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)     

Effect of steady ampoule rotation on radial dopant segregation in vertical Bridgman growth of GaSe

For vertical Bridgman growth of the nonlinear optical material GaSe in an ampoule sufficiently long that flow and dopant transport are not significantly influenced by the upper free surface, we show computationally that steady rotation about the ampoule axis strongly affects the flow and radial solid-phase dopant segregation. Radial segregation depends strongly on both growth rate U and rotation rate Ω over the ranges 0.25 μms⁻¹U3.0 μms⁻¹ and 0Ω270 rpm. For each growth rate considered, the overall radial segregation passes through two local maxima as Ω increases, before ultimately decreasing at large Ω. Rotation has only modest effects on interface deflection. Radial segregation computed using a model with isotropic conductivity (one-third the trace of the conductivity tensor) predicts much less radial segregation than the “correct” model using the anisotropic conductivity, with the segregation decreasing monotonically with Ω. Consideration of a model in which centrifugal acceleration is deliberately omitted shows that, as Ω increases, diminution and ultimately disappearance of the “secondary” vortex lying immediately above the interface is due to centrifugal buoyancy, while axial distension of the larger “primary” vortex above is due to Coriolis effects. These results, which are qualitatively different from those accounting for centrifugal buoyancy, suggest that several earlier computational and analytical predictions of rotating vertical Bridgman growth are either limited to rotation rates sufficiently low that centrifugal buoyancy is unimportant, or are artifacts associated with its neglect. The overall radial segregation depends approximately linearly on the product of and the growth rate U for the conditions considered, where is the segregation coefficient.     

Thermocapillary‐buoyancy flow of silicon melt in a shallow annular pool

In order to understand the nature of surface spoke patterns on silicon melt in industrial Czochralski furnaces, a series of unsteady three‐dimensional numerical simulations were conducted for thermocapillary‐buoyancy flow of silicon melt in annular pool (inner radius r_i = 15 mm, outer radius r_o = 50 mm, depth d = 3 mm). The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces either are adiabatic or allow heat transfer in the vertical direction. Results show that a small temperature difference in the radial direction generates steady roll‐cell thermocapillary‐buoyancy flow. With large temperature difference, the simulation can predict three‐dimensional oscillatory flow, which is characterized by spoke patterns traveling in the azimuthal direction. The small vertical heat flux (3 W/cm²) does not have significant effects on the characteristics of this oscillatory flow. Details of the flow and temperature disturbances are discussed and the critical conditions for the onset of the oscillatory flow are determined. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulation of Temperature and Flow Field in the Melt for the Vapour‐pressure‐ controlled Czochralski Growth of GaAs

The influence of the melt flow on the temperature field and interface during the vapour‐pressure‐controlled growth of GaAs was studied numerically with the commercial general‐purpose program FIDAP^TM. The thermal boundary conditions for the domain of seed, crystal, boron oxide and crucible were taken from a global calculation for an equipment used at the IKZ to grow 6″ crystals. Due to the large melt volume the buoyancy forces become rather strong and have to be counteracted by reasonable rotation rates. Preliminary results have been obtained for iso‐ and counter‐rotation showing that the flow field exhibits structures on small scales. High rotation rates are needed to counteract the buoyancy flow efficiently and to achieve a smooth flat interface. Even if the the flow structure is not resolved in detail, the interface shape can be deduced form the calculations.     

Static magnetic fields in semiconductor floating-zone growth

Heat and mass transfer in semiconductor float-zone processing are strongly influenced by convective flows in the zone, originating from sources such as buoyancy convection, thermocapillary (Marangoni) convection, differential rotation, or radio frequency heating. Because semiconductor melts are conducting, flows can be damped by the use of static magnetic fields to influence the interface shape and the segregation of dopants and impurities. An important objective is often the suppression of time-dependent flows and the ensuing dopant striations. In RF-heated Si-FZ-crystals, fields up to 0.5Tesla show some flattening of the interface curvature and a reduction of striation amplitudes. In radiation-heated (small-scale) Si-FZ crystals, fields of 0.2–0.5Tesla already suppress the majority of the dopant striations. The uniformity of the radial segregation is often compromised by using a magnetic field, due to the directional nature of the damping. Transverse fields lead to an asymmetric interface shape and thus require crystal rotation (resulting in rotational dopant striations) to achieve a radially symmetric interface, whereas axial fields introduce a coring effect. A complete suppression of dopant striations and a reduction of the coring to insignificant values, combined with a shift of the axial segregation profile towards a more diffusionlimited case, are possible with axial static fields in excess of 1Tesla. Strong static magnetic fields, however, can also lead to the appearance of thermoelectromagnetic convection, caused by the interaction of thermoelectric currents with the magnetic field.     

Analysis of SiC crystal sublimation growth by fully coupled compressible multi-phase flow simulation

A fully coupled compressible multi-phase flow solver was developed to effectively design a large furnace for producing large-size SiC crystals. Compressible effect, convection and buoyancy effects, flow coupling between argon gas and species, and the Stefan effect are included. A small and experimental furnace is used to validate the solver. First, the essentiality of 2D flow calculation and the significance of incorporating buoyancy effect and gas convection, the Stefan effect, and flow interaction between argon gas and species were investigated by numerical results. Then the effects of argon gas on deposition rate, growth rate, graphitization on the powder source, and supersaturation and stoichiometry on the seed were analyzed. Finally, the advantages of an extra chamber design were explained, and improvement of growth rate was validated by the present solver.     

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.     

数值模拟自然对流对直拉单晶硅的影响

在直拉单晶硅生长的过程中,自然对流对晶体界面的形状、温度场及应力分布影响很大。本文采用二维模型对熔体内自然对流对单晶硅的影响作了数值模拟,在低雷诺数时采用层流模型,高雷诺数时采用紊流模型,Gr的变化范围从3×106到3×1010,这样涵盖了从小尺寸到大尺寸的直拉单晶硅生长系统。数值结果表明熔体的流动状态不仅与熔体的Gr有关,还与熔体高度和坩埚半径的比值密切相关。当Gr>108时,熔体内确实存在紊流现象,层流模型不再适合,随着Gr的增大,紊流现象加剧,轴心处的等温线变得更为陡峭,不利于晶体生长。     

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.     

Influence of transverse magnetic field on thermocapillary flow in liquid bridge

For exploring the optimizing convection control technique by external magnetic field in floating zone crystal growth of semiconductor under microgravity, thermocapillary flow in a floating half‐zone model is simulated numerically, and the influences of both the transversal uniform magnetic field and the magnetic field generated by transversal four coils on thermocapillary flow are investigated. The results indicate that the transversal uniform magnetic field is likely to break the axisymmetrical structure of thermocapillary flow, which is unfavorable to the growth of high‐quality crystal; under the magnetic field generated by transversal four coils, both the mean and the maximum velocities increase with the increment of the distance between coils or the decrement of coil radius; and the convection tends to be more axisymmetrical with increasing coil radius. Compared to the transversal uniform magnetic field, the magnetic field generated by transversal four coils of appropriate radius and relative distance may not only suppress convection, but also enhance the axisymmetry of convection at the same time, and finally, the better convection control can be achieved. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of gravitation on the processes of heat and mass transfer in solution crystal growth by the travelling heater method (THM) (III)

The effect of the natural and thermocapillary convection on the vortex configuration in solution during growth of PbTe crystal by the travelling heater method is considered. The estimation of the parameters of growth process (i. e. axial temperature gradients, gravitational acceleration, degree of the solution's surface contact with ampoule), when the vortex configuration undergoes qualitative variation, is given. In terms of the one-dimensional thermodiffusive problem solution the effect produced by the convective stirring on the position of growing and dissolving interfaces is described.     

On the morphological instability of the growing crystals (III). Micro-interferometric investigations of the morphological stability of KDP crystals growing from aqueous solutions

The paper is aimed at the elucidation of the limits to which the faceted forms of KDP crystals preserve their stability and beyond which a skeletal growth from stagnant solutions takes place. In order to avoid the natural convections the experiments were performed in two-dimensional cells. On the basis of quantitative criteria, it has been shown that the kinetic regime of growth has been replaced by a diffusion one when the concentration on the crystal surface drops with respect to the bulk value with 0.1% to 0.2%. The critical size above which the KDP crystal no longer retains its polyhedral growth mode in a diffusion regime since a gross morphological defect in the form of a depression appears on its (010) face has been compared with the theoretical expressions derived by Cahn and Chernov.     

The effect of buoyancy on the weight gain signal in LEC crystal growth

Crystal diameter monitoring in liquid encapsulated Czochralski (LEC) growth using the weighing method is investigated. An analytical expression for the differential weight gain signal is derived for stationary growth conditions accounting for bouyancy. For the GaAs/B₂O₃ system the buoyancy component of the signal can easily amount to 30%. The formula can be applied to determine the crystal/melt interface area, provided the radius of the part emerging from the encapsulant is known, either in advance as the seed dimension, or by separate monitoring using, e.g., conventional optical methods.     

Thermal radiation and low-temperature-vapour growth of HgI2 crystal in production furnace

Heat exchanges in a sealed ampoule in the LTVG (low temperature vapour growth) furnace have been modelled in order to compute temperature fields and control the growth of HgI₂ crystals from vapour phase at low temperatures. We use a coupled conductive-radiative model to determine the shapes of the source and the crystal at different equilibrium states (i.e. without growth rate). The model involves conductivity anisotropy in the crystal and radiative exchanges between grey and diffuse surfaces (source and crystal interfaces, Pyrex walls), which are considered as opaque. Internal buoyancy effect is not taken into account as the pressure inside the ampoule is very small. The source temperature is fixed. For different undercoolings, i.e. for different cold finger temperatures, the “equilibrium” isotherm between the source/gas and crystal/gas interface has been numerically obtained. This “equilibrium” isotherm, which is associated with the stop of the growing process, gives a crystal shape. This shape is compared with experimental results given by the ETH-Zürich group. The model would permit a better understanding and control of the future HgI₂ crystal growth experiment. The computations are performed using a finite element package (FIDAP).     

Solutal convection in crystal growth: effect of interface curvature on flow structuration in a three-dimensional cylindrical configuration

Due to temperature and concentration gradients in the molten phase, it is well known that convective flows can develop in the bulk under normal conditions of gravity. These motions modify the shape of the growing interface and the concentration distribution along it. This study will only focus on the case of pure solutal convection and the effect of a given interface curvature on the flow. In particular, the transition from a 3D-flow to a 2D one as a function of the interface curvature is examined in order to investigate possible values of the operating parameters and fluid properties. The main aim is to justify the use of 2D-simulation tools for predicting the convective transport in cylindrical crystal growth ampoules.     

生长气氛和速度在金红石(TiO2)单晶体生长中的作用研究

本文研究了生长气氛和生长速度在焰熔法金红石单晶体生长中的作用,对比了晶体在空气中与在氧气中退火的结果,测定了晶体试样的摇摆曲线和透过率.研究表明:金红石单晶体的生长受炉膛气氛、生长界面温度和生长速度的影响;炉膛气氛决定晶体能否形成,是关键因素;炉膛气氛中的氧分压大于液固界面(即生长界面)处熔体的氧离解压是生长完整晶体的前提条件;晶体在退火过程中消除热应力,但更重要的是通过氧化反应消除氧空位,在氧气氛中退火,可明显缩短退火时间.在所优化的实验条件下制备的晶体,完整性较好,透过率为70～72;,与商用晶体的透过率基本一致.     

Numerical simulation of the LEC-growth of GaAs crystals with account of high-pressure gas convection

The influence of the inert gas pressure on the growth of 4 GaAs crystals by the liquid encapsulated Czochralski method (LEC) process is studied for a range of the Ar gas pressure up to 10 bar by using our finite-volume computer code STHAMAS. Up to the pressure of 0.6 bar we are considering laminar convection. For the pressure range from 5 to 10 bar we are using the buoyancy extended standard k-ε turbulence model with wall functions to simulate the gas flow. The numerical results show that the Argon gas pressure has a strong influence on the consumption of heater power in qualitative agreement with our experimental results. The convex curvature of the growth interface and the maximum thermal stress (von Mises criterion) are found to increase with increasing gas pressure both in the laminar and turbulent evaluations.     

The role of surface and interface structure in crystal growth

Crystal growth occurs at the interface of a crystal and its growth medium. Due to the abrupt termination at the surface, at the interface the properties of the crystal will typically deviate from the bulk and this can affect the growth behaviour. Also the properties of the growth medium at the interface will typically differ from the bulk. In growth from solution, for example, the liquid will show ordering induced by the crystal surface or have a different composition. Here techniques to study such growth interfaces will be discussed together with examples of the effect that the properties of the interface can have on the growth.     

Numerical calculation of marangoni convection in a rectangular open boat

Numerical calculations of the flow caused by thermocapillary forces at the free surface in NaNO₃ and Si melts in an open boat geometry have been carried out. The results agree well with measured flow velocities in model experiments for crystal growth configurations with NaNO₃ melts. For Si melts a transition to a more complicated multiple eddy flow occurs. The projection-iteration method used for the calculations is much more effective than the finite difference methods applied until now and is especially advantageous for more complicated flow patterns.     

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)