The variations of crystallization front shape during the growth of oxide crystals greatly affect the optical quality of the crystal so it is absolutely necessary to know the details of the growth process clearly. Thus, most oxide crystals exhibit some degree of transparency to infrared radiation, and this can greatly influence the solid-liquid interface shape, temperature distribution in a crystal and flow patterns in a melt.
The heat transfer in growing oxide crystals is characterized by the following phenomena:
Internal radiation through the crystal depends largely on the absorption coefficient and refraction index of the crystal. The first parameter determines the radiative heat absorption and emission inside the crystal while the second reflection and refraction of radiation at the crystal side surface. Absorption coefficient of oxide crystals, as a rule, is wavelength dependent. Every so often this dependence takes the form that the crystal is highly transparent in some wavelength range (transparence window) and practically opaque outside it. In latter case both internal and surface radiation of the crystal affects the heat transfer in a furnace.
- Absorption coefficient of a melt is generally much greater than that of a crystal. Therefore, radiation is a crucial in heat removal from the melt through the crystallization front.
- Refractive index of oxide crystals significantly exceeds unity, and, consequently, multiple reflections and refractions of radiation at the crystal side surface take place.
- The side surface of oxide crystals is rather specular than diffusely reflective.
- Oxide crystals are distinguished by a small thermal conductivity both in solid and liquid phases. As a result, Prandtl number turns out to be high and heat transport in a melt is determined by convection.
- Finally, oxide crystals demonstrate the pronounced tendency toward faceting of the solidification front, and as a result, the shape of the crystals can be strongly deviated from a circular cylinder and presents an irregular prism.
It is obvious that consideration all these peculiarities is a formidable task at present. Therefore at this stage we have neglected faceted growth and considered heat transfer in axisymmetrical approximation. For calculation of radiation heat transfer we have developed own approach, which allows one to solve axisymmetric transfer problem in arbitrary domains with both diffuse and specular (Fresnel) boundaries .
A global analysis of heat transfer in growing BGO (Bi4Ge3O12) crystals in low thermal gradient Czochralski process was carried out to explain observed in practice significant variation of the solid-liquid interface shape during the crystal growth [2,3]. This technique was developed in the Institute of Inorganic Chemistry (Novosibirsk, Russia) and at present allows one to obtain nearly perfect BGO crystals up to 140 mm in diameter and up to 400 mm in length. Schematic diagram of an experimental set-up, which allows one to grow crystals up to 80 mm in diameter, is presented in left side of Fig.1 and calculation domain is shown in the right side of Figure1.
The whole problem was divided into two subproblems: convection and heat transfer in the melt and heat transfer in crystal and gap between crystal and crucible. For the solution of the first subproblem the common model of heat transfer in a melt was exploited. We considered conduction and convection, with the fluid velocity determined by the solution of Navier-Stokes and continuity equations written for an incompressible fluid with the Boussinesq approximation. The crucible was stationary, while crystal was rotated at a constant rate. Marangoni convection was not taken into account. The temperature distribution along the crucible wall was given and the meniscus shape near triple point was neglected, as a rule.
For the calculation of heat transfer in crystal and gap between crystal and crucible we used own approach mentioned above. The main difficulty in simulating radiative heat transport in domains with the Fresnel boundaries lays in the fact that specular reflection coefficient is strongly dependent on the incidence angle. Particularly, for BGO crystals this coefficient is varied more than a factor of six from 0.15 to 1 in the narrow range from 26 to 28 degrees (Figure 2). Each problem had its own grid. Subdivision of the domain above the melt is shown in Figure 3.
In simulation diameters of crystal and crucible were equal to 77 and 100 mm, and the height of crucible and the initial height of a melt were equal to 250 and 160 mm, respectively. The rotational velocity of crystal was equal to 15 rpm and pulling rate 0.5 mm/hour. With respect to the decrease of the melt level, growth rate proves out equal to 1.5 mm per hour. Special attention was given to the consideration of specular (Fresnel) reflection at the crystal side surface. Up to now this phenomenon has not been taken into account in crystal growth simulation. Fig. 4 shows the evolution of the solid-melt interface and the temperature field in crystal as it is pulled for diffuse and specular conical part of crystal side surface.
In the case of diffuse reflection the deflection of crystallization front toward the melt during the whole process turns out small and does not exceed 10 mm and fail to reproduce the observed shapes of solid-liquid interface presented. By the contrast, in the case of specular reflection the shape of solid-liquid interface and its variations with crystal growth are surprisingly similar to observed in experiment. The results shown here demonstrate the important role of specular reflection at the conical part of the crystal (its shoulder) in setting the shape of solid/liquid interface in LTG Czochralski growth of BGO (Bi4Ge3O12) crystals. Models based on the assumption of diffuse reflection turned out to be unsuitable to simulate significant convexity of the interface toward the melt observed in this process. One could expect also that specular reflection play significant role during the growth of other oxide crystals.
The statement of the problem, development of software for treatment of radiation problem and simulation of crystal growth process was performed in cooperation with Laboratory of Applied Mathematics and Mathematical Physics Ioffe Physico-Technical Institute of Russian Academy of Science.
 “Numerical solution of axisymmetric radiative transfer problems in arbitrary domains using the characteristic method”, S.A.Rukolaine, M.G.Vasiliev, V.S.Yuferev, A.O.Galyukov, J. Quant. Spectr. Radiat. Transfer 73 (2002) 205.
 “Global analysis of heat transfer in growing BGO crystals (Bi4Ge3O12) by low-thermal gradient Czochralski method”, I.Yu.Evstratov, S.A.Rukolaine, V.S.Yuferev, M.G.Vasiliev et al, J.Crystal Growth, 235 (2002) 371.
 “Variations of solid-liquid interface in the BGO low thermal gradients Cz growth for diffuse and specular crystal side surface”, 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) 383-397.