experimental ke with op3 rpm is higher than the theoretical result. It can be concluded that natural convection increases ke and should be considered to get exact ke in future theoretical study. Even with very low o, RRG is in 2.2%.
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