V.S. Yuferev

Variations of solid-liquid interface in the BGO low thermal gradients Cz growth for diffuse and specular crystal side surface

Abstract A global analysis of heat transfer in growing BGO (Bi4Ge3O12) crystals was carried out to explain the significant variation of the solid–liquid interface shape observed in practice during crystal growth. Special attention was given to the accurate treatment of radiative heat transfer in crystal and the gap between crystal and crucible wall. Crystal side surface was assumed to be transparent and diffuse or specular (Fresnel) reflective while melt was opaque. Spectral absorptivity of crystal was approximated by a three-band model. It is shown that rotationally driven vortex dominates under the solid–liquid interface during the whole growth process and no reconstruction of flow pattern, which could lead to interface inversion, takes place. As a result, in the case of diffuse crystal surfaces, the calculated deflection of the crystallization front does not exceed several millimeters and that completely disagrees with the observations. In contrast, for the specular crystal surface the solid–liquid interface turns out to be deeply convex toward the melt at the initial stage of the growth, and then, as the crystal is pulled, its convexity strongly diminishes. Such behavior is in good agreement with the experiment and mainly determined by specular reflection at the conical part of the crystal (its shoulder) while the effect of specular reflectivity of the cylindrical part and melt-free surface is significantly less. It is also shown that radiation transfer in the wavelength band of crystal opacity in conjunction with the specular reflection of radiation in the band of crystal transparency result in the appearance of “convex–concave” isotherms in the upper part of the crystal.

Process optimisation of MOVPE growth by numerical modelling of transport phenomena including thermal radiation

Abstract A global transport model for the MOVPE of III–V growth based on the finite volume solution of coupled flow, heat and mass transfer, including detailed radiative transfer, multicomponent diffusion and homogeneous and heterogeneous chemical reactions, is presented. For radiative transfer modelling, a combined approach is used of grey-diffuse view-factor based heat flux exchange between the semi-transparent reactor walls through the transparent reactor interior, and a spherical harmonics approximation for the radiative-conductive heat transfer problem in participating massive quartz elements with complex shapes. The described modelling approach is applied to the horizontal multiwafer radial flow Planetary Reactor™, validated experimentally and used for process improvements. The mutual interaction of changing radiation properties at internal solid boundaries due to semiconductor coatings and thermal behaviour in that particular MOVPE reactor is discussed.

An experimental and theoretical study of temperature distribution in sapphire crystals grown from the melt by Stepanov's method

Abstract The axial temperature distribution in shaped sapphire single crystals was determined experimentally by growing thermocouples into crystals. The data obtained are compared with a theory based on a lightguide approximation for radiative-conductive heat exchange in a thin transparent crystal. Including several simplifying assumptions permits one to obtain a one-dimensional equation for the temperature averaged over crystal cross section. A nummerical solution of this equation yields the axial temperature distribution.

Development of advanced mathematical models for numerical calculations of radiative heat transfer in metalorganic chemical vapour deposition reactors

Abstract The effect of radiative heat transfer in a horizontal chemical vapour deposition (CVD) reactor on the upper wall temperature is studied in detail. A three-band model for the quartz absorption coefficient is introduced and the wall emittance, reflectance, and transmittance are calculated for the cases of specular and diffuse walls, and also for walls covered by a film. Numerical simulation of the heat transfer in the horizontal reactor has shown that the upper wall temperature varies about up to 40–70 K depending on the type of the wall and the emissivity of the susceptor.

Advanced mathematical models for simulation of radiative heat transfer in CVD reactors

The effect of radiative heat transfer, in horizontal chemical vapor deposition reactors, on the upper wall temperature and the formation of parasitic GaAs and As deposits is studied in detail. A multi-band model is developed, including the interference of thermal radiation. It is shown that the interference phenomenon leads to a significant additional decrease of the wall temperature that, in turn, results in a substantial diminution of the deposition rate of the GaAs layer. The As film deposited on the reactor wall results in a more significant decrease of the wall temperature, of about 130 K.

Global analysis of heat transfer in growing BGO crystals (Bi4Ge3O12) by low-gradient Czochralski method

A numerical analysis of heat transfer in a set-up for growing BGO crystals is undertaken. Consideration is given to the correct treatment of radiative heat transport inside and outside the crystal. Effect of internal radiation is demonstrated. Conjugate heat transfer in crystal, melt and crucible was studied for different crystal lengths. Violation of the fully faceted growth at the initial stage of pulling is discussed.

Control of radiative-conductive heat transfer in shaped semi-transparent crystals pulled from the melt

Abstract Theoretical investigation of heat exchange in a model (system) consisting of a shaped cylindrical semi-transparent crystal with a set of screens surrounding the crystal and an outer heater is considered. It is shown that the temperature field in the crystal is very sensitive to the temperature distribution along the surface of the heater and to a lesser extent depends on the parameters of the set of screens. Solutions to direct and inverse problems are obtained. In the first case, the temperature of the heater is given and the crystal temperature is found. In the second case, in contrast, the temperature of the heater which will ensure the wanted temperature in the crystal is sought. Special attention is given to the case when the linear temperature distribution in the crystal is needed.

Investigation of the variations in the crystallization front shape during growth of gadolinium gallium and terbium gallium crystals by the Czochralski method

Numerical investigation of the variations in the crystallization front shape during growth of gadolinium gallium and terbium gallium garnet crystals in the same thermal zone and comparison of the obtained results with the experimental data have been performed. It is shown that the difference in the behavior of the crystallization front during growth of the crystals is related to their different transparency in the IR region. In gadolinium gallium garnet crystals, which are transparent to thermal radiation, a crystallization front, strongly convex toward the melt, is formed in the growth stage, which extremely rapidly melts under forced convection. Numerical analysis of this process has been performed within the quasistationary and nonstationary models. At the same time, in terbium gallium garnet crystals, which are characterized by strong absorption of thermal radiation, the phase boundary shape changes fairly smoothly and with a small amplitude. In this case, as the crystal is pulled, the crystallization front tends to become convex toward the crystal bulk.

The formation of a dislocation structure in ribbon-shaped germanium single crystals under thermal stress

Abstract The formation of the dislocation structure in germanium single crystal ribbons grown from melt by Stepanovs method has been studied. It is shown that the relatively high dislocation density in the ribbons is due to thermal stresses present during growth and that the dislocation density can be controlled by properly varying the temperature of a radiation shield. Data are presented on the temperature distribution along the ribbon pulling axis, the tangential thermoelastic stress pattern, and the distribution of dislocations for three different shield geometries. The distribution of dislocations over the ribbon width is shown to correlate well with that of the rms tangential stresses. Samples of low dislocation density germanium ribbon were produced.

New approach to simulation of radiative-conductive heat transfer in semi-transparent crystals being pulled from a melt

Abstract A method is suggested for the solution of multi-dimensional radiative heat transfer problems arising in growing crystals. The basic idea of the approach lies in the construction of a special division of the unit sphere into a set of solid angles (cells) and in the approximation of the radiation intensity in each solid angle by the P1 approximation. The radiant transport equation is satisfied in the mean over each elementary cell and the system of partial differential equations of the second order relative to the local zeroth-moments of radiation intensity are obtained. It is shown that the solid angle subdivision can be carried out in different ways with respect to specific features of the heat transfer problem under consideration. As a result even a very rough partition permits satisfactory accuracy of the numerical solutions. One of the main advantages of the method consists in using solid angle subdivisions which can be varied from point to point of the spatial domain. The latter gave possibility to simulate the radiative heat transfer in a circular cylinder of finite length with specular side surface. On this basis the calculation of the temperature field in a cylindrical sapphire crystal being pulled from the melt has been carried out without any restriction on the size of the crystal