Oxide Crystals
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 solidliquid 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 [1].
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 solidliquid 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 setup,
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 Fig.1.
Fig.1. Left side: scheme of growth setup; rightside: scheme of computational domain.
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 NavierStokes 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 (Fig. 2).
Fig.2. "Fresnel" specular reflection coefficient of the boundary BGO/gas. 1  for
radiation incident from the side of crystal, 2  from the side of gas.
Each problem had its own grid. Subdivision of the domain above the melt is shown in Fig.3.
\
Fig.3. Computational grid for radiation problem. 1  discretization in a meridional plane
(r,z), 2  discretization in a horizontal crosssection,
3  the shape of a computational
grid cell.
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 solidmelt interface and the temperature field in crystal as it is pulled for
diffuse and specular conical part of crystal side surface.
Fig.4. Evolution of crystallization front and temperature field in crystal during
the growth. Left side (a) of each figure corresponds to diffuse reflection, and
right side (b) to the specular one.
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 solidliquid interface presented.
By the contrast, in the case of specular reflection the shape of solidliquid
interface and its variations with crystal growth are surprisingly similar to observed in experiment.
The results presented 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 PhysicoTechnical
Institute of Russian Academy of Science.
References:
[1] "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.
[2] "Global analysis of heat transfer in growing BGO crystals (Bi4Ge3O12) by lowthermal
gradient Czochralski method", I.Yu.Evstratov, S.A.Rukolaine, V.S.Yuferev, M.G.Vasiliev
et al, J.Crystal Growth, 235 (2002) 371.
[3] "Variations of solidliquid 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) 383397.