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A process model for silicon carbide growth by physical vapor transport
Journal of Crystal Growth 229 (2001) 510–515
A process model for silicon carbide growth by physical vapor transport V. Prasad*, Q.-S. Chen, H. Zhang Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2300, USA
Abstract A heat and mass transfer model is proposed for the physical vapor transport growth of SiC crystals with the kinetics represented by the Hertz–Knudsen equation. The growth rate is assumed to be proportional to the supersaturation of the rate-limiting agent. A numerical algorithm has been developed that consists of the calculations of radio-frequency, time-harmonic magnetic field by induction heating, radiation and conduction heat transfer in the system, as well as the growth rate. The generated heat power density in the graphite susceptor is obtained by solving the magnetic vector potential equations. Radiative heat transfer is obtained by solving the integrated equations for radiation. The theoretical growth rate compares well with the published experimental data. # 2001 Elsevier Science B.V. All rights reserved. PACS: 07.05.T; 52.75.R; 77.84.B; 81.10.A; 81.05.C Keywords: A1. Computer simulation; A1. Diffusion; A1. Growth models; A1. Mass transfer; A2. Growth from vapor; B2. Semiconducting silicon carbide
1. Introduction Silicon carbide (SiC) is a promising wide bandgap material that can function under hightemperature, high-frequency and intensive-radiation conditions [1]. The diameter of SiC single crystal grown by physical vapor transport has been recently increased from 50 to 100 mm [2]. The growth process includes sublimation of sintered polycrystalline silicon carbide charge in the bottom region of a graphite crucible, transport of the vaporized species}Si, SiC, SiC2 and Si2C, and condensation and deposition of SiC on a pre-
oriented seed mounted at the bottom of the lid of the crucible. Process modeling is an important way to help industry to design and develop large diameter crystal growth systems. In spite of many publications concerning the SiC bulk crystal growth modeling, the growth mechanism at elevated inert gas pressures is still not clear. Some studies indicated that the growth could be performed at high pressures, while the majority of experimental works report only on the low pressure growth.
2. Mathematical model *Corresponding author. Tel.: +1-631-632-8350; fax: +1631-632-8205. E-mail address: [email protected] (V. Prasad).
The present numerical algorithm consists of calculation of the radio-frequency (RF) time-
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 2 1 9 - 2
V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
harmonic magnetic field by induction heating, radiation and conduction heat transfer in the system and temperature distribution, as well as the growth kinetics [3,4]. Electromagnetic field is calculated to predict the generated heat power in the graphite heater using RF induction heating. For low frequency ( f 51 MHz), the Maxwell equations can be simplified using a quasi-steady approximation. It assumes that the current in the induction coil is time harmonic and the eddy currents in the graphite susceptor generate heat. The magnetic potential vector equation for an axisymmetric system is as follows: 2 q 1q 1 q2 A0 þ þ iosc A0 ¼ J0 ; 2 2 2 r qr r qr qz mm ð1Þ where A0 is the azimuthal component of the magnetic potential vector, mm is the magnetic permeability, o ¼ 2pf is the angular frequency of the AC current, sc is the electrical conductivity, and J0 is the current density in the azimuthal direction. The radiation–conduction equation is as follows: ðrcp Þe
qT _ sv þ DHvs M _ vs ¼ r ðkeff rTÞ DHsv M qt 1 þ sc o2 A0 A0* ðqradi þ qinsu ÞdA=dV; 2 ð2Þ
where DHsv and DHvs are the latent heat absorbed during sublimation and released during the growth _ sv and M _ vs are phase process, respectively. M change rates from solid to vapor and from vapor to solid, respectively. Due to complexities of the growth system geometry, the heat flux on the radiation enclosure, qradi , and on the outer wall of the furnace, qinsu , are set as source terms in the energy equation. The convective and radiative heat transfer is considered on the outer wall 4 of the furnace, qinsu ¼ esðT 4 Tamb Þ. The model accounts for the radiative heat transfer from the inner surfaces of the growth domain and the holes. The radiation enclosure is broken into a number of rings, the heat flux on each ring is expressed as, ðqradi Þj ¼ ðA1 BÞjk ðsTk4 Þ, where ek is the emissivity of the material of
511
ring k, Ajk ¼ djk =ek Fj; k ð1 ek Þ=ek ; Bjk ¼ djk Fj; k ; Fj; k is the view factor from ring j to ring k, and dj; k is the Kronecker delta. dA and dV are the area over a finite-volume face and finite volume near gas/solid interface. According to Lilov’s thermodynamic analysis [5], during the sublimation process, the SiC vapor pressure is much lower than that of Si and SiC2. Hence, only the reaction of vapor species Si and SiC2 to form SiC crystal is considered here. The vapor becomes silicon rich at temperatures lower than 2546 K, and becomes SiC2 rich at temperatures above 2900 K. The growth rate of SiC crystal can be expressed as a function of the molecular transport rate of SiC to the crystal [6]: MSiC GSiC ¼ JSiC ; ð3Þ rSiC where the molecular transport rate of SiC is related to the supersaturation of Si and SiC2, JSiC ¼ 2wA ½pA ðLÞ pA* ðLÞ . The subscript A denotes the rate-determining species, and is chosen as SiC2 at temperatures below 2546 K and as Si at temperatures above 2900 K. The subscript A can be either Si or SiC2 at temperatures between 2546 and 2900 K. A coordinate z is introduced and is set as 0 at the charge surface and L at the seed surface. pA* is the equilibrium vapor pressure of the species A, and pA is the actual vapor p pressure of the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi species A. The coefficient wA ¼ 1= 2pMA RT . The multiple 2 in the expression of JSiC is introduced because one SiC2 molecule and one Si molecule form two SiC molecules. The distribution of the vapor pressure in the grow chamber can be obtained by solving a onedimensional mass transfer equation for Stefan flow and is expressed as [6] pSiC2 ðzÞ þ pSi ðzÞ ¼ p ½p pSiC2 ð0Þ pSi ð0Þ
expðPe z=LÞ;
ð4Þ
where Peclet number, Pe ¼ UL=D, U is the advective velocity, and p is the inert gas pressure, or the total vapor pressure, which includes vapor pressures of species Si, SiC, SiC2 and Si2C inside the growth chamber. The diffusion coefficient is taken as D ¼ D0 ðT=T0 Þn ðp0 =pÞ [6], with 5 106 5D0 52 104 m2 =s; n ¼ 1:8; T0 ¼ 273 K, and p0 ¼ 1 atm. The advective velocity can be
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V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
expressed as a function of transport rate of SiC2 and Si, U ¼ JSiC RT=p [6]. The vapor pressures of SiC species can be found in Ref. [5].
3. Results and discussions The growth rate of SiC crystal depends on both the mass transfer by Stefan flow and the growth kinetics. The growth rate dependence on the pressure is shown in Fig. 1 for a temperature gradient of 20 K/cm. The gap between charge and seed, L, is chosen as 50 mm. As can be seen from Fig. 1, the predicted growth rates compare well with the experimental data for different growth temperatures. The experimental data were obtained at growth temperatures of 2073, 2273 and 2398 K [7,8]. The axial and radial temperature gradients are very important for the growth and enlargement of crystals. The growth rate is related
not only to the growth temperature but also to the axial temperature gradient. It is experimentally observed that the growth rate increases with the axial temperature gradient at fixed growth temperature and inert gas pressure [9]. The axial temperature gradient is related to the diameter of the growth chamber and the thickness of the susceptor. The growth curves shift towards the high temperature region when the axial temperature gradient becomes smaller, which occurs during the scale up of the growth system. The low temperature growth is associated with smallscale systems with high axial temperature gradients. A typical case with five turns of RF coil, current of 1200 A, and frequency of 10 kHz has been considered. The magnetic potential contours, ðA0 Þreal , and temperature distribution are shown in Figs. 2a and b, respectively. The inert gas pressure is set as 26,666 Pa (200 Torr). The radius
Fig. 1. Growth rate dependence on the pressure for a temperature gradient of 20 K/cm and a gap of 5 cm between seed and source. The experimental data are taken from Refs. [7,8].
V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
513
Fig. 2. (a) Magnetic potential contours, ðA0 Þreal , and (b) temperature distribution for a system with five turns of coil, a current of 1200 A, a frequency of 10 kHz, and an inert gas pressure of 26,666 Pa.
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V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
of the graphite susceptor Rs is taken as 70 mm. The system has a small axial temperature gradient near the seed, about 4 K/cm (Fig. 1). The experimental data by Balkas et al. [9] indicate that the growth rate is a linear function of DT and exponential function of the inverse temperature and has the following expression: Gr ¼ Aðp; S; tÞDT expðU=RTÞ;
ð5Þ
where R is the gas constant, DT ¼ Tcharge Tseed is the temperature difference between the seed and source, U is the activation energy, Aðp; S; tÞ is a constant defined by the inert gas pressure p, the effective source surface area S and the duration of the growth run t. The dependence of the growth rate on the temperature of the growth and the temperature on the bottom of the crucible is also shown in Fig. 3 for the system in Fig. 2. The coil positions for the three points (I ¼ 1000, 1100 and 1200 A) in Fig. 3 are the same, and the temperature differences between the charge and the seed, Tcharge Tseed , are about 66, 72, 81 K, respectively. The current prediction is based on the assumption that there is no leak in the crucible. The PVT
Fig. 3. Theoretical prediction of growth rate versus the temperature on the seed and the temperature on the bottom of the crucible at inert gas pressure of 26,666 Pa. The three points from left to right corresponds to I ¼ 1200, 1100, 1000 A, respectively. Experimental data are from Ref. [9].
process depends on the temperature of the growth as well as on the temperature at the source, which together determines the pressure drop between the source and the crystal surfaces, actual supersaturation and resulting growth rate. The pressure drop balances the mass transfer resistance from the source to the crystal interface and the resistance of the Knudsen layer on the crystal interface.
4. Conclusions A global model has been developed to obtain the magnetic and thermal fields by solving the magnetic potential vector equation and the energy equation, respectively. The growth rate is assumed to be proportional to the supersaturation of the rate-determining species and can be obtained by solving both the Hertz–Knudsen equation and mass transfer equation. The kinetics model represented by the Hertz–Knudsen equation and supersaturation theory works well for high supersaturation at the growth interface or where the growth is mainly controlled by the mass transfer resistance from the source to the crystal interface and the kinetics of incorporation into the solid phase do not play a significant role in the resulting growth rates. The magnetic potential field, temperature field and growth rate are obtained for different currents. The predicted growth rate compares well with the experimental data. The growth mechanism at elevated inert gas pressures is explained. Desirable growth temperature and inert gas pressure are obtained to achieve certain growth rate profile across the seed surface. The enlargement of the crystal can be realized by designing a system with a large radial temperature gradient. The growth system can thus be optimized by arranging its components, such as, induction coil, graphite susceptor and hotzone.
Acknowledgement This work was supported by a SBIR project, contract #N00014-98-C-0176 (BMDO/
V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
IST-managed by ONR), and by an NSF award, CTS-9876198 for the third author. [5] [6]
References [1] Yu.M. Tairov, V.F. Tsvetkov, J. Crystal Growth 43 (1978) 209. [2] D. Hobgood, M. Brady, W. Brixius, G. Fechko, R. Glass, . D. Henshall, J. Jenny, R. Leonard, D. Malta, St.G. Muller, V. Tsvetkov, C. Carter, Jr, Mater. Sci. Forum 338–342 (2000) 3. [3] H. Zhang, V. Prasad, M.K. Moallemi, Num. Heat Transfer B 29 (1996) 399. [4] Q.-S. Chen, H. Zhang, V. Prasad, C.M. Balkas, N.K. Yushin, Modelling of heat transfer and kinetics of physical
[7]
[8]
[9]
515
vapor transport growth of silicon carbide crystals, 2001, J. Heat Transfer, in press. S.K. Lilov, Mater. Sci. Eng. B 21 (1993) 65. E. Kaldis, M. Piechotka, Bulk Crystal Growth by Physical Vapour Transport, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 2, Elsevier Science, Amsterdam, 1994, pp. 615–658. G. Augustine, H.McD. Hobgood, V. Balakrishna, G. Dunne, R.H. Hopkins, Phys. Stat. Sol. B 202 (1997) 137. A.S. Segal, A.N. Vorob’ev, S.Yu. Karpov, Yu.N. Makarov, E.N. Mokhov, M.G. Ramm, M.S. Ramm, A.D. Roenkov, Yu.A. Vodakov, A.I. Zhmakin, Mater. Sci. Eng. B 61–62 (1999) 40. C.M. Balkas, A.A. Maltsev, M.D. Roth, N.K. Yushin, Mater. Sci. Forum 338–342 (2000) 79.
A process model for silicon carbide growth by physical vapor transport V. Prasad*, Q.-S. Chen, H. Zhang Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2300, USA
Abstract A heat and mass transfer model is proposed for the physical vapor transport growth of SiC crystals with the kinetics represented by the Hertz–Knudsen equation. The growth rate is assumed to be proportional to the supersaturation of the rate-limiting agent. A numerical algorithm has been developed that consists of the calculations of radio-frequency, time-harmonic magnetic field by induction heating, radiation and conduction heat transfer in the system, as well as the growth rate. The generated heat power density in the graphite susceptor is obtained by solving the magnetic vector potential equations. Radiative heat transfer is obtained by solving the integrated equations for radiation. The theoretical growth rate compares well with the published experimental data. # 2001 Elsevier Science B.V. All rights reserved. PACS: 07.05.T; 52.75.R; 77.84.B; 81.10.A; 81.05.C Keywords: A1. Computer simulation; A1. Diffusion; A1. Growth models; A1. Mass transfer; A2. Growth from vapor; B2. Semiconducting silicon carbide
1. Introduction Silicon carbide (SiC) is a promising wide bandgap material that can function under hightemperature, high-frequency and intensive-radiation conditions [1]. The diameter of SiC single crystal grown by physical vapor transport has been recently increased from 50 to 100 mm [2]. The growth process includes sublimation of sintered polycrystalline silicon carbide charge in the bottom region of a graphite crucible, transport of the vaporized species}Si, SiC, SiC2 and Si2C, and condensation and deposition of SiC on a pre-
oriented seed mounted at the bottom of the lid of the crucible. Process modeling is an important way to help industry to design and develop large diameter crystal growth systems. In spite of many publications concerning the SiC bulk crystal growth modeling, the growth mechanism at elevated inert gas pressures is still not clear. Some studies indicated that the growth could be performed at high pressures, while the majority of experimental works report only on the low pressure growth.
2. Mathematical model *Corresponding author. Tel.: +1-631-632-8350; fax: +1631-632-8205. E-mail address: [email protected] (V. Prasad).
The present numerical algorithm consists of calculation of the radio-frequency (RF) time-
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 2 1 9 - 2
V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
harmonic magnetic field by induction heating, radiation and conduction heat transfer in the system and temperature distribution, as well as the growth kinetics [3,4]. Electromagnetic field is calculated to predict the generated heat power in the graphite heater using RF induction heating. For low frequency ( f 51 MHz), the Maxwell equations can be simplified using a quasi-steady approximation. It assumes that the current in the induction coil is time harmonic and the eddy currents in the graphite susceptor generate heat. The magnetic potential vector equation for an axisymmetric system is as follows: 2 q 1q 1 q2 A0 þ þ iosc A0 ¼ J0 ; 2 2 2 r qr r qr qz mm ð1Þ where A0 is the azimuthal component of the magnetic potential vector, mm is the magnetic permeability, o ¼ 2pf is the angular frequency of the AC current, sc is the electrical conductivity, and J0 is the current density in the azimuthal direction. The radiation–conduction equation is as follows: ðrcp Þe
qT _ sv þ DHvs M _ vs ¼ r ðkeff rTÞ DHsv M qt 1 þ sc o2 A0 A0* ðqradi þ qinsu ÞdA=dV; 2 ð2Þ
where DHsv and DHvs are the latent heat absorbed during sublimation and released during the growth _ sv and M _ vs are phase process, respectively. M change rates from solid to vapor and from vapor to solid, respectively. Due to complexities of the growth system geometry, the heat flux on the radiation enclosure, qradi , and on the outer wall of the furnace, qinsu , are set as source terms in the energy equation. The convective and radiative heat transfer is considered on the outer wall 4 of the furnace, qinsu ¼ esðT 4 Tamb Þ. The model accounts for the radiative heat transfer from the inner surfaces of the growth domain and the holes. The radiation enclosure is broken into a number of rings, the heat flux on each ring is expressed as, ðqradi Þj ¼ ðA1 BÞjk ðsTk4 Þ, where ek is the emissivity of the material of
511
ring k, Ajk ¼ djk =ek Fj; k ð1 ek Þ=ek ; Bjk ¼ djk Fj; k ; Fj; k is the view factor from ring j to ring k, and dj; k is the Kronecker delta. dA and dV are the area over a finite-volume face and finite volume near gas/solid interface. According to Lilov’s thermodynamic analysis [5], during the sublimation process, the SiC vapor pressure is much lower than that of Si and SiC2. Hence, only the reaction of vapor species Si and SiC2 to form SiC crystal is considered here. The vapor becomes silicon rich at temperatures lower than 2546 K, and becomes SiC2 rich at temperatures above 2900 K. The growth rate of SiC crystal can be expressed as a function of the molecular transport rate of SiC to the crystal [6]: MSiC GSiC ¼ JSiC ; ð3Þ rSiC where the molecular transport rate of SiC is related to the supersaturation of Si and SiC2, JSiC ¼ 2wA ½pA ðLÞ pA* ðLÞ . The subscript A denotes the rate-determining species, and is chosen as SiC2 at temperatures below 2546 K and as Si at temperatures above 2900 K. The subscript A can be either Si or SiC2 at temperatures between 2546 and 2900 K. A coordinate z is introduced and is set as 0 at the charge surface and L at the seed surface. pA* is the equilibrium vapor pressure of the species A, and pA is the actual vapor p pressure of the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi species A. The coefficient wA ¼ 1= 2pMA RT . The multiple 2 in the expression of JSiC is introduced because one SiC2 molecule and one Si molecule form two SiC molecules. The distribution of the vapor pressure in the grow chamber can be obtained by solving a onedimensional mass transfer equation for Stefan flow and is expressed as [6] pSiC2 ðzÞ þ pSi ðzÞ ¼ p ½p pSiC2 ð0Þ pSi ð0Þ
expðPe z=LÞ;
ð4Þ
where Peclet number, Pe ¼ UL=D, U is the advective velocity, and p is the inert gas pressure, or the total vapor pressure, which includes vapor pressures of species Si, SiC, SiC2 and Si2C inside the growth chamber. The diffusion coefficient is taken as D ¼ D0 ðT=T0 Þn ðp0 =pÞ [6], with 5 106 5D0 52 104 m2 =s; n ¼ 1:8; T0 ¼ 273 K, and p0 ¼ 1 atm. The advective velocity can be
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V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
expressed as a function of transport rate of SiC2 and Si, U ¼ JSiC RT=p [6]. The vapor pressures of SiC species can be found in Ref. [5].
3. Results and discussions The growth rate of SiC crystal depends on both the mass transfer by Stefan flow and the growth kinetics. The growth rate dependence on the pressure is shown in Fig. 1 for a temperature gradient of 20 K/cm. The gap between charge and seed, L, is chosen as 50 mm. As can be seen from Fig. 1, the predicted growth rates compare well with the experimental data for different growth temperatures. The experimental data were obtained at growth temperatures of 2073, 2273 and 2398 K [7,8]. The axial and radial temperature gradients are very important for the growth and enlargement of crystals. The growth rate is related
not only to the growth temperature but also to the axial temperature gradient. It is experimentally observed that the growth rate increases with the axial temperature gradient at fixed growth temperature and inert gas pressure [9]. The axial temperature gradient is related to the diameter of the growth chamber and the thickness of the susceptor. The growth curves shift towards the high temperature region when the axial temperature gradient becomes smaller, which occurs during the scale up of the growth system. The low temperature growth is associated with smallscale systems with high axial temperature gradients. A typical case with five turns of RF coil, current of 1200 A, and frequency of 10 kHz has been considered. The magnetic potential contours, ðA0 Þreal , and temperature distribution are shown in Figs. 2a and b, respectively. The inert gas pressure is set as 26,666 Pa (200 Torr). The radius
Fig. 1. Growth rate dependence on the pressure for a temperature gradient of 20 K/cm and a gap of 5 cm between seed and source. The experimental data are taken from Refs. [7,8].
V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
513
Fig. 2. (a) Magnetic potential contours, ðA0 Þreal , and (b) temperature distribution for a system with five turns of coil, a current of 1200 A, a frequency of 10 kHz, and an inert gas pressure of 26,666 Pa.
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V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
of the graphite susceptor Rs is taken as 70 mm. The system has a small axial temperature gradient near the seed, about 4 K/cm (Fig. 1). The experimental data by Balkas et al. [9] indicate that the growth rate is a linear function of DT and exponential function of the inverse temperature and has the following expression: Gr ¼ Aðp; S; tÞDT expðU=RTÞ;
ð5Þ
where R is the gas constant, DT ¼ Tcharge Tseed is the temperature difference between the seed and source, U is the activation energy, Aðp; S; tÞ is a constant defined by the inert gas pressure p, the effective source surface area S and the duration of the growth run t. The dependence of the growth rate on the temperature of the growth and the temperature on the bottom of the crucible is also shown in Fig. 3 for the system in Fig. 2. The coil positions for the three points (I ¼ 1000, 1100 and 1200 A) in Fig. 3 are the same, and the temperature differences between the charge and the seed, Tcharge Tseed , are about 66, 72, 81 K, respectively. The current prediction is based on the assumption that there is no leak in the crucible. The PVT
Fig. 3. Theoretical prediction of growth rate versus the temperature on the seed and the temperature on the bottom of the crucible at inert gas pressure of 26,666 Pa. The three points from left to right corresponds to I ¼ 1200, 1100, 1000 A, respectively. Experimental data are from Ref. [9].
process depends on the temperature of the growth as well as on the temperature at the source, which together determines the pressure drop between the source and the crystal surfaces, actual supersaturation and resulting growth rate. The pressure drop balances the mass transfer resistance from the source to the crystal interface and the resistance of the Knudsen layer on the crystal interface.
4. Conclusions A global model has been developed to obtain the magnetic and thermal fields by solving the magnetic potential vector equation and the energy equation, respectively. The growth rate is assumed to be proportional to the supersaturation of the rate-determining species and can be obtained by solving both the Hertz–Knudsen equation and mass transfer equation. The kinetics model represented by the Hertz–Knudsen equation and supersaturation theory works well for high supersaturation at the growth interface or where the growth is mainly controlled by the mass transfer resistance from the source to the crystal interface and the kinetics of incorporation into the solid phase do not play a significant role in the resulting growth rates. The magnetic potential field, temperature field and growth rate are obtained for different currents. The predicted growth rate compares well with the experimental data. The growth mechanism at elevated inert gas pressures is explained. Desirable growth temperature and inert gas pressure are obtained to achieve certain growth rate profile across the seed surface. The enlargement of the crystal can be realized by designing a system with a large radial temperature gradient. The growth system can thus be optimized by arranging its components, such as, induction coil, graphite susceptor and hotzone.
Acknowledgement This work was supported by a SBIR project, contract #N00014-98-C-0176 (BMDO/
V. Prasad et al. / Journal of Crystal Growth 229 (2001) 510–515
IST-managed by ONR), and by an NSF award, CTS-9876198 for the third author. [5] [6]
References [1] Yu.M. Tairov, V.F. Tsvetkov, J. Crystal Growth 43 (1978) 209. [2] D. Hobgood, M. Brady, W. Brixius, G. Fechko, R. Glass, . D. Henshall, J. Jenny, R. Leonard, D. Malta, St.G. Muller, V. Tsvetkov, C. Carter, Jr, Mater. Sci. Forum 338–342 (2000) 3. [3] H. Zhang, V. Prasad, M.K. Moallemi, Num. Heat Transfer B 29 (1996) 399. [4] Q.-S. Chen, H. Zhang, V. Prasad, C.M. Balkas, N.K. Yushin, Modelling of heat transfer and kinetics of physical
[7]
[8]
[9]
515
vapor transport growth of silicon carbide crystals, 2001, J. Heat Transfer, in press. S.K. Lilov, Mater. Sci. Eng. B 21 (1993) 65. E. Kaldis, M. Piechotka, Bulk Crystal Growth by Physical Vapour Transport, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 2, Elsevier Science, Amsterdam, 1994, pp. 615–658. G. Augustine, H.McD. Hobgood, V. Balakrishna, G. Dunne, R.H. Hopkins, Phys. Stat. Sol. B 202 (1997) 137. A.S. Segal, A.N. Vorob’ev, S.Yu. Karpov, Yu.N. Makarov, E.N. Mokhov, M.G. Ramm, M.S. Ramm, A.D. Roenkov, Yu.A. Vodakov, A.I. Zhmakin, Mater. Sci. Eng. B 61–62 (1999) 40. C.M. Balkas, A.A. Maltsev, M.D. Roth, N.K. Yushin, Mater. Sci. Forum 338–342 (2000) 79.