Cover Picture: Cryst. Res. Technol. 1/2011

The cover picture shows an isosurface of the difference between instantaneous and averaged temperature, colored by geometric height using a large eddy simulation of Marangoni convection in Czochralski crystal growth. (see pages 14–22) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Combined effects of crucible geometry and Marangoni convection on silicon Czochralski crystal growth

In order to understand the influence of crucible geometry combined with natural convection and Marangoni convection on melt flow pattern, temperature and pressure fields in silicon Czochralski crystal growth process, a set of numerical simulations was conducted. We carry out calculation enable us to determine temperature, pressure and velocity fields in function of Grashof and Marangoni numbers. The essential results show that the hemispherical geometry of crucible seems to be adapted for the growth of a good quality crystal and the pressure field is strongly affected by natural and Marangoni convection and it is more sensitive than temperature. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time-dependent simulation of the growth of large silicon crystals by the Czochralski technique using a turbulent model for melt convection

A numerical model is developed to perform the dynamic and global simulation of Czochralski growth. The effect of melt convection is taken into account by means of an eddy viscosity flow model, which can represent the mixing effect of flow oscillations on the heat transfer. Our method is used to investigate the dynamics of the growth of a 40 cm silicon crystal.     

Influence of solidification of binary eutectic lamellar systems on surface tension driven convection in zero gravity conditions

The effect of solidification of the lamellar binary eutectic system on the onset of surface tension driven convection (so-called Marangoni instability) is studied in a zero gravity environment. Some main general conclusions concerning the possibility for onset of the Marangoni convection within the melt can be drawn from the analysis, viz., the curvature of the free surface has a destabilizing effect on the onset of the flow; the increased perturbed heat transfer rate from the system stabilizes the melt; the onset of the solutal Marangoni convection closely depends on the ratio of the lamellae half-widths and the solutal Marangoni instability is more sensitive against the perturbations than the temperature Marangoni one.     

Solutocapillary convection in germanium‐silicon melts

Surface tension gradients in free crystal growth melts give rise to convective flow. If these gradients are due to thermal gradients, the well known thermocapillary (Marangoni) convection ensues. Concentration gradients due to segregation at the interface during growth can lead to additional solutocapillary convection. A system with large solutocapillary convection is Ge‐Si due to the pronounced segregation and the strong difference in surface tension; solutal buoyancy convection is also present due to the large density difference between Ge and Si. Solutocapillary convection will oppose thermocapillary convection in the Ge‐Si system since Si, having the higher surface tension, is preferentially incorporated into the crystal. A set of experiments directly proving and partially quantifying the effect has been conducted under microgravity during a parabolic flight campaign by recrystallizing Ge‐Si mixtures of different compositions, between 3% and 9% Si, in a crucible with tracers to visualize the movement. Solutocapillary flow with initial flow rates in excess of 5.5 cm/s at the onset of crystallization was measured. A slight dependence of the flow velocity on the initial Si content has been found. Experiments on the ground showed the same effect but with overall smaller speeds; this difference can be explained by the additional action of solutal buoyancy convection. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Marangoni Convective Effect during Crystal Growth in Space

GaSb:Te and GaInSb samples have been solidified under microgravity conditions during the D2 Spacelab mission. Experimental design and parameters are described. Analysis of the thermal data taken during the flight, associated to numerical simulations of heat transfer in the experiment, with the help of FIDAP, gave the experimental conditions (thermal gradients and growth rate). Quantitative chemical analyses of the samples show a chemical segregation characteristic of strong mixing in the melt during crystal growth. Silica crucibles with an internal screw thread groove on the inner wall were used in order to get dewetting of samples from the crucible. It was therefore supposed that Marangoni convection on the free surface associated to the groove might have been the source of convection. This hypothesis has been studied by numerical simulation using FIDAP and the velocity field obtained is in agreement with a strong perturbation of the solutal boundary layer ahead the solid-liquid interface. This can explain the observed chemical segregation.     

Global heat loss and thermal stress analysis in Czochralski crystal growth

Based on our invention of an energy‐efficient Czochalski crystal growth furnace, a 2D‐axisymmetric numerical simulation model of LiNbO₃ crystal growth is developed. The heat transfer, melt and gas flow, radiation and the interface deflection have been examined. Heat losses in the furnace and the insulator, as well as the heating power and thermal stress distribution at three stages of crystal growth are calculated in detail. It is found that a large proportion of heat dissipates through the water‐cooling system, and at the steel shell of the furnace, gas convection heat transfer is the major cooling mechanism. Less heat dissipation by radiation and more heat flux by gas convection to the crystal sidewall results in a larger concentrated thermal stress, which may induce large crystal cracks in the growth process. The simulation results of heating power are in coincidence with the actual power of our furnace, which verifies the feasibility of our model. The detailed information with respect to the device obtained from simulation can help to optimize the energy‐saving design and growth process.     

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)     

Marangoni flow in half-zone liquid bridge of molten tin under ramped temperature difference

A long-run numerical simulation was carried out on a realistic half-zone liquid bridge model of molten tin, which is identical to JAXA's (former NASDA) liquid bridge experiment apparatus. Using the time-dependent temperature difference imposed on both ends of supporting iron rods, the simulation numerically reproduces the experiment for a time period of 3100 s and enables the study of the two-step bifurcation behavior of Marangoni flow. The present study also evaluates the effect of heating velocity on the bifurcations of the Marangoni flow and indicates that the case with a higher heating velocity gives larger critical Marangoni numbers. Moreover, in this study, we investigate the cause of the second critical Marangoni number and critical frequency disagreement between the experimental results and numerical results, and indicate that the second critical Marangoni number determined through free surface temperature oscillations in the experiment may not correspond to the exact onset of oscillatory Marangoni flow.     

微重力条件下Marangoni数对分离结晶过程的影响

为了了解微重力条件下新型分离结晶生长过程中熔体热毛细对流的基本特征,利用有限差分法进行了三维数值模拟.当熔体顶部分别为自由表面及固壁边界条件时,得到了新型分离结晶Bridgrnan生长过程中熔体热毛细对流的速度分布和温度分布.结果表明:熔体顶部为自由表面时,当Marangoni数较小时,在上自由表面和下部狭缝处自由表面的表面张力的驱动下,熔体内部产生了逆时针和顺时针两个流动方向相反的流胞,此时熔体内的流动状态为稳态;随着Marangoni数进一步的增大,流胞的流动逐渐增强并逐步向熔体内部扩展,熔体内部温度分布非线性增强,上自由表面和下部狭缝处自由表面处速度增大;当Marangoni数超过某一临界值后,流动转化为非稳态流动.当熔体顶部为固壁时,与熔体顶部为自由表面时相比,临界Marangoni数增大.流动失稳的物理机制是流速的变化和阻力的变化之间存在滞后.     

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.     

分离结晶垂直Bridgman法生长CdZnTe晶体的全局数值模拟

采用FLUENT软件对分离结晶Bridgman法生长CdZnTe晶体进行了全局数值模拟.模拟对象为:熔体上部边界条件分别为固壁和自由表面时两种晶体生长系统.重点考虑坩埚和晶体之间狭缝宽度e和重力对分离结晶过程的影响.在计算中分别取e=0 mm、0.5 mm和1 mm三种狭缝宽度,得到了在微重力和常重力条件下的温度分布、结晶界面形状以及流函数分布图.结果表明:在微重力条件下,当熔体上部为固壁时,随着狭缝宽度的增大,热毛细力作用增强,流动强度增强;当熔体上部为自由表面时,则与之相反.在常重力条件下,由于浮力-热毛细对流的共同作用,随着狭缝宽度的增加,流动强度逐渐减弱,有助于提高晶体生长质量.     

Melt Flow and Species Transport in μg‐Gradient‐Freeze Growth of Germanium

The result of a μg‐experiment on the Gradient‐Freeze growth of Ge:Zn with doping from the vapour phase shows a homogeneous distribution of the zinc in the melt, indicating the dominating role of a gravity‐independent transport mechanism. This effect is investigated numerically on the basis of a global model of the growth setup. The numerical simulation includes the melt flow and the transport of the dopant taking into account buoyant and thermocapillary forces. The results confirm the minor influence of gravity on the species transport. The complete mixing of the melt can be explained by thermocapillary (Marangoni) convection only.     

Gas bubbles in shaped sapphire

Shaped sapphire single crystals almost always contain a typical defect: small bubbles also known as “micro-voids”. The goal of this paper is to review the existing literature and give a survey of our recent work on this specific defect. As a conclusion to our review we propose a mechanism for bubble formation and its occurrence and distribution in shaped sapphire.The bubbles contain CO which originates as a result of complex reactions between the molten alumina, Mo crucible and graphite pieces. The CO dissolves in the molten alumina and is probably present as a saturated solution whatever the growth conditions. Then the dissolved gas is rejected at the solid–liquid interface and bubble nucleation occurs where the concentration of CO is highest. Therefore, the location of the bubbles in the crystal is governed by the hydrodynamics in the meniscus, which include Marangoni convection. The bubble is finally incorporated in the crystal when it reaches a critical size which depends only on the growth rate.When a central capillary channel is used, the bubbles are located in a layer close to the surface of the crystal from where they can be easily removed by polishing. Also, all the literature agrees on the fact that interface destabilization at high velocity results in the presence of bubbles throughout the entire crystal and should be avoided.     

On the significance of surface tension driven flow in floating zone melting experiments

The predominance of the surface tension driven (Marangoni) flow over the buoyancy driven free convection for small zone lengths in electron beam floating zone melting of doped molybdenum rods is pointed out. The observed different patterns of microsegregation and the critical conditions of their occurrence, i.e. B-cores below a critical zone length of l_c = 3.5 mm and B-striations above this value, are shown to be in qualitative agreements with theoretical predictions made from convection models.     

Three dimensional simulation of melt flow in Czochralski crystal growth with steady magnetic fields

Three-dimensional transient numerical simulations were carried out to investigate the melt convection and temperature fluctuations within an industrial Czochralski crucible. To study the magnetic damping effects on the growth process, a vertical magnetic field and a cusp magnetic field were considered. Due to our special interest in the melt convection, only local simulation was conducted. The melt flow was calculated by large-eddy simulation (LES) and the magnetic forces were implemented in the CFD code by solving a set of user-defined scalar (UDS) functions. In the absence of magnetic fields, the numerical results show that the buoyant plumes rise from the crucible to the free surface and the crystal–melt interface, which indicates that the heat and mass transfer phenomena in Si melt can be characterized by the turbulent flow patterns. In the presence of a vertical magnetic field, the temperature fluctuations in the melt are significantly damped, with the buoyant plumes forming regular cylindrical geometries. The cusp magnetic field could also markedly reduce the temperature fluctuations, but the buoyant plumes would break into smaller vortical structures, which gather around the crystal as well as in the center of the crucible bottom. With the present crucible configurations, it is found that the vertical magnetic field with an intensity of 128 mT can damp the temperature fluctuations more effectively than the 40 mT cusp magnetic field, especially in the region near the growing crystal.     

Numerical investigation of natural and forced solutal convection above the surface of a growing crystal

Numerical simulation of the solution dynamics (natural and forced convection) above the surface of growing aqueous-soluble KDP crystals is carried out. A flow of supersaturated solution falling on the crystal surface through the multi-nozzle feeder is considered. An effective method based on time-dependent Navier–Stokes and energy equations in Boussinesq approximation has been developed for numerical investigation. Parametrical calculations of different regimes of hydrodynamics and mass transfer in a crystallization chamber are performed. Some important quantitative characteristics for high-rate growth technology development are obtained.     

Time-dependent simulation of Czochralski silicon crystal growth

We have developed a detailed mathematical model and numerical simulation tools based on the streamline upwind/Petrov-Galerkin (SUPG) finite element formulation for the Czochralski silicon crystal growth. In this paper we consider the mathematical modeling and numerical simulation of the time-dependent melt flow and temperature field in a rotationally symmetric crystal growth environment. Heat inside the Czochralski furnace is transferred by conduction, convection and radiation, Radiating surfaces are assumed to be opaque, diffuse and gray. Hence the radiative heat exchange can be modeled with a non-local boundary condition on the radiating part of the surface. The position of the crystal-melt interface is solved by the enthalpy method. The melt flow is assumed to be laminar and governed by the cylindrically symmetric and incompressible Navier-Stokes equations coupled with the calculation of temperature.     

Simulation of temperature and flow fields in an inductively heated melt growth system

The goal of the research presented here is to apply a global analysis of an inductively heated Czochralski furnace for a real sapphire crystal growth system and predict the characteristics of the temperature and flow fields in the system. To do it, for the beginning stage of a sapphire growth process, influence of melt and gas convection combined with radiative heat transfer on the temperature field of the system and the crystal‐melt interface have been studied numerically using the steady state two‐dimensional finite element method. For radiative heat transfer, internal radiation through the grown crystal and surface to surface radiation for the exposed surfaces have been taken into account. The numerical results demonstrate that there are a powerful vortex which arises from the natural convection in the melt and a strong and large vortex that flows upwards along the afterheater side wall and downwards along the seed and crystal sides in the gas part. In addition, a wavy shape has been observed for the crystal‐melt interface with a deflection towards the melt. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Radial segregation due to weak convection in a floating zone

The influence of weak convection, caused by surface tension forces, on radial dopant segregation occurring in crystals grown under microgravity conditions is studied numerically. The geometry considered corresponds to a floating-zone configuration with partially coated melt surfaces consisting of small evenly distributed spots of free surfaces. In order to distinguish dopant distribution due to weak convection clearly from distribution due to diffusion the spots only cover one quarter of the periphery. Thus, surface tension-driven convection is allowed only over one quarter of the floating-zone configuration resulting in an asymmetric dopant distribution. The percentage of free surfaces present is varied in order to alter the Marangoni flow rates. The maximum dopant concentration due to radial segregation is plotted as a function of a certain convection level. The results of the present numerical study are supposed to be used to design corresponding space experiments launched at the end of the year 2000.