Three‐dimensional unsteady thermocapillary flow under rotating magnetic field

Under a rotating magnetic filed (RMF), the instability of thermocapillary flow and its evolution with increasing Marangoni number (Ma) for semiconductor melt (Pr = 0.01) in a floating liquid bridge model (As = 1) are investigated numerically. Under 5 mT RMF, the thermocapillary flow is steady and axisymmetric with Ma < Ma_c, and the critical Marangoni number Ma_c for convection instability is 29.5, which is obtained by the direct numerical simulation. When the Ma is a little bit beyond the Ma_c, the thermocapillary flow loses stability to become a three‐dimensional rotating oscillatory convection, and a periodic oscillation is confirmed by the fast Fourier transform analysis, the oscillatory main frequency decays with increasing Ma. Under 1 mT–6 mT RMF, the Ma_c increases roughly with the magnetic strength except the Ma_c at 4 mT, where the corresponding change of flow mode after the instability is observed. The oscillatory convection occurs with a smaller Ma in the RMF than that without magnetic field. In addition, no instability toward a three‐dimensional steady convection, which is the state of thermocapillary flow without magnetic field after the first instability, is observed under the RMF.     

Three‐dimensional flow in a thin annular layer of silicon melt with bidirectional temperature gradients

Bidirectional temperature gradients coexist virtually in surface tension driven flows. However, the simulations have been performed to the flow with only one temperature gradient. A series of 3 D numerical simulations are conducted to investigate the Marangoni‐thermocapillary flow of silicon melt in a thin annular layer with bidirectional temperature gradients. The temperature gradients are produced by the temperature difference ΔT between walls and the constant heat flux q on the bottom, respectively. When changing q, the melt presents different state evolutions at different ΔT. Furthermore, two critical q are found, one makes the minimum melt temperature higher than the crystallization temperature and the other makes the flow unsteady. Both of the critical heat fluxes decrease with increasing ΔT. q contributes more to the elevation of the melt temperature, while ΔT contributes more to the enhancement of the melt instability. In addition, the melt on the free surface flows mainly along the radial direction.     

Three dimensional numerical study of the effect of large Grashof number on HEM crystal growth

Heat transfer and fluid flow in HEM crystal growth of silicon in cylindrical cavity is studied numerically. The walls of the crucible are heated to a fixed temperature. The exchanger that causes and induces natural convection is seated at the middle‐bottom of the crucible. The finite‐volume method is employed to solve the governing equations with proper boundary conditions. The effects of transport mechanism on the temperature distribution, melt flow, pressure and stream function are presented. We focus our work on the pressure field which has not yet been studied in HEM crucible. Also, we extend our work on a wide range Grashof number and for large numbers until 10¹² not yet studied in HEM furnace. It is found that the onset of flow fluctuations appears at Gr = 10¹⁰. Uniform temperature is observed in the entire melt at high Grashof number with development of a thermal boundary layer close to the exchanger. The thermal boundary layer thickness is calculated for strong buoyancy regime. Besides, for very high Gr number, buoyancy has less effect on temperature and then on melt‐crystal interface shape. During enlarging Gr, pressure evolution is related to temperature variation more than flow pattern.     

Numerical investigation of heat transport and fluid flow during the seeding process of oxide Czochralski crystal growth Part 2: rotating seed

In this paper, the role of seed rotation on the characteristics of the two‐dimensional temperature and flow field in the oxide Czochralski crystal growth system has been studied numerically for the seeding process. Based on the finite element method, a set of two‐dimensional quasi‐steady state numerical simulations were carried out to analyze the seed‐melt interface shape and heat transfer mechanism in a Czochralski furnace with different seed rotation rates: ω_seed = 5‐30 rpm. The results presented here demonstrate the important role played by the seed rotation for influencing the shape of the seed‐melt interface during the seeding process. The seed‐melt interface shape is quite sensitive to the convective heat transfer in the melt and gaseous domain. When the local flow close to the seed‐melt interface is formed mainly due to the natural convection and the Marangoni effect, the interface becomes convex towards the melt. When the local flow under the seed‐melt interface is of forced convection flow type (seed rotation), the interface becomes more concave towards the melt as the seed rotation rate (ω_seed) is increased. A linear variation of the interface deflection with respect to the seed rotation rate has been found, too. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical and analytical modelling of the MHD buoyancy-driven flow in a Bridgman crystal growth configuration

We present analytical and numerical models of magnetohydrodynamic(MHD) buoyancy-driven flow within the liquid pool of a horizontal Bridgman crystal growth furnace, under the influence of a uniform vertical magnetic field B₀. A horizontal differentially heated cylinder, whose aspect ratio (radius to length) is small enough for a fully developed regime to be established in the central core, is considered. With Hartmann layers remaining electrically inactive, a modified Rayleigh number Ra_G, which is the ration of the ordinary Rayleigh number to the square of the Hartmann number, is found to control the MHD reorganisation of the flow. This modified Rayleigh number is a measure of the importance of thermal convection relative to diffusion if velocity is estimated from the balance between the torques of buoyancy and the Laplace force. When Ra_G is much smaller than unity (quasi-diffusive regime), an analytical modelling of the flow, based on a power series of Ra_G, demonstrates that this balance requires secondary vortices within vertical mid-planes of the cylinder, both within the core flow and near the end walls. A 3-D numerical calculation of the flow provides evidence of the transition from a convective MHD flow (when Ra_G is still of the order of unity) to the quasi-diffusive flow, analytically studied. Indeed, this transition takes the form of a rather complex 3-D MHD organisation of the flow which is due to the nonuniformity of the axial temperature gradient along the cylinder.     

Characterization of large‐sized Nd:YAG single crystals grown by horizontal directional solidification

Nd³⁺‐doped Y₃Al₅O₁₂ single crystals have been grown by the horizontal directional solidification (HDS) method in different thermal zone. The Grashof (G_r), Prandtl (P_r), Marangoni (M_a) and Rayleigh (R_a) numbers of melt in HDS system have been discussed for our experimental system to understand the mechanism of melt flow patterns and concentration gradient of dopant. The concentration gradient of Nd³⁺ ions was explained with melt flow processes during crystal growth in different thermal zone, and results indicated that high growth temperature will be helpful for uniformity of dopant in HDS‐grown single crystal. The main microscopic growth defects such as bubbles and irregular inclusions in HDS‐grown Nd:YAG crystals were observed, and the causes were discussed as well. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modification of Fluid Flow and Heat Transport in Vertical Bridgman Configurations by Rotating Magnetic Fields

Applying a rotating magnetic field to an electrically conducting liquid, a Lorentz force is induced which generates a melt rotation of a certain angular velocity. A cylindrical gallium melt (aspect ratio 2.5) has been used as a model liquid. The melt has been heated from the bottom (Ra = 10⁶) or from the top (Ra = −10⁶) and the resulting temperature fluctuations in the melt have been measured in dependence on the rotating field strength (B_max = 30 mT). In the case of the unstable gradient 0.8 mT are sufficient to dominate the buoyancy driven convection and to reduce the amplitude of the buoyancy caused temperature oscillations for more than one order of magnitude. At the same time, the fluctuation frequency increases with the field strength. In the case of the stabilizing temperature gradient, low amplitude/high frequency temperature fluctuations are generated by the rotating magnetic field, indicating the transition to a time-dependent flow. In both cases we see an increase of the convective heat transport for magnetic inductions higher than approximately 5 m T. Applying the rotating magnetic field to the Bridgman growth of gallium doped germanium, the same behavior can be seen: Growing with a top-seeded arrangement, the intensity of the dopant striations is decreased and their frequency is increased. Growing with a bottom-seeded arrangement, the interface curvature changes from concave to convex and the flow becomes time-dependent.     

Experimental investigations of buoyancy-driven convection in vertical (BixSb1−x)2Te3 molten zones

Experimental results for various states of buoyancy driven flow in vertical (Bi_0.23Sb_0.75)₂Te₃ molten zones with covered surface are presented. Critical thermal wall Rayleigh numbers Ra for the onset of time-dependent convection have been determined by means of temperature measurements. The stability diagram obtained for the existing buoyancy driven convection shows the increase of Ra with increasing aspect ratio. This relation is also known from other crystal growth configurations and is due to the damping influence of container walls. At the beginning in the oscillatory region of convection extremely long periods of oscillation (maximum 850 s) were observed, which are caused by another mechanism than periods (25 … 37 s) registered at increasing melt heights. Furthermore, Bi_0.5Sb_1.5Te₃ crystals were grown by using the vertical zone melting technique. The microscopic striations observed in the grown crystals correlate exactly with the temperature signals caused by time dependent convection. However, the fluctuations of the tellurium distribution in axial direction measured by scanning the Seebeck coefficient are presumably generated by unsteady solutal convection during growth.     

Effect of surface tension‐driven flow on BaB2O4 crystal growth from high temperature melt‐solution

The surface tension driven‐flow in BaB₂O₄ (BBO) melt‐solution is visualized by differential interference microscope coupled with Schlieren technique, and the streamline of the steady thermocapillary convection is found to be in form of an axially symmetric pattern. Based on the observation of BBO crystal rotation caused by the convective vortex, the widths of interfacial concentration, heat and momentum boundary layer are calculated. The effect of thermocapillary convection on boundary layer thickness is also investigated. Results show that the width of boundary layer decreases linearly with the increasing of dimensionless Marangoni number. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical modelling of instability and supercritical oscillatory states in a Czochralski model system of oxide melts

The motivation for this study is the need for accurate numerical models of melt flow instabilities during Czochralski growth of oxides. Such instabilities can lead to undesirable spiralling shapes of the bulk crystals produced by the growing process. The oxide melts are characterized by Prandtl numbers in the range 5<Pr <20, which makes the oxide melt flow qualitatively different from the intensively studied flows of semiconductors characterized by smaller Prandtl numbers Pr <0.1. At the same time, these flows can be modelled experimentally by many transparent test fluids (e.g. water, silicon oils, salt melts), which have similar Prandtl numbers, but allow one to avoid the extremely high melting‐point temperatures of the oxide materials. Most previous studies of melt instabilities for Prandtl numbers larger than unity suffer from a lack of accuracy that is caused by the use of coarse grids. Recent convergence studies made for a series of simplified problems and for a hydrodynamic model of Czochralski growth showed that for a second order finite volume method reliable stability results can be obtained on grids having at least 100 nodes in the shortest spatial direction. The obvious numerical difficulties call for an extensive benchmark exercise, which is proposed here on the basis of recently published experimental and numerical data, as well as some preliminary results of this study. The calculations presented are performed by two independent numerical approaches, which are based on second‐order finite volume and finite element discretizations. We start our comparison from the steady states, whose parametric dependencies sometimes exhibit turning points and multiplicity. We then compare the critical temperature differences corresponding to the onset of instability, and finally compare calculated supercritical oscillatory states and phase plots. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of heat transport and fluid flow during the seeding process of oxide Czochralski crystal growth Part 1: non‐rotating seed

For the seeding process of oxide Czochralski crystal growth, the flow and temperature field of the system as well as the seed‐melt interface shape have been studied numerically using the finite element method. The configuration usually used initially in a real Czochralski crystal growth process consists of a crucible, active afterheater, induction coil with two parts, insulation, melt, gas and non‐rotating seed crystal. At first the volumetric distribution of heat inside the metal crucible and afterheater inducted by the RF coil was calculated. Using this heat source the fluid flow and temperature field were determined in the whole system. We have considered two cases with respect to the seed position: (1) before and (2) after seed touch with the melt. It was observed that in the case of no seed rotation (ω_seed = 0), the flow pattern in the bulk melt consists of a single circulation of a slow moving fluid. In the gas domain, there are different types of flow motion related to different positions of the seed crystal. In the case of touched seed, the seed‐melt interface has a deep conic shape towards the melt. It was shown that an active afterheater and its location with respect to the crucible, influences markedly the temperature and flow field of the gas phase in the system and partly in the melt. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pattern transition of temperature distribution at Czochralski silicon melt surface

The melt surface temperature in Czochralski silicon growth was studied by CCD camera observation. The thermal radiation energy from the melt surface was converted into temperature by the blackbody calibration method and was recorded with a VCR as two-dimensional color images. The experimental results without a crystal revealed that the temperature distribution at the melt surface can change in four patterns depending on the crucible rotation rate: axisymmetric spoke pattern at low rotation rates, n-folded and island patterns at medium rotation rates, and cellular patterns at high rotation rates. To predict the fluid motion from the experimental observations, three-dimensional time-dependent numerical simulations of the silicon melt flow were executed. As a result, a qualitative transition model for the temperature distribution and the Czochralski silicon melt flow was derived.     

The radial selectivity of in-situ core-doped crystal rods grown by the double die EFG method

The radial selectivity of in-situ core-doped Cr, Nd: LiNbO₃ crystal rods, grown by the double die EFG (edge-defined film fed growth) method, has been investigated. Two crucibles are combined with an outer and inner die for ascending of different doped melts. The critical growth parameters affected by the diffusion depended spreading effect of the dopant within the melt meniscus have been estimated. The first experimental tests show a good agreement with the theoretical prediction. Single crystalline LiNbO₃ rods with a length of about 100 mm and an outer diameter of 5 mm consisting of sharp separated inner Cr- or Nd-doped core region with diameters between 1.5 and 4 mm were grown successfully.     

Numerical modeling of frequency influence on the electromagnetic stirring of semiconductor melts

Alternating magnetic fields can be used in order to increase the level of convection and to mix the doped semiconductor alloys. A numerical analysis of the electromagnetic induced convection in GaInSb semiconductor melts is performed by using the software package CrysVUn. The magnetic field parameters are varied in order to obtain a maximum efficiency of the induced convection with a minimum quantity of the heat released in the melt. The influence of the electrical current frequency on the convection intensity is analyzed for samples with various radii (R = 0.5 – 3cm). Numerical procedure is validated by comparing the numerical results obtained in mercury samples with the experimental data given from the literature, which show a maximum stirring for a magnetic skin depth δ = 0.2R , in the case of a mercury sample with the radius R = 10 cm. This maximum corresponds to a shielding parameter R _ω = 40. Our numerical results show that the value of the shielding parameter for which the convection intensity reaches the maximum depends on the sample radius and increases when the sample radius increases. The results of this analysis are important in the case of samples with small radius, when a good mixing of the melt can be obtained for frequencies much lower than those corresponding to a shielding parameter R_ω = 40. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study of influences of buoyancy and solutal Marangoni convection on flow structures in a germanium‐silicon floating zone

This paper presents a numerical study of Marangoni flows in a floating zone of germanium‐silicon crystals, which was performed by using a commercial finite element program FIDAD^TM. The numerical results point out that for fluids with a small Pr number the influence of buoyancy forces cannot be ignored in the numerical model. Furthermore, the competition between the thermocapillary (TC) and solutocapillary (SC) flows in the floating zones was qualitatively examined. If the TC flow is as strong as that in the Si‐rich floating zone, the SC flow may be restricted to the bottom area near the free surface. Otherwise, the SC flow may overcome the TC flow and induce a surface transfer of species. The numerical predictions agree well with the previous experiment results. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of internal radiation on the heat transfer during growth of YAG single crystals by the Czochralski method

Heat and mass transfer taking place during growth of Y₃Al₅O₁₂ (YAG) crystals by the Czochralski method, including inner radiation, is analyzed numerically using a Finite Element Method. For inner radiative heat transfer through the crystal the band approximation model and real transmission characteristics, measured from obtained crystals, are used. The results reveal significant differences in temperature and melt flow for YAG crystals doped with different dopands influencing the optical properties of the crystals. When radiative heat transport through the crystal is taken into account the melt‐crystal interface shape is different from that when the radiative transport is not included. Its deflection remains constant over a wide range of crystal rotation rates until it finally rapidly changes in a narrow range of rotation rates. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Role of marangoni tension effects on the melt convection in directional solidification process for multi-crystalline silicon ingots

We carried out global simulations to investigate the marangoni tension effect on the thermal and flow fields in the silicon melt of the directional solidification process for multi-crystalline silicon ingots. The argon flow rate was varied to provide different solidification conditions and to change the relative values between the argon shear stress and the marangoni tension at the melt free surface. We found that the marangoni tension together with the shear stress mainly influences the upper layer melt convection while the thermal buoyancy force dominates the bulk flow of the melt. At low argon flow rates, the argon shear stress can be neglected and the marangoni tension alone enhances the melt convection intensity near the gas–melt–crucible triple junction point. The marangoni tension is so weak that it cannot modify the melt flow pattern in this case. For medium flow rate, the marangoni tension can significantly weaken the shear stress effect at the outer part of the melt free surface, leading to a distinctive flow pattern in the silicon melt. With further increase in argon flow rate, the shear stress sharply increases and dominates the upper layer melt flow, limiting the marangoni tension effect to the triple point. The numerical results are helpful for better understanding and controlling of the directional solidification process for high quality multi-crystalline silicon ingots.     

Nature of large-scale oscillations in oxygen concentration along the growth axis in Czochralski-grown silicon crystals

Oxygen distribution in a Si crystal (100 mm in diameter) has been studied by the absorption method in the range of the absorption band of interstitial oxygen, λ = 5.81 μm. Large-scale fluctuations (～1 cm) of the oxygen concentration (N ₀) along the growth axis were determined. Depending on the melt height, the regions of the chaotic and quasiperiodic changes were established, as well as the region of the constant N ₀ value, and their relation to turbulent, quasiperiodic, and stationary modes of melt convection in crystallization. The values of the critical Rayleigh number for the melt transition from stationary to quasiperiodic (3 × 10³) and from quasiperiodic to turbulent (1.7 × 10⁴) convection modes are determined for growth of silicon crystals by the Czochralski method. The dominating modes of N ₀ concentration oscillations at two incommensurable frequencies, f ₁ = 1.3 × 10^?3 and f ₂ = 6 × 10^?4 Hz, are assumed to be related to the oscillatory transfer of oxygen from the walls of the quartz crucible to the crystallization front and restructurization of the convective flow pattern of the melt in the course of crystal growth.     

Crystal Growth and Properties of the Compounds CuGa3Se5 and CuIn3Se5

CuIn₃Se₅ and CuGa₃Se₅ uniform single crystals 12 mm in diameter and 40 mm in length with the chalcopyrite‐related structure were prepared by directed crystallization of the melt. The melting points of these compounds were defined by means of the differential thermal analysis (DTA). The lattice parameters a and c as well as the axial thermal expansion coefficients α_a and α_c were determined as a function of temperature in the range from 90 to 650 K by the X‐ray diffraction method (XRD). It is found that for both the compounds the coefficients of expansion along the a ‐axis are larger than those along the c ‐axis over the entire temperature range studied.     

3D simulation of the effects of growth parameters on the growth of sapphire crystals using heat exchanger method

3D simulations using the commercial CFDRC and FIDAP code, which are based on finite element techniques, were performed to investigate the effects of anisotropic conductivity on the convexity of the melt–crystal interface and the hot spots of sapphire crystal in a heat‐exchanger‐method crystal growth system. The convection boundary conditions of both the energy input to the crucible by the radiation as well as convection inside the furnace and the energy output through the heat exchanger are modeled. The cross‐sectional flow pattern and the shape of the melt–crystal interface are confirmed by comparing the 3‐D modeling results with previous 2D simulation results. In the 3D model, the “hot spots” in the corners of the crucible are donut shaped, and the shape changes with the value of the conductivity of anisotropic crystal. The outline of the crystal becomes more convex as the conductivity in the z direction (k_sz) increases. The outline of melt–crystal interface is elliptical when the anisotropic conductivity is moving in the radial direction (k_sx and k_sy). The portion at the outline touching the bottom of the crucible is smaller than the maximum outline of the crystal, meaning that the shape at the “hot spot”, changes with the value of the conductivities of anisotropic crystal. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)