- Email: [email protected]
The behavior of powder sublimation in the long-term PVT growth of SiC crystals
ARTICLE IN PRESS Journal of Crystal Growth 312 (2010) 1486–1490
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
The behavior of powder sublimation in the long-term PVT growth of SiC crystals Xi Liu a,b,c, Bo-yuan Chen a,b, Li-Xin Song a,c, Er-Wei Shi a, Zhi-Zhan Chen a,n a
Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, China Graduate University of Chinese Academy of Sciences, Beijing 100049, China c Key Laboratory of Inorganic Coating Materials, Chinese Academy of Sciences, Shanghai 200050, China b
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 July 2009 Received in revised form 14 December 2009 Accepted 17 January 2010 Communicated by M. Skowronski Available online 22 January 2010
The effect of different temperature distribution in powder on the powder sublimation was investigated by experiments and simulations. On the one hand, appropriately large temperature difference in the powder will contribute to high growth rate when the mass transportation between the powder and the seed proceeds smoothly. Nevertheless, the recrystallization at the bottom of powder will reduce the available powder in the growth and should be avoided. On the other hand, when the temperature difference in the powder is so large that the rate of sublimation of the powder increases beyond the normal limit of the rate of mass transportation between the powder and the seed, the mass transportation in the powder will be obstructed and even the crystal growth will be interrupted. & 2010 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulation A1. Growth model A1. Heat transfer A2. Single crystal growth B1. Growth from vapor B2. Semiconducting materials
1. Introduction Silicon carbide (SiC) is a promising wide bandgap semiconductor that can be used to fabricate electronic and optoelectronic devices. Presently, physical vapor transport (PVT) is a commercial method to produce large bulk SiC single crystals. SiC powder is used as the source material, in which physical and chemical processes occur during the long-term growth. In the last few years experimental study of the powder density during growth by X-ray technique demonstrated the formation of a dense disk and compressed core as well as the graphitization of the SiC granules [1–7]. Numerical modeling is a powerful tool to study the powder evolution. However, many questions, like the effects of growth conditions on the long-term stability of powder sublimation and crystal growth rate, still remain open and encourage more in-depth investigations. This paper presents experimental results and numerical analysis to clarify the obstructed sublimation of the powder.
2. Experiments and numerical model 6H-SiC bulk single crystals were grown in an inductively heated graphite crucible with graphite thermal insulators by the PVT n
Corresponding author: Tel: 86 21 52411109; fax: 86 21 52413903. E-mail address: [email protected] (Z.-Z. Chen).
0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.01.029
method. The growth apparatus and growth parameters are reported in Ref. [8]. In order to investigate the effects of growth conditions on the stability of sublimation and growth rate, three different positions of heating coils were used (high position, middle position and low position). In experiment B, the position of heating coils was the same as usual (the bottom of the crucible is 5 cm above the bottom coil). In experiment A, the position of heating coils was 15 mm higher than that of experiment B. In experiment C, the position of heating coils was 15 mm lower than that of experiment B. The initial powder porosity in the three experiments was about 55%. After the growth, powder graphitization was measured by an X-ray diffractometer (XRD, D/max 2550 V, Japan). Numerical modeling of the temperature field evolution and growth rate is given in Ref. [8]. Furthermore, the sublimated mass of the SiC powder and its graphitization degree during the growth run are calculated, assuming the shell structure of SiC granules characterized mean granule (R) and porosity (e). As a result of SiC granule sublimation, multicomponent gas mixture (Si, Si2C and SiC2) is produced and the granule becomes a SiC core with radius (r) surrounded by the graphite layer, which is permeable to the gas mixture. The sketch of the shell structure of the granule model is shown in Fig. 1 [5]. The gas mixture transports through the powder by diffusion and convection. At the same time, the condensation of the gas mixture on the granule induces the recrystallization. The global mass transport is coupled with heterogeneous chemical reactions proceeding on the granule surface. The details of the model can be found in Ref. [5].
ARTICLE IN PRESS X. Liu et al. / Journal of Crystal Growth 312 (2010) 1486–1490
1487
Fig. 1. The sketch of the shell structure of granule model [5].
Table 1 The results of the three experiments. Experiments
Tup (1C)
Tdown (1C)
t (h)
PAr (Torr)
L (mm)
v (mm/h)
A B C
2100 2100 2100
2231 2257 2285
50 50 50
20 20 20
12.0 13.3 4
0.240 0.266 0.08
Tup is the temperature at the top temperature hole; Tdown is the temperature at the bottom temperature hole; t is the total growth time; PAr is the pressure of Ar; L is the length of the crystal; and v is the average growth rate.
3. Results and discussion Three experiments were carried out under the same growth conditions except for the position of the heating coils. The detailed parameters and experimental results are listed in Table 1. After growth, the ingot lengths were 12.0, 13.3 and 4 mm, respectively. The average growth rates were 0.240 mm/h for experiment A, 0.266 mm/h for experiment B and 0.08 mm/h for experiment C. It is easily found that the crystal growth was interrupted in experiment C. Consequently, the residual powders of three experiments were investigated carefully. After growth experiments, the positions of each powder did not change and each residual powder consists of the compressed disk, the core and the graphite foam. Fig. 2 shows the photos of the compressed disks in the three experiments and the XRD patterns are inserted. The diffraction peaks of 6H-SiC marked (1 0 1), (0 0 6), (1 0 3) and graphite marked (0 0 2) can illustrate that three compressed disks are partly graphitized. Furthermore, there are a number of large hexagonal crystals at the bottom of the disk in experiment C and their diffraction peaks demonstrate that 6H-SiC crystals grow spontaneously at the bottom of the disk. Fig. 3 shows the XRD patterns of the powder in the bottom of the crucible in the three experiments. In experiments B and C, they look like graphite foam and the sole existence of diffraction peaks of graphite proves they are fully graphitized. However, the existences of diffraction peaks of 6H-SiC and graphite in experiment A illustrate that it is just partly graphitized. In order to further investigate the powder sublimation, three simulations a, b and c, corresponding to experiments A, B and C, are carried out to calculate the temperature distribution, the graphitization degree distribution and the mass transportation in the powder.
Fig. 2. The photo and the XRD pattern of the compressed disk in experiment A (a); the photo and the XRD pattern of the compressed disk in experiment B (b); and the photo and XRD patterns of the compressed disk in experiment C (c).
Fig. 4 shows the steady temperature distribution of the powder in the three simulations. It can be found that the hottest area of powder along the central axis is at the bottom in simulations b and c. However, it is a little above the bottom in simulation a. Table 2 lists the highest/lowest temperatures and the maximum temperature differences in the powder. The highest/lowest temperatures in the powder are 2338/2243 1C for simulation a, 2363/2245 1C for simulation b and 2397/2247 1C for simulation c. The maximum temperature differences in the powder are 95, 118 and 150 1C. Fig. 4 also shows the final graphitization degree
ARTICLE IN PRESS 1488
X. Liu et al. / Journal of Crystal Growth 312 (2010) 1486–1490
Fig. 3. The XRD patterns of the powder in the bottom of the crucible in the three experiments.
distribution of the powder in the three simulations. It shows that the powder at the disk area is partly graphitized in the three simulations. It also illustrates that the bottom of the powder is only a little graphitized in simulation a. However, the bottom of the powder is completely graphitized in simulations b and c. The computed results are in excellent agreement with the XRD patterns. Based on the modeling of SiC powder sublimation and the computed temperature field evolution, both the total sublimated powder mass during the whole growth and the crystal growth rate are calculated and shown in Fig. 5. As shown in Fig. 5(a), it can be observed that the ascending trend of three curves (a,b,c) becomes more and more flat and the efficiency of sublimation in simulation c is the highest compared to the other simulations. As shown in Fig. 5(b), the crystal growth rate in simulation c is the maximum and the rate in simulation a is the minimum during the initial stage. The total sublimated mass and the average growth rate are also listed in Table 2. They are 97.87 g (0.257 mm/h) for simulation a , 107.05 g (0.284 mm/h ) for simulation b and 115.53 g (0.310 mm/h) for simulation c. According to the above calculated results of simulations a and b, it can be easily understood that the whole temperature distribution and the largest temperature difference in the powder play a very important role in the powder sublimation and the crystal growth rate, when the pressure of argon is invariable. On the one hand, the large temperature difference in the powder is helpful to enhance the efficiency of sublimation and the growth rate. On the other hand, the temperature distribution in the powder like simulation a induces recrystallization at the bottom of powder and reduces the available powder that should have been transported to the surface of the growing crystal. The calculated results of simulations a and b are in good agreement with experiments A and B. However, the calculated results of simulation c and experiment C seem to be contradictory. According to Ref. [5], the contribution of powder sublimation into the crystal growth rate increases with the temperature difference in the powder. The maximum temperature difference in the powder is 150 1C in simulation c and its crystal growth rate is supposed to be the maximum. As shown in Fig. 5, the crystal growth rate in simulation c is maximum in the fifth hour and the total mass
Fig. 4. The steady temperature distribution (left) and final graphitization degree distribution (right) of the source powder in simulation a (a); the steady temperature distribution (left) and final graphitization degree distribution (right) of the source powder in simulation b (b); and the steady temperature distribution (left) and final graphitization degree distribution (right) of the source powder in simulation c (c).
Table 2 The calculated temperature distribution and the average growth rate for the three simulations. Simulations
Tmax (1C)
Tmin (1C)
DT (1C)
PAr (Torr)
M (g)
v¯ (mm/h)
a b c
2338 2363 2397
2243 2245 2247
95 118 150
20 20 20
97.87 107.05 115.53
0.257 0.284 0.310
Tmax/Tmin is the maximum/minimum temperature of the powder; DT= Tmax-Tmin; PAr is the pressure of Ar; M is the total sublimated mass of the powder; and v¯ is the average growth rate.
transported from the powder in simulation c is also maximum. Nevertheless, it is proved that both the sublimation of the powder and the growth rate are unstable by the crystal length and the residual powder (Fig. 2). Another hidden factor that affects the
ARTICLE IN PRESS X. Liu et al. / Journal of Crystal Growth 312 (2010) 1486–1490
1489
Table 3 The calculated v and v0 for the three simulations. Simulations
tpeak (h)
Tp (1C)
Ts (1C)
PAr (Torr)
v (mm/h)
v0 (mm/h)
a b c
6 6 5
2329 2352 2373
2192 2192 2188
20 20 20
0.424 0.572 0.809
0.480 0.609 0.767
tpeak is the peak time; Tp is the maximum temperature in the powder; Ts is the temperature at the center of the seed surface; v is the maximum growth rate calculated by the sublimation of the powder; and v0 is the growth rate controlled by the mass diffusion in the gap between the powder and the crystal at the peak time.
powder surface, D is the diffusion coefficient given by Kaldis and Piechotka [11] and R is the gas constant. When the pressure of decomposed SiC vapor at the powder surface is greater than the equilibrium pressure, the powder will stop decomposing. When the pressure of decomposed SiC vapor at the seed is less than the equilibrium pressure, the seed growth will be interrupted. Therefore, the upper normal limit of the mass transport rate can be obtained from Eq. (1) and can be expressed as follows: J¼
Fig. 5. The total sublimated powder mass during the whole growth in the three simulations (a); and the calculated growth rate based on the model of the sublimation of the powder in the three simulations (b).
stability of the sublimation and growth rate is possibly neglected by the modeling of SiC powder sublimation. During the initial stage of crystal growth, the powder close to the hot wall of the crucible decomposes and then moves upwards to the crystal seed along the wall of the crucible. With the evolution of the temperature field in the powder, the whole powder begins to react. Most of the decomposed SiC vapor flows through the center of the powder by the repetitive process of sublimation and recrystallization and then moves upwards to the growing crystal [3]. If the argon pressure is much higher than that of other vapor species in the crucible as in the case of our work, the upwards movement of the SiC vapor is diffusion dominated and Steafan flow can be neglected [9,10]. The mass transport rate can be expressed by the following equations [10]: J¼
PSiC2 ðcÞPSiC2 ðsÞ LRðTc þ Ts =2Þ=D
PSiC ðcÞPSiC ðsÞ 2 2
LRðTc þ Ts =2Þ=D
ð2Þ
ðcÞ and PSiC ðsÞare the equilibrium pressures of the ratewhere PSiC 2 2 determining specie at the powder surface and the growth interface, respectively. Consequently, the maximum mass transport rate is the upper normal limit of the crystal growth rate. As shown in Fig. 5(b), it can be observed that the maximum growth rate appears at about the fifth hour when the powder close to the hot wall of the crucible decomposes. Both the growth rate at the peak time (v0 ) according to the growth model reported in Ref. [8] and the maximum growth rate (v) based on the modeling of SiC powder sublimation are computed and listed in Table 3. It can be found that v0 (0.480, 0.609 mm/h) is larger than v (0.424, 0.572 mm/h) in simulations a and b. Consequently, the rate of the powder sublimation is less than the mass transportation rate between the powder and the seed and the crystal growth proceeds smoothly. Therefore, the growth rates calculated in simulations a and b, which are shown in Fig. 5(b), can predict the experimental results. However, v0 (0.767 mm/h) is less than v (0.809 mm/h) in simulation c. On the one hand, the rate of powder sublimation is larger than the mass transportation rate between the powder and the seed and it makes the SiC vapor gradually accumulate above the disk and then the pressure of the SiC vapor above the disk becomes so large that the channels of mass transportation in the powder are obstructed seriously. One the other hand, the large temperature gradient in the powder will induce the growth of the crystal granule in the low temperature gradient zone [3], and then, the permeability of the powder decreases if the particle diameter of the SiC powder increases [7]. Low permeability of the powder not only decreases the sublimed powder mass but also increases the SiC vapor pressure in the low temperature gradient zone of the powder. Consequently, the SiC vapor under the disk becomes supersaturated and begins to deposit on the bottom of the disk spontaneously. In case c, extremely large temperature difference in the powder goes against the sublimation of the powder and even interprets the whole crystal growth like experiment C.
ð1Þ
where PSiC2 ðcÞ and PSiC2 ðsÞ are the pressures of the rate-determining specie at the powder surface and the growth interface, respectively, Tc and Ts are the temperatures at the powder surface and the seed, respectively, L is the distance between the seed and
4. Conclusion The temperature distribution of the powder plays an important role in the stability of powder sublimation and the crystal
ARTICLE IN PRESS 1490
X. Liu et al. / Journal of Crystal Growth 312 (2010) 1486–1490
growth rate. When the channels of mass transportation in the powder are unobstructed, the large temperature difference in the powder will be helpful in enhancing the efficiency of sublimation and the growth rate. The recrystallization at the bottom of the powder should be avoided. When the sublimation of the powder has an advantage over the SiC vapor diffusion in the gap between the powder and the growing crystal, the channels of mass transportation in the powder will be obstructed and it will even interrupt the crystal growth.
Acknowledgements This work was supported by ‘‘863’’ Project (2006AA03A146), Knowledge Innovation Program of the Chinese Academy of Sciences (KGCX2-YW-206) and Natural Science Foundation of Shanghai (06ZR14096) (Science and Technology Commission of Shanghai Municipality (09DZ1141400, 09520714900)).
References ¨ [1] P.J. Wellmann, Z. Herro, A. Winnacker, R. PUsche, M. Hundhausen, P. Masri, A. Kulik, M. Bogdanov, S. Karpov, M. Ramm, Y. Makarov, J. Crys. Growth 275 (2005) e1807. [2] P.J. Wellmann, D. Hofmann, L. Kadinski, M. Selder, T.L. Straubinger, A. Winnacker, J. Crys. Growth 225 (2001) 312. ¨ [3] P.J. Wellmann, Z. Herro, M. Selder, F. Durst, R. PUsche, M. Hundhausen, L. Ley, A. Winnacker, Mater. Sci. Forum 433–436 (2003) 9. [4] D.S. Karpov, O.V. Bord, S.Y. Karpov, A.I. Zhmakin, M.S. Ramm, Y.N. Makarov, Mater. Sci. Forum 353–356 (2001) 37. [5] A.V. Kulik, M.V. Bogdanov, S.Y. Karpov, M.S. Ramm, Y.N. Makarov, Mater. Sci. Forum 457–460 (2004) 67. [6] D. Cai, L.L. Zheng, H. Zhang, D. Zhuang, Z.G. Herro, R. Schlesser, Z. Sitar, J. Crys. Growth 306 (2007) 39. [7] X.L. Wang, D. Cai, H. Zhang, J. Crys. Growth 305 (2007) 122. [8] X. Liu, E.W. Shi, L.X. Song, Z.Z. Chen, J. Crys. Growth 310 (2008) 4314. [9] R.H. Ma, H. Zhang, V. Prasad, M. Dudley, Cryst. Growth Des. 2 (2002) 213. [10] R.H. Ma, H. Zhang, S. Ha, M. Skowrinski, J. Crys. Growth 252 (2003) 523. [11] E. Kaldis, E.M. Piechotka, Bulk crystal growth by physical vapor transport, in: D.I.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 2, Elsevier Science, Amsterdam, 1994, p. 615.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
The behavior of powder sublimation in the long-term PVT growth of SiC crystals Xi Liu a,b,c, Bo-yuan Chen a,b, Li-Xin Song a,c, Er-Wei Shi a, Zhi-Zhan Chen a,n a
Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 200050, China Graduate University of Chinese Academy of Sciences, Beijing 100049, China c Key Laboratory of Inorganic Coating Materials, Chinese Academy of Sciences, Shanghai 200050, China b
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 July 2009 Received in revised form 14 December 2009 Accepted 17 January 2010 Communicated by M. Skowronski Available online 22 January 2010
The effect of different temperature distribution in powder on the powder sublimation was investigated by experiments and simulations. On the one hand, appropriately large temperature difference in the powder will contribute to high growth rate when the mass transportation between the powder and the seed proceeds smoothly. Nevertheless, the recrystallization at the bottom of powder will reduce the available powder in the growth and should be avoided. On the other hand, when the temperature difference in the powder is so large that the rate of sublimation of the powder increases beyond the normal limit of the rate of mass transportation between the powder and the seed, the mass transportation in the powder will be obstructed and even the crystal growth will be interrupted. & 2010 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulation A1. Growth model A1. Heat transfer A2. Single crystal growth B1. Growth from vapor B2. Semiconducting materials
1. Introduction Silicon carbide (SiC) is a promising wide bandgap semiconductor that can be used to fabricate electronic and optoelectronic devices. Presently, physical vapor transport (PVT) is a commercial method to produce large bulk SiC single crystals. SiC powder is used as the source material, in which physical and chemical processes occur during the long-term growth. In the last few years experimental study of the powder density during growth by X-ray technique demonstrated the formation of a dense disk and compressed core as well as the graphitization of the SiC granules [1–7]. Numerical modeling is a powerful tool to study the powder evolution. However, many questions, like the effects of growth conditions on the long-term stability of powder sublimation and crystal growth rate, still remain open and encourage more in-depth investigations. This paper presents experimental results and numerical analysis to clarify the obstructed sublimation of the powder.
2. Experiments and numerical model 6H-SiC bulk single crystals were grown in an inductively heated graphite crucible with graphite thermal insulators by the PVT n
Corresponding author: Tel: 86 21 52411109; fax: 86 21 52413903. E-mail address: [email protected] (Z.-Z. Chen).
0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.01.029
method. The growth apparatus and growth parameters are reported in Ref. [8]. In order to investigate the effects of growth conditions on the stability of sublimation and growth rate, three different positions of heating coils were used (high position, middle position and low position). In experiment B, the position of heating coils was the same as usual (the bottom of the crucible is 5 cm above the bottom coil). In experiment A, the position of heating coils was 15 mm higher than that of experiment B. In experiment C, the position of heating coils was 15 mm lower than that of experiment B. The initial powder porosity in the three experiments was about 55%. After the growth, powder graphitization was measured by an X-ray diffractometer (XRD, D/max 2550 V, Japan). Numerical modeling of the temperature field evolution and growth rate is given in Ref. [8]. Furthermore, the sublimated mass of the SiC powder and its graphitization degree during the growth run are calculated, assuming the shell structure of SiC granules characterized mean granule (R) and porosity (e). As a result of SiC granule sublimation, multicomponent gas mixture (Si, Si2C and SiC2) is produced and the granule becomes a SiC core with radius (r) surrounded by the graphite layer, which is permeable to the gas mixture. The sketch of the shell structure of the granule model is shown in Fig. 1 [5]. The gas mixture transports through the powder by diffusion and convection. At the same time, the condensation of the gas mixture on the granule induces the recrystallization. The global mass transport is coupled with heterogeneous chemical reactions proceeding on the granule surface. The details of the model can be found in Ref. [5].
ARTICLE IN PRESS X. Liu et al. / Journal of Crystal Growth 312 (2010) 1486–1490
1487
Fig. 1. The sketch of the shell structure of granule model [5].
Table 1 The results of the three experiments. Experiments
Tup (1C)
Tdown (1C)
t (h)
PAr (Torr)
L (mm)
v (mm/h)
A B C
2100 2100 2100
2231 2257 2285
50 50 50
20 20 20
12.0 13.3 4
0.240 0.266 0.08
Tup is the temperature at the top temperature hole; Tdown is the temperature at the bottom temperature hole; t is the total growth time; PAr is the pressure of Ar; L is the length of the crystal; and v is the average growth rate.
3. Results and discussion Three experiments were carried out under the same growth conditions except for the position of the heating coils. The detailed parameters and experimental results are listed in Table 1. After growth, the ingot lengths were 12.0, 13.3 and 4 mm, respectively. The average growth rates were 0.240 mm/h for experiment A, 0.266 mm/h for experiment B and 0.08 mm/h for experiment C. It is easily found that the crystal growth was interrupted in experiment C. Consequently, the residual powders of three experiments were investigated carefully. After growth experiments, the positions of each powder did not change and each residual powder consists of the compressed disk, the core and the graphite foam. Fig. 2 shows the photos of the compressed disks in the three experiments and the XRD patterns are inserted. The diffraction peaks of 6H-SiC marked (1 0 1), (0 0 6), (1 0 3) and graphite marked (0 0 2) can illustrate that three compressed disks are partly graphitized. Furthermore, there are a number of large hexagonal crystals at the bottom of the disk in experiment C and their diffraction peaks demonstrate that 6H-SiC crystals grow spontaneously at the bottom of the disk. Fig. 3 shows the XRD patterns of the powder in the bottom of the crucible in the three experiments. In experiments B and C, they look like graphite foam and the sole existence of diffraction peaks of graphite proves they are fully graphitized. However, the existences of diffraction peaks of 6H-SiC and graphite in experiment A illustrate that it is just partly graphitized. In order to further investigate the powder sublimation, three simulations a, b and c, corresponding to experiments A, B and C, are carried out to calculate the temperature distribution, the graphitization degree distribution and the mass transportation in the powder.
Fig. 2. The photo and the XRD pattern of the compressed disk in experiment A (a); the photo and the XRD pattern of the compressed disk in experiment B (b); and the photo and XRD patterns of the compressed disk in experiment C (c).
Fig. 4 shows the steady temperature distribution of the powder in the three simulations. It can be found that the hottest area of powder along the central axis is at the bottom in simulations b and c. However, it is a little above the bottom in simulation a. Table 2 lists the highest/lowest temperatures and the maximum temperature differences in the powder. The highest/lowest temperatures in the powder are 2338/2243 1C for simulation a, 2363/2245 1C for simulation b and 2397/2247 1C for simulation c. The maximum temperature differences in the powder are 95, 118 and 150 1C. Fig. 4 also shows the final graphitization degree
ARTICLE IN PRESS 1488
X. Liu et al. / Journal of Crystal Growth 312 (2010) 1486–1490
Fig. 3. The XRD patterns of the powder in the bottom of the crucible in the three experiments.
distribution of the powder in the three simulations. It shows that the powder at the disk area is partly graphitized in the three simulations. It also illustrates that the bottom of the powder is only a little graphitized in simulation a. However, the bottom of the powder is completely graphitized in simulations b and c. The computed results are in excellent agreement with the XRD patterns. Based on the modeling of SiC powder sublimation and the computed temperature field evolution, both the total sublimated powder mass during the whole growth and the crystal growth rate are calculated and shown in Fig. 5. As shown in Fig. 5(a), it can be observed that the ascending trend of three curves (a,b,c) becomes more and more flat and the efficiency of sublimation in simulation c is the highest compared to the other simulations. As shown in Fig. 5(b), the crystal growth rate in simulation c is the maximum and the rate in simulation a is the minimum during the initial stage. The total sublimated mass and the average growth rate are also listed in Table 2. They are 97.87 g (0.257 mm/h) for simulation a , 107.05 g (0.284 mm/h ) for simulation b and 115.53 g (0.310 mm/h) for simulation c. According to the above calculated results of simulations a and b, it can be easily understood that the whole temperature distribution and the largest temperature difference in the powder play a very important role in the powder sublimation and the crystal growth rate, when the pressure of argon is invariable. On the one hand, the large temperature difference in the powder is helpful to enhance the efficiency of sublimation and the growth rate. On the other hand, the temperature distribution in the powder like simulation a induces recrystallization at the bottom of powder and reduces the available powder that should have been transported to the surface of the growing crystal. The calculated results of simulations a and b are in good agreement with experiments A and B. However, the calculated results of simulation c and experiment C seem to be contradictory. According to Ref. [5], the contribution of powder sublimation into the crystal growth rate increases with the temperature difference in the powder. The maximum temperature difference in the powder is 150 1C in simulation c and its crystal growth rate is supposed to be the maximum. As shown in Fig. 5, the crystal growth rate in simulation c is maximum in the fifth hour and the total mass
Fig. 4. The steady temperature distribution (left) and final graphitization degree distribution (right) of the source powder in simulation a (a); the steady temperature distribution (left) and final graphitization degree distribution (right) of the source powder in simulation b (b); and the steady temperature distribution (left) and final graphitization degree distribution (right) of the source powder in simulation c (c).
Table 2 The calculated temperature distribution and the average growth rate for the three simulations. Simulations
Tmax (1C)
Tmin (1C)
DT (1C)
PAr (Torr)
M (g)
v¯ (mm/h)
a b c
2338 2363 2397
2243 2245 2247
95 118 150
20 20 20
97.87 107.05 115.53
0.257 0.284 0.310
Tmax/Tmin is the maximum/minimum temperature of the powder; DT= Tmax-Tmin; PAr is the pressure of Ar; M is the total sublimated mass of the powder; and v¯ is the average growth rate.
transported from the powder in simulation c is also maximum. Nevertheless, it is proved that both the sublimation of the powder and the growth rate are unstable by the crystal length and the residual powder (Fig. 2). Another hidden factor that affects the
ARTICLE IN PRESS X. Liu et al. / Journal of Crystal Growth 312 (2010) 1486–1490
1489
Table 3 The calculated v and v0 for the three simulations. Simulations
tpeak (h)
Tp (1C)
Ts (1C)
PAr (Torr)
v (mm/h)
v0 (mm/h)
a b c
6 6 5
2329 2352 2373
2192 2192 2188
20 20 20
0.424 0.572 0.809
0.480 0.609 0.767
tpeak is the peak time; Tp is the maximum temperature in the powder; Ts is the temperature at the center of the seed surface; v is the maximum growth rate calculated by the sublimation of the powder; and v0 is the growth rate controlled by the mass diffusion in the gap between the powder and the crystal at the peak time.
powder surface, D is the diffusion coefficient given by Kaldis and Piechotka [11] and R is the gas constant. When the pressure of decomposed SiC vapor at the powder surface is greater than the equilibrium pressure, the powder will stop decomposing. When the pressure of decomposed SiC vapor at the seed is less than the equilibrium pressure, the seed growth will be interrupted. Therefore, the upper normal limit of the mass transport rate can be obtained from Eq. (1) and can be expressed as follows: J¼
Fig. 5. The total sublimated powder mass during the whole growth in the three simulations (a); and the calculated growth rate based on the model of the sublimation of the powder in the three simulations (b).
stability of the sublimation and growth rate is possibly neglected by the modeling of SiC powder sublimation. During the initial stage of crystal growth, the powder close to the hot wall of the crucible decomposes and then moves upwards to the crystal seed along the wall of the crucible. With the evolution of the temperature field in the powder, the whole powder begins to react. Most of the decomposed SiC vapor flows through the center of the powder by the repetitive process of sublimation and recrystallization and then moves upwards to the growing crystal [3]. If the argon pressure is much higher than that of other vapor species in the crucible as in the case of our work, the upwards movement of the SiC vapor is diffusion dominated and Steafan flow can be neglected [9,10]. The mass transport rate can be expressed by the following equations [10]: J¼
PSiC2 ðcÞPSiC2 ðsÞ LRðTc þ Ts =2Þ=D
PSiC ðcÞPSiC ðsÞ 2 2
LRðTc þ Ts =2Þ=D
ð2Þ
ðcÞ and PSiC ðsÞare the equilibrium pressures of the ratewhere PSiC 2 2 determining specie at the powder surface and the growth interface, respectively. Consequently, the maximum mass transport rate is the upper normal limit of the crystal growth rate. As shown in Fig. 5(b), it can be observed that the maximum growth rate appears at about the fifth hour when the powder close to the hot wall of the crucible decomposes. Both the growth rate at the peak time (v0 ) according to the growth model reported in Ref. [8] and the maximum growth rate (v) based on the modeling of SiC powder sublimation are computed and listed in Table 3. It can be found that v0 (0.480, 0.609 mm/h) is larger than v (0.424, 0.572 mm/h) in simulations a and b. Consequently, the rate of the powder sublimation is less than the mass transportation rate between the powder and the seed and the crystal growth proceeds smoothly. Therefore, the growth rates calculated in simulations a and b, which are shown in Fig. 5(b), can predict the experimental results. However, v0 (0.767 mm/h) is less than v (0.809 mm/h) in simulation c. On the one hand, the rate of powder sublimation is larger than the mass transportation rate between the powder and the seed and it makes the SiC vapor gradually accumulate above the disk and then the pressure of the SiC vapor above the disk becomes so large that the channels of mass transportation in the powder are obstructed seriously. One the other hand, the large temperature gradient in the powder will induce the growth of the crystal granule in the low temperature gradient zone [3], and then, the permeability of the powder decreases if the particle diameter of the SiC powder increases [7]. Low permeability of the powder not only decreases the sublimed powder mass but also increases the SiC vapor pressure in the low temperature gradient zone of the powder. Consequently, the SiC vapor under the disk becomes supersaturated and begins to deposit on the bottom of the disk spontaneously. In case c, extremely large temperature difference in the powder goes against the sublimation of the powder and even interprets the whole crystal growth like experiment C.
ð1Þ
where PSiC2 ðcÞ and PSiC2 ðsÞ are the pressures of the rate-determining specie at the powder surface and the growth interface, respectively, Tc and Ts are the temperatures at the powder surface and the seed, respectively, L is the distance between the seed and
4. Conclusion The temperature distribution of the powder plays an important role in the stability of powder sublimation and the crystal
ARTICLE IN PRESS 1490
X. Liu et al. / Journal of Crystal Growth 312 (2010) 1486–1490
growth rate. When the channels of mass transportation in the powder are unobstructed, the large temperature difference in the powder will be helpful in enhancing the efficiency of sublimation and the growth rate. The recrystallization at the bottom of the powder should be avoided. When the sublimation of the powder has an advantage over the SiC vapor diffusion in the gap between the powder and the growing crystal, the channels of mass transportation in the powder will be obstructed and it will even interrupt the crystal growth.
Acknowledgements This work was supported by ‘‘863’’ Project (2006AA03A146), Knowledge Innovation Program of the Chinese Academy of Sciences (KGCX2-YW-206) and Natural Science Foundation of Shanghai (06ZR14096) (Science and Technology Commission of Shanghai Municipality (09DZ1141400, 09520714900)).
References ¨ [1] P.J. Wellmann, Z. Herro, A. Winnacker, R. PUsche, M. Hundhausen, P. Masri, A. Kulik, M. Bogdanov, S. Karpov, M. Ramm, Y. Makarov, J. Crys. Growth 275 (2005) e1807. [2] P.J. Wellmann, D. Hofmann, L. Kadinski, M. Selder, T.L. Straubinger, A. Winnacker, J. Crys. Growth 225 (2001) 312. ¨ [3] P.J. Wellmann, Z. Herro, M. Selder, F. Durst, R. PUsche, M. Hundhausen, L. Ley, A. Winnacker, Mater. Sci. Forum 433–436 (2003) 9. [4] D.S. Karpov, O.V. Bord, S.Y. Karpov, A.I. Zhmakin, M.S. Ramm, Y.N. Makarov, Mater. Sci. Forum 353–356 (2001) 37. [5] A.V. Kulik, M.V. Bogdanov, S.Y. Karpov, M.S. Ramm, Y.N. Makarov, Mater. Sci. Forum 457–460 (2004) 67. [6] D. Cai, L.L. Zheng, H. Zhang, D. Zhuang, Z.G. Herro, R. Schlesser, Z. Sitar, J. Crys. Growth 306 (2007) 39. [7] X.L. Wang, D. Cai, H. Zhang, J. Crys. Growth 305 (2007) 122. [8] X. Liu, E.W. Shi, L.X. Song, Z.Z. Chen, J. Crys. Growth 310 (2008) 4314. [9] R.H. Ma, H. Zhang, V. Prasad, M. Dudley, Cryst. Growth Des. 2 (2002) 213. [10] R.H. Ma, H. Zhang, S. Ha, M. Skowrinski, J. Crys. Growth 252 (2003) 523. [11] E. Kaldis, E.M. Piechotka, Bulk crystal growth by physical vapor transport, in: D.I.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 2, Elsevier Science, Amsterdam, 1994, p. 615.