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Simulation of global heat transfer in the Czochralski process for BGO sillenite crystals
ARTICLE IN PRESS
Journal of Crystal Growth 266 (2004) 103–108
Simulation of global heat transfer in the Czochralski process for BGO sillenite crystals O.N. Budenkovaa,*, M.G. Vasilieva, V.S. Yufereva, E.N. Bystrovab, V.V. Kalaevb, c ! V. Bermudez , E. Die! guezc, Yu.N. Makarovd a
A.F.Ioffe Physico-Technical Institute, Russian Academy of Sciences, 26 Polytechnicheskaya, 194021 St. Petersburg, Russia b Softimpact Ltd, 29 Engels Av., P.O. Box 29, 194156 St. Petersburg, Russia c ! Departamento de Fisica de Materiales, Universidad Autonoma de Madrid, 28049 Madrid, Spain d STR Inc., P.O. Box 70604, Richmond, VA 23255, USA
Abstract The influence of radiative heat transfer on the temperature fields in the crystal and the shape of the melt/liquid interface in Czochralski sillenite BGO growth is considered. It is shown that specular reflection at the shoulder side surface results in a large convexity of the interface and the appearance of a thin cool region near the crystal axis. The latter phenomenon is explained by poor conductivity and rather appreciable absorption coefficient of sillenite crystals and may be responsible for the formation of a dark core in the crystal center. r 2004 Elsevier B.V. All rights reserved. PACS: 61.50.Ah; 68.45.v; 81.10.Aj; 81.10.Fq; 44.40.+a Keywords: A1. Computer simulation; A1. Heat transfer; A1. Interfaces; A1. Radiation; A2. Czochralski method; B1. Oxides
1. Introduction Sillenite BGO and BSO crystals are widely used for various applications in optical devices. One of the main problems related with the crystal quality of these compounds is the presence of a central core area, which is column shaped and darker in color compared with the surrounding material in the crystal boule. The analysis of the grown crystals shows that the core is a more strained area compared to the core-free region [1]. This suggests the existence of certain peculiarities in the *Corresponding author. Tel.: +7-812-247-91-75; fax: +7812-247-1017. E-mail address: [email protected] (O.N. Budenkova).
temperature distribution in the core area during the growth, which, however, have not been revealed in simulations so far. Specific features of sillenite crystals are sufficient transparency in the infrared range, very high refraction index and practical opacity of their melts. It means that internal radiation has to play a decisive role in the heat removal from the solid/liquid interface and, consequently, influences on the shape of the solid/ melt interface. Besides, according to Ref. [2] the absorption coefficient of Bi12GeO20 crystals is not too small and reaches 0.3–0.4 cm1 in the range of wavelengths from 2 to 6 mm. Therefore, one can expect that absorption and emanation of radiation by a crystal have to appreciably influence the heat transfer process. Simulation of heat transfer in
0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2004.02.035
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growing sillenite crystals by the Cz method was performed in Refs. [3–7]. However, internal radiation was treated either using simplified model when crystalline transparency was assumed [3], or neglected at all [4–7], while the interface was believed to be flat. Effect of internal radiation on the shape of the crystallization front in Czochralski oxides growth was considered repeatedly (see, for example, Refs. [8–10] and references within). Yet, in all the cited papers the side surface of a crystal was assumed either opaque or transparent and diffusely reflecting. At the same time, the side surface of oxide crystals grown by the Cz method is more likely specular than diffuse. In growing sillenite crystals the effect of specular reflection has to be especially significant owing to large value of refraction index (n > 2:3) and small angle of the total inner reflection (of the order of 25 ). Simulation of heat transfer with respect to specular reflection at the crystal side surface was performed for the first time in Ref. [11] using the new approach to the solution of the radiative heat transport equation [12,13]. The growth of Bi4Ge3O12 crystals was considered. It was shown that specular reflection greatly increases the deflection of the solid/liquid interface toward the melt and leads to the more significant distortions of isotherms that can even take the ‘‘convex–concave’’ shape. The present paper is focused on the global heat transfer simulation in growing the BGO sillenite crystals that are far less transparent in the infrared range than eulithine compounds. Besides, in comparison with Ref. [11] another version of Czochralski process is considered. As in Ref. [11] special attention is given to the accurate treatment of radiative heat transfer through the crystal and the influence of the specular reflection (refraction) at the crystal surface on the thermal fields inside the crystal and the shape of a solid/ melt interface.
2. Statement of the problem Schematic diagram of the experimental setup is presented in Fig. 1. The system employs resistive heating to melt the charge. The platinum crucible is immovable while the crystal is rotated. Pulling
Fig. 1. Schematic diagram of the system and temperature distribution in the modeling domain at the first stage of simulation (A) and temperature distribution along the crucible wall, afterheater and heater (B). Solid lines in (B) are calculation results, while squares are the measured values in the reference points.
rates and crystal rotation are 0.5–5 mm/h and 5– 60 rpm and for normal conditions these values are equal to 3 mm/h and 30 rpm, respectively. The problem was considered in axisymmetric geometry and quasi-steady-state behavior was assumed. Because global modeling of heat exchange inside the growth setup with respect to volume radiation transfer through the crystal is a formidable task at present, simulation was divided into two stages. At the first one all processes of heat transfer, including convection in the melt, conduction in the gas phase and solid blocks of a furnace and radiative exchange between melt and solid blocks were treated accurately with the exception of radiative transfer inside the crystal. The latter in this stage of simulation was assumed either totally transparent with non-reflecting surface or, on the contrary, absolutely opaque with the averaged emissivity equal to 0.8. For calculation of convection in a melt the common model based on Navier–Stokes equations with the Boussinesq approximation was exploited, while Marangoni convection was not taken into account. Heat generation over the heater was assumed uniform and was fitted in such a way that the calculated temperature at the reference points was as close to that measured by thermocouples as possible. For realization of this stage of simulation, as well as for the further calculation of the convection in the melt, the CGSim package [14] was used. As a result, temperature distributions over the crucible
ARTICLE IN PRESS O.N. Budenkova et al. / Journal of Crystal Growth 266 (2004) 103–108
wall, afterheater and heater were obtained and used further as boundary conditions at the second stage of simulation. The latter consisted of solution of the two subproblems: convection and heat transfer in the melt and heat transfer in crystal and gap between crystal, crucible, heater and afterheater. Both subproblems were matched along the melt-free surface and the crystallization front. The shape of the latter was calculated in a way similar to that given in Ref. [11]. The key point of radiative problem is a solution of radiant transport equation with respect to reflection and refraction at the crystal surface. As it was mentioned above, we used approach [12,13], which allows one to calculate radiative heat transport in domains of complex shape with transparent both diffuse and specular boundaries in a unified manner independently of the values of absorption coefficient and refraction index. One of the specific features of the growth setup shown in Fig. 1 is that there is a gap between crucible and afterheater through which the direct exchange of heat radiation between the crystal and the heater takes place. To simulate this feature we used two approaches. In the first approximate one we replaced the gap between crucible and afterheater by a black (nonreflective) boundary with the appropriate temperature distribution, taken from the calculation at the first stage of simulation, which practically coincides with the temperature of gas. So, in this case the calculation domain did not include the heater at all (Fig. 2A). In the second approach, the calculation domain included additionally a part of the heater (Fig. 2B) and radiation passing through the gap was calculated directly in the course of solution of radiant transport problem. The temperature of the black boundary involved in domain B also was equal to that of gas. Further, we will denote these domains as A and B, respectively. Radiative properties of materials and boundaries appearing in the radiative problem are given in Table 1. The melt was opaque, the crystallization front was black with the melting temperature Tm ¼ 1203 K. Spectral absorptivity of the crystal was calculated from the infrared transmission spectra given in Ref. [2] and was approximated by the three-band model. As a result, absorption coefficient was taken
105
Fig. 2. Domains used in calculation of radiative heat transfer in crystal and gap between the crystal, crucible, afterheater and heater: (A) The gap between crucible and afterheater is replaced by a black (radiant, non-reflective) surface and (B) the gap is transparent for radiation exchange between the crystal and the heater.
Table 1 Radiative properties used in simulation Radiative properties
Values
Crucible emissivity Heater emissivity Afterheater emissivity Emissivity of the free melt surface Lid emissivity Seedholder emissivity
0.15 0.8 0.8 0.75 0.8 0.15
equal to 0.482 cm1 for wavelengths lo6:89 mm, 5.9834 cm1 for the range 6:89olo9:16 mm and infinity beyond 9.16 mm. Corresponding Planck’s weight functions taking into account the contributions of the different bands were equal to 0.8672, 0.0649 and 0.0679, respectively. Other thermophysical properties used in simulation are given in Table 2.
3. Numerical results and discussion As it was mentioned above, at the first stage of simulation the temperature distributions along the heater, crucible, afterheater and upper lid were found. It is to be noted that the temperature distributions obtained during the first stage of global simulation for totally transparent and
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Table 2 Thermophysical properties used in simulation Thermophysical properties
Values
Melt Density (g/cm3) Expansion coefficient (1/K) Viscosity (cP) Thermal conductivity (W/(cm K)) Specific heat (J/(g K))
8.13 12 105 17 3.45 103 0.39
Crystal Thermal conductivity (W/(cm K)) Fusion latent heat (J/cm3)
1.8 103 430
Growth setup Thermal conductivity of the crucible (W/ (cm K)) Thermal conductivity of the seedholder (W/ (cm K)) Thermal conductivity of the afterheater (W/ (cm K))
1.5 2 103 2 103
opaque crystal were not differed appreciably, only by several degrees and results of calculation for opaque crystal are presented in Fig. 1B. It is seen that the temperature distribution over the crucible wall and afterheater satisfactorily coincides with the measured values of temperature at the reference points that demonstrate a validity of the developed model of global heat transfer. The small gas gap between the seedholder and the upper lid was treated in further simulation as a black boundary with temperature equal to that of the gas phase. At the second stage convection and heat transfer in the melt, radiative transport and conduction in the crystal and the shape of solid/ melt interface were calculated. Simulation showed that convection is forced and rotationally driven vortex occupies the whole crucible (Fig. 3). This is explained by the high rotation rate of the crystal, rather modest temperature gradients along the crucible wall and relatively small height of the melt. Fig. 3 also shows that in the case of diffusely reflecting crystal surface the temperature fields inside the crystal are practically identical for both domains A and B. It is seen that the crystallization front is slightly curved and convex toward the melt (per 0.5 mm), while temperature decreases mono-
Fig. 3. (A) Temperature distribution in solid and liquid phase and flow pattern in the melt for the diffuse crystal surface and calculation domain B; contour spacing for temperature is equal to 1 K. (B) Comparison of the temperature distributions in the crystal with diffuse surface for the calculation domains A (left) and B (right); contour spacing for temperature is equal to 0.5 K.
tonically from the crystallization front to the shoulder without appreciable variations in radial direction. This picture is similar to that obtained in Ref. [11] for BGO eulithine crystal and, probably, is typical for all cases when crystal side surface is diffusely reflecting and convection is controlled by a crystal rotation. However, the results change drastically if the crystal surface is assumed to be specularly reflecting. More exactly, in accordance with [11,15] only conical part of the crystal has to be considered as specular, while the cylindrical part may remain to be diffusely reflecting. It is explained by the fact that in practice the surface of actual crystals is deflected from an ideal (cylindrical or conical) surface. As it was shown in Ref. [15], even small long wavelength perturbations of the cylindrical part of crystal surface lead to appreciable scattering of the axial radiation heat flux and this phenomenon can be taken into account in the framework of diffuse reflection. At the same time, the shoulder surface has to be treated as specular one, even, if the similar perturbations occur. Results of calculations for specular reflection are shown in Fig. 4A. It is seen that, firstly, as in our previous work [11], specular reflection causes the
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Fig. 4. Temperature fields in the crystal in the case of specular reflection at the shoulder surface: (A) temperature isolines for calculation domains A (left) and B (right) and (B) radial temperature distributions for several values of z-coordinate. Solid symbols correspond to calculation performed for domain A and open symbols for domain B.
crystallization front to deflect deeper toward the melt (up to 7 mm) that is explained by increasing of non-uniformity in the radiation heat flux distribution at the solid/liquid interface. Secondly, in both cases the distortion of isotherms becomes much more significant not only as compared with the previous case of diffuse reflection, but in relation to Bi4Ge3O12 crystal with specular surface, which was considered in Ref. [11]. The maximum of the distortion of the temperature field takes place near the axis in the upper part of the crystal where a thin cool region appears. As a result, the temperature distribution in radial direction becomes non-monotonic (Fig. 4B). This effect is especially distinctly manifested in a strict consideration of the radiation exchange between the crystal and the heater (calculation domain B). In this case, the temperature field turns out nonmonotonic also along the crystal axis (Fig. 4B) and isotherms near the axis take the fusiform shape. This phenomenon is explained probably by a greater absorptivity and smaller conductivity of sillenite crystals as compared with eulithine one. Small thermal conductivity cannot smooth the irregularities of the temperature fields caused by the radiative heat removing. To verify this explanation we have performed model calculations with thermal conductivity raised as high as 0.018 W/(cm K) and with absorption coefficient
Fig. 5. Effect of absorption coefficient (a) in the first wavelength band and thermal conductivity (l) on the temperature field in crystal. (A) a ¼ 0:482 cm1, l ¼ 0:0018 W/cm K, as in Table 1; (B) a ¼ 0:03 cm1, l ¼ 0:0018 W/cm K; and (C) a ¼ 0:03 cm1, l ¼ 0:018 W/cm K.
in the first wavelength band reduced as low as 0.03 cm1. In doing so, the shape of solid/melt interface was fixed. Results of calculations presented in Fig. 5 confirmed the explanation given above and showed that in transparent and sufficiently thermal conductive crystals a cool region along the axis is not actually formed. Naturally, this is valid for opaque crystals too. Fig. 6 shows that increase of the crystal length does not lead to essential changes in the temperature distribution. Thus, this phenomenon is inherent to the present growth process and is directly related to the specular reflection at the shoulder surface because in the case of diffuse reflection the temperature field inside the crystal is monotone and the cool region is not formed. The temperature fields, which are obtained in the case of specular reflection, are surprisingly well correlate with the dark core observed in the center of grown crystals. The curvature of temperature field near the axis in Fig. 4B is rather significant and, in principle, can be responsible for the appearance in this area elevated thermal stresses revealed in Ref. [1].
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responsible for its formation. Note also that the replacement of the gap between crucible and afterheater by a radiant and non-reflecting boundary does not result in significant errors and can be used in simulation.
Acknowledgements This work was supported by INTAS grant, project number INTAS-00-263.
References
Fig. 6. Temperature field in the crystal of length 3 cm (to the shoulder) in the case of specular reflection at the shoulder conical surface. Contour spacing for temperature is equal to 1 K.
4. Conclusions The results presented here demonstrate that in the case of BGO sillenite crystals that are distinguished by a greater absorptivity and smaller conductivity in comparison with the BGO eulithine ones, specular reflection at the shoulder surface does not only increase the deflection of the crystallization front toward the melt, but drastically changes the temperature field in crystal and results in the appearance of a thin cool area near the crystal axis. The latter correlates with a dark core observed in grown crystals and may be
[1] M.T. Santos, L. Arizmendi, D. Bravo, E. Di!eguez, Mater. Res. Bull. 31 (1996) 389. [2] S. Fu, H. Ozoe, J. Mater. Sci. 34 (1999) 371. [3] C. Lin, S. Motakeff, J. Crystal Growth 128 (1993) 834. [4] J.P. Fontaine, G.P. Extr!emet, V. Chevrier, J.C. Launay, J. Crystal Growth 139 (1994) 67. [5] M.T. Santos, J.C. Rojo, L. Arizmendi, E. Di!eguez, J. Crystal Growth 142 (1994) 103. [6] M.T. Santos, J.C. Rojo, A. Cintas, L. Arizmendi, E. Di!eguez, J. Crystal Growth 156 (1995) 413. [7] J.C. Rojo, C. Mar!ın, J.J. Derby, E. Di!eguez, J. Crystal Growth 183 (1998) 604. [8] Q. Xiao, J.J. Derby, J. Crystal Growth 139 (1994) 147. [9] M. Kobayashi, T. Hagino, T. Tsukada, M. Hozawa, J. Crystal Growth 235 (2002) 258. [10] E.M. Nunes, M.H.N. Naraghi, H. Zhang, V. Prasad, J. Crystal Growth 236 (2002) 596. [11] V.S. Yuferev, O.N. Budenkova, M.G. Vasiliev, S.A. Rukolaine, V.N. Shlegel, Ya.V. Vasiliev, A.I. Zhmakin, J. Crystal Growth 253 (2003) 371. [12] S.A. Rukolaine, M.G. Vasiliev, V.S. Yuferev, A.O. Galyukov, J. Quant. Spectrosc. Radiat. Transfer 73 (2002) 205. [13] S.A. Rukolaine, M.G. Vasilyev, V.S. Yuferev, V. Mamedov, J. Quant. Spectrosc. Radiat. Transfer 84 (2004) 371. [14] V.V. Kalaev, I.Yu. Evstratov, Yu.N. Makarov, J. Crystal Growth 249 (1–2) (2003) 87. [15] O.N. Budenkova, V.M. Mamedov, M.G. Vasiliev, V.S. Yuferev, Yu.N. Makarov, J. Crystal Growth, in press.
Journal of Crystal Growth 266 (2004) 103–108
Simulation of global heat transfer in the Czochralski process for BGO sillenite crystals O.N. Budenkovaa,*, M.G. Vasilieva, V.S. Yufereva, E.N. Bystrovab, V.V. Kalaevb, c ! V. Bermudez , E. Die! guezc, Yu.N. Makarovd a
A.F.Ioffe Physico-Technical Institute, Russian Academy of Sciences, 26 Polytechnicheskaya, 194021 St. Petersburg, Russia b Softimpact Ltd, 29 Engels Av., P.O. Box 29, 194156 St. Petersburg, Russia c ! Departamento de Fisica de Materiales, Universidad Autonoma de Madrid, 28049 Madrid, Spain d STR Inc., P.O. Box 70604, Richmond, VA 23255, USA
Abstract The influence of radiative heat transfer on the temperature fields in the crystal and the shape of the melt/liquid interface in Czochralski sillenite BGO growth is considered. It is shown that specular reflection at the shoulder side surface results in a large convexity of the interface and the appearance of a thin cool region near the crystal axis. The latter phenomenon is explained by poor conductivity and rather appreciable absorption coefficient of sillenite crystals and may be responsible for the formation of a dark core in the crystal center. r 2004 Elsevier B.V. All rights reserved. PACS: 61.50.Ah; 68.45.v; 81.10.Aj; 81.10.Fq; 44.40.+a Keywords: A1. Computer simulation; A1. Heat transfer; A1. Interfaces; A1. Radiation; A2. Czochralski method; B1. Oxides
1. Introduction Sillenite BGO and BSO crystals are widely used for various applications in optical devices. One of the main problems related with the crystal quality of these compounds is the presence of a central core area, which is column shaped and darker in color compared with the surrounding material in the crystal boule. The analysis of the grown crystals shows that the core is a more strained area compared to the core-free region [1]. This suggests the existence of certain peculiarities in the *Corresponding author. Tel.: +7-812-247-91-75; fax: +7812-247-1017. E-mail address: [email protected] (O.N. Budenkova).
temperature distribution in the core area during the growth, which, however, have not been revealed in simulations so far. Specific features of sillenite crystals are sufficient transparency in the infrared range, very high refraction index and practical opacity of their melts. It means that internal radiation has to play a decisive role in the heat removal from the solid/liquid interface and, consequently, influences on the shape of the solid/ melt interface. Besides, according to Ref. [2] the absorption coefficient of Bi12GeO20 crystals is not too small and reaches 0.3–0.4 cm1 in the range of wavelengths from 2 to 6 mm. Therefore, one can expect that absorption and emanation of radiation by a crystal have to appreciably influence the heat transfer process. Simulation of heat transfer in
0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2004.02.035
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growing sillenite crystals by the Cz method was performed in Refs. [3–7]. However, internal radiation was treated either using simplified model when crystalline transparency was assumed [3], or neglected at all [4–7], while the interface was believed to be flat. Effect of internal radiation on the shape of the crystallization front in Czochralski oxides growth was considered repeatedly (see, for example, Refs. [8–10] and references within). Yet, in all the cited papers the side surface of a crystal was assumed either opaque or transparent and diffusely reflecting. At the same time, the side surface of oxide crystals grown by the Cz method is more likely specular than diffuse. In growing sillenite crystals the effect of specular reflection has to be especially significant owing to large value of refraction index (n > 2:3) and small angle of the total inner reflection (of the order of 25 ). Simulation of heat transfer with respect to specular reflection at the crystal side surface was performed for the first time in Ref. [11] using the new approach to the solution of the radiative heat transport equation [12,13]. The growth of Bi4Ge3O12 crystals was considered. It was shown that specular reflection greatly increases the deflection of the solid/liquid interface toward the melt and leads to the more significant distortions of isotherms that can even take the ‘‘convex–concave’’ shape. The present paper is focused on the global heat transfer simulation in growing the BGO sillenite crystals that are far less transparent in the infrared range than eulithine compounds. Besides, in comparison with Ref. [11] another version of Czochralski process is considered. As in Ref. [11] special attention is given to the accurate treatment of radiative heat transfer through the crystal and the influence of the specular reflection (refraction) at the crystal surface on the thermal fields inside the crystal and the shape of a solid/ melt interface.
2. Statement of the problem Schematic diagram of the experimental setup is presented in Fig. 1. The system employs resistive heating to melt the charge. The platinum crucible is immovable while the crystal is rotated. Pulling
Fig. 1. Schematic diagram of the system and temperature distribution in the modeling domain at the first stage of simulation (A) and temperature distribution along the crucible wall, afterheater and heater (B). Solid lines in (B) are calculation results, while squares are the measured values in the reference points.
rates and crystal rotation are 0.5–5 mm/h and 5– 60 rpm and for normal conditions these values are equal to 3 mm/h and 30 rpm, respectively. The problem was considered in axisymmetric geometry and quasi-steady-state behavior was assumed. Because global modeling of heat exchange inside the growth setup with respect to volume radiation transfer through the crystal is a formidable task at present, simulation was divided into two stages. At the first one all processes of heat transfer, including convection in the melt, conduction in the gas phase and solid blocks of a furnace and radiative exchange between melt and solid blocks were treated accurately with the exception of radiative transfer inside the crystal. The latter in this stage of simulation was assumed either totally transparent with non-reflecting surface or, on the contrary, absolutely opaque with the averaged emissivity equal to 0.8. For calculation of convection in a melt the common model based on Navier–Stokes equations with the Boussinesq approximation was exploited, while Marangoni convection was not taken into account. Heat generation over the heater was assumed uniform and was fitted in such a way that the calculated temperature at the reference points was as close to that measured by thermocouples as possible. For realization of this stage of simulation, as well as for the further calculation of the convection in the melt, the CGSim package [14] was used. As a result, temperature distributions over the crucible
ARTICLE IN PRESS O.N. Budenkova et al. / Journal of Crystal Growth 266 (2004) 103–108
wall, afterheater and heater were obtained and used further as boundary conditions at the second stage of simulation. The latter consisted of solution of the two subproblems: convection and heat transfer in the melt and heat transfer in crystal and gap between crystal, crucible, heater and afterheater. Both subproblems were matched along the melt-free surface and the crystallization front. The shape of the latter was calculated in a way similar to that given in Ref. [11]. The key point of radiative problem is a solution of radiant transport equation with respect to reflection and refraction at the crystal surface. As it was mentioned above, we used approach [12,13], which allows one to calculate radiative heat transport in domains of complex shape with transparent both diffuse and specular boundaries in a unified manner independently of the values of absorption coefficient and refraction index. One of the specific features of the growth setup shown in Fig. 1 is that there is a gap between crucible and afterheater through which the direct exchange of heat radiation between the crystal and the heater takes place. To simulate this feature we used two approaches. In the first approximate one we replaced the gap between crucible and afterheater by a black (nonreflective) boundary with the appropriate temperature distribution, taken from the calculation at the first stage of simulation, which practically coincides with the temperature of gas. So, in this case the calculation domain did not include the heater at all (Fig. 2A). In the second approach, the calculation domain included additionally a part of the heater (Fig. 2B) and radiation passing through the gap was calculated directly in the course of solution of radiant transport problem. The temperature of the black boundary involved in domain B also was equal to that of gas. Further, we will denote these domains as A and B, respectively. Radiative properties of materials and boundaries appearing in the radiative problem are given in Table 1. The melt was opaque, the crystallization front was black with the melting temperature Tm ¼ 1203 K. Spectral absorptivity of the crystal was calculated from the infrared transmission spectra given in Ref. [2] and was approximated by the three-band model. As a result, absorption coefficient was taken
105
Fig. 2. Domains used in calculation of radiative heat transfer in crystal and gap between the crystal, crucible, afterheater and heater: (A) The gap between crucible and afterheater is replaced by a black (radiant, non-reflective) surface and (B) the gap is transparent for radiation exchange between the crystal and the heater.
Table 1 Radiative properties used in simulation Radiative properties
Values
Crucible emissivity Heater emissivity Afterheater emissivity Emissivity of the free melt surface Lid emissivity Seedholder emissivity
0.15 0.8 0.8 0.75 0.8 0.15
equal to 0.482 cm1 for wavelengths lo6:89 mm, 5.9834 cm1 for the range 6:89olo9:16 mm and infinity beyond 9.16 mm. Corresponding Planck’s weight functions taking into account the contributions of the different bands were equal to 0.8672, 0.0649 and 0.0679, respectively. Other thermophysical properties used in simulation are given in Table 2.
3. Numerical results and discussion As it was mentioned above, at the first stage of simulation the temperature distributions along the heater, crucible, afterheater and upper lid were found. It is to be noted that the temperature distributions obtained during the first stage of global simulation for totally transparent and
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Table 2 Thermophysical properties used in simulation Thermophysical properties
Values
Melt Density (g/cm3) Expansion coefficient (1/K) Viscosity (cP) Thermal conductivity (W/(cm K)) Specific heat (J/(g K))
8.13 12 105 17 3.45 103 0.39
Crystal Thermal conductivity (W/(cm K)) Fusion latent heat (J/cm3)
1.8 103 430
Growth setup Thermal conductivity of the crucible (W/ (cm K)) Thermal conductivity of the seedholder (W/ (cm K)) Thermal conductivity of the afterheater (W/ (cm K))
1.5 2 103 2 103
opaque crystal were not differed appreciably, only by several degrees and results of calculation for opaque crystal are presented in Fig. 1B. It is seen that the temperature distribution over the crucible wall and afterheater satisfactorily coincides with the measured values of temperature at the reference points that demonstrate a validity of the developed model of global heat transfer. The small gas gap between the seedholder and the upper lid was treated in further simulation as a black boundary with temperature equal to that of the gas phase. At the second stage convection and heat transfer in the melt, radiative transport and conduction in the crystal and the shape of solid/ melt interface were calculated. Simulation showed that convection is forced and rotationally driven vortex occupies the whole crucible (Fig. 3). This is explained by the high rotation rate of the crystal, rather modest temperature gradients along the crucible wall and relatively small height of the melt. Fig. 3 also shows that in the case of diffusely reflecting crystal surface the temperature fields inside the crystal are practically identical for both domains A and B. It is seen that the crystallization front is slightly curved and convex toward the melt (per 0.5 mm), while temperature decreases mono-
Fig. 3. (A) Temperature distribution in solid and liquid phase and flow pattern in the melt for the diffuse crystal surface and calculation domain B; contour spacing for temperature is equal to 1 K. (B) Comparison of the temperature distributions in the crystal with diffuse surface for the calculation domains A (left) and B (right); contour spacing for temperature is equal to 0.5 K.
tonically from the crystallization front to the shoulder without appreciable variations in radial direction. This picture is similar to that obtained in Ref. [11] for BGO eulithine crystal and, probably, is typical for all cases when crystal side surface is diffusely reflecting and convection is controlled by a crystal rotation. However, the results change drastically if the crystal surface is assumed to be specularly reflecting. More exactly, in accordance with [11,15] only conical part of the crystal has to be considered as specular, while the cylindrical part may remain to be diffusely reflecting. It is explained by the fact that in practice the surface of actual crystals is deflected from an ideal (cylindrical or conical) surface. As it was shown in Ref. [15], even small long wavelength perturbations of the cylindrical part of crystal surface lead to appreciable scattering of the axial radiation heat flux and this phenomenon can be taken into account in the framework of diffuse reflection. At the same time, the shoulder surface has to be treated as specular one, even, if the similar perturbations occur. Results of calculations for specular reflection are shown in Fig. 4A. It is seen that, firstly, as in our previous work [11], specular reflection causes the
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Fig. 4. Temperature fields in the crystal in the case of specular reflection at the shoulder surface: (A) temperature isolines for calculation domains A (left) and B (right) and (B) radial temperature distributions for several values of z-coordinate. Solid symbols correspond to calculation performed for domain A and open symbols for domain B.
crystallization front to deflect deeper toward the melt (up to 7 mm) that is explained by increasing of non-uniformity in the radiation heat flux distribution at the solid/liquid interface. Secondly, in both cases the distortion of isotherms becomes much more significant not only as compared with the previous case of diffuse reflection, but in relation to Bi4Ge3O12 crystal with specular surface, which was considered in Ref. [11]. The maximum of the distortion of the temperature field takes place near the axis in the upper part of the crystal where a thin cool region appears. As a result, the temperature distribution in radial direction becomes non-monotonic (Fig. 4B). This effect is especially distinctly manifested in a strict consideration of the radiation exchange between the crystal and the heater (calculation domain B). In this case, the temperature field turns out nonmonotonic also along the crystal axis (Fig. 4B) and isotherms near the axis take the fusiform shape. This phenomenon is explained probably by a greater absorptivity and smaller conductivity of sillenite crystals as compared with eulithine one. Small thermal conductivity cannot smooth the irregularities of the temperature fields caused by the radiative heat removing. To verify this explanation we have performed model calculations with thermal conductivity raised as high as 0.018 W/(cm K) and with absorption coefficient
Fig. 5. Effect of absorption coefficient (a) in the first wavelength band and thermal conductivity (l) on the temperature field in crystal. (A) a ¼ 0:482 cm1, l ¼ 0:0018 W/cm K, as in Table 1; (B) a ¼ 0:03 cm1, l ¼ 0:0018 W/cm K; and (C) a ¼ 0:03 cm1, l ¼ 0:018 W/cm K.
in the first wavelength band reduced as low as 0.03 cm1. In doing so, the shape of solid/melt interface was fixed. Results of calculations presented in Fig. 5 confirmed the explanation given above and showed that in transparent and sufficiently thermal conductive crystals a cool region along the axis is not actually formed. Naturally, this is valid for opaque crystals too. Fig. 6 shows that increase of the crystal length does not lead to essential changes in the temperature distribution. Thus, this phenomenon is inherent to the present growth process and is directly related to the specular reflection at the shoulder surface because in the case of diffuse reflection the temperature field inside the crystal is monotone and the cool region is not formed. The temperature fields, which are obtained in the case of specular reflection, are surprisingly well correlate with the dark core observed in the center of grown crystals. The curvature of temperature field near the axis in Fig. 4B is rather significant and, in principle, can be responsible for the appearance in this area elevated thermal stresses revealed in Ref. [1].
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responsible for its formation. Note also that the replacement of the gap between crucible and afterheater by a radiant and non-reflecting boundary does not result in significant errors and can be used in simulation.
Acknowledgements This work was supported by INTAS grant, project number INTAS-00-263.
References
Fig. 6. Temperature field in the crystal of length 3 cm (to the shoulder) in the case of specular reflection at the shoulder conical surface. Contour spacing for temperature is equal to 1 K.
4. Conclusions The results presented here demonstrate that in the case of BGO sillenite crystals that are distinguished by a greater absorptivity and smaller conductivity in comparison with the BGO eulithine ones, specular reflection at the shoulder surface does not only increase the deflection of the crystallization front toward the melt, but drastically changes the temperature field in crystal and results in the appearance of a thin cool area near the crystal axis. The latter correlates with a dark core observed in grown crystals and may be
[1] M.T. Santos, L. Arizmendi, D. Bravo, E. Di!eguez, Mater. Res. Bull. 31 (1996) 389. [2] S. Fu, H. Ozoe, J. Mater. Sci. 34 (1999) 371. [3] C. Lin, S. Motakeff, J. Crystal Growth 128 (1993) 834. [4] J.P. Fontaine, G.P. Extr!emet, V. Chevrier, J.C. Launay, J. Crystal Growth 139 (1994) 67. [5] M.T. Santos, J.C. Rojo, L. Arizmendi, E. Di!eguez, J. Crystal Growth 142 (1994) 103. [6] M.T. Santos, J.C. Rojo, A. Cintas, L. Arizmendi, E. Di!eguez, J. Crystal Growth 156 (1995) 413. [7] J.C. Rojo, C. Mar!ın, J.J. Derby, E. Di!eguez, J. Crystal Growth 183 (1998) 604. [8] Q. Xiao, J.J. Derby, J. Crystal Growth 139 (1994) 147. [9] M. Kobayashi, T. Hagino, T. Tsukada, M. Hozawa, J. Crystal Growth 235 (2002) 258. [10] E.M. Nunes, M.H.N. Naraghi, H. Zhang, V. Prasad, J. Crystal Growth 236 (2002) 596. [11] V.S. Yuferev, O.N. Budenkova, M.G. Vasiliev, S.A. Rukolaine, V.N. Shlegel, Ya.V. Vasiliev, A.I. Zhmakin, J. Crystal Growth 253 (2003) 371. [12] S.A. Rukolaine, M.G. Vasiliev, V.S. Yuferev, A.O. Galyukov, J. Quant. Spectrosc. Radiat. Transfer 73 (2002) 205. [13] S.A. Rukolaine, M.G. Vasilyev, V.S. Yuferev, V. Mamedov, J. Quant. Spectrosc. Radiat. Transfer 84 (2004) 371. [14] V.V. Kalaev, I.Yu. Evstratov, Yu.N. Makarov, J. Crystal Growth 249 (1–2) (2003) 87. [15] O.N. Budenkova, V.M. Mamedov, M.G. Vasiliev, V.S. Yuferev, Yu.N. Makarov, J. Crystal Growth, in press.