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.     

旋转磁场对空间浮区内流场及浓度场的影响

采用全浮区模型数值研究了旋转磁场作用下熔区内热毛细对流流动特性,分析了磁场强度对流场及浓度场的影响.研究发现,无磁场时,熔体内杂质浓度场和流场呈现三涡胞对称振荡特征;温度场主要由扩散作用决定,呈对称分布.旋转磁场作用下,Ma数基本保持不变.当磁场强度B0≤1 mT时,熔体内杂质浓度场和流场与无磁场时结构类似,但旋转磁场的搅拌作用使得熔体内周期性振荡提前出现,且当旋转磁场产生的洛伦兹力相对较大时,表面张力产生的三维振荡对流得到很好地抑制.B0=5 mT时,周向波动被完全抑制,熔区内流场和浓度场呈二维轴对称分布.旋转磁场对熔体流动产生的轴向抑制作用和周向搅拌作用,都有助于熔体流动的稳定性、浓度分布以及温度分布的均匀性,从而有利于高质量晶体的生长.     

Experiments on the oscillatory convection of low Prandtl number liquid in Czochralski configuration for crystal growth with cusp magnetic field

This paper presents results of experiments on the oscillatory convection of mercury in a Czochralski configuration with cusp magnetic field. Temperature fluctuation measurements are carried out to determine the critical Rayleigh number for the onset of time dependent natural convection. The effects of a cusp magnetic field on the supercritical natural convection coupled with rotation of crystal disk are investigated. In the presence of a rotating flow it is found that a cusp magnetic field can induce a new long wave instability and can amplify the temperature fluctuation depending on the magnitude of the relevant flow similarity parameters and the melt aspect ratios. A flow regime diagram for the amplification and damping of the temperature fluctuations is presented to provide an experimental data base for finding optimum growth conditions in the cusp magnetic field Czochralski process.     

轴向磁场对环形浅液池内硅熔体热毛细对流的影响

为了更好地了解轴向磁场对温度梯度作用下Marangoni-热毛细对流的影响,采用有限差分法对环形浅液池内硅熔体制单晶的流动进行了数值模拟.研究了三种不同边界条件下,Ha数分别为0、10、20、30对应下硅熔体内部流动强度和自由表面速度.结果表明,轴向磁场对浅液池内的Marangoni对流、热毛细对流和耦合的Marangoni-热毛细对流都有较好的抑制作用,且随着磁场强度的增强,抑制作用增加,更有利于提高晶体的结晶质量.当磁场强度和底部热流密度一定时,随着水平温度梯度的增加,靠近内壁的流动得到增强,外壁附近流动反而减弱.     

Effect of a static magnetic field on silicon transport in liquid phase diffusion growth of SiGe

Liquid phase diffusion experiments have been performed without and with the application of a 0.4 T static magnetic field using a three‐zone DC furnace system. SiGe crystals were grown from the germanium side for a period of 72 h. Experiments have led to the growth of single crystal sections varying from 0 to 10 mm thicknesses. Examination of the processed samples (single and polycrystalline sections) has shown that the effect of the applied static magnetic field is significant. It alters the temperature distribution in the system, reduces mass transport in the melt, and leads to a much lower growth rate. The initial curved growth interface was slightly flattened under the effect of magnetic field. There were no growth striations in the single crystal sections of the samples. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

不同高径比下浮区晶体生长熔体内对流不稳定性分析

本文通过数值模拟的方法,研究了零重力条件下半浮区液桥内熔体热毛细对流的演化规律。在液桥的高度L和温差ΔT保持不变的情况下,通过改变液桥的半径R来改变液桥的高径比(Ar=L/R)。随着高径比Ar的变化,液桥内的对流表现出不同的流动特征。在Ar=0.5时,热毛细对流处于三维稳态;在Ar=1时,流场和温度场从稳态模式向非稳态周期多频振荡模式转变,它们之间的频率关系满足倍频关系(f_n=nf₁);在Ar=1.25时,监测点的速度振荡频率增大,表现为较小幅度的振荡模式,且温度振荡消失。     

Effect of magnetic field on melt flow and crystal growth of oxide crystals

The flow in an oxide melt such as LiNbO, and TiO₂ in a high magnetic field was observed by using magnetic-field-applied Czochralski equipment for oxide crystals. It was found that the flows in oxides melts were very much different from these in a semiconductor melt. The single crystals of TiO₂ were grown in a magnetic field by using this equipment.     

Bifurcation analysis of the thermocapillary convection in cylindrical liquid bridges

We consider the instability of the steady, axisymmetric thermocapillary convection in cylindrical liquid bridges. Finite-difference method is applied to compute the steady axisymmetric basic solutions, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for liquid bridges of O(1) aspect ratio Γ (= length/radius) the first instability of the basic state is through either a regular bifurcation (stationary) or Hopf bifurcation (oscillatory), depending on the Prandtl number of the liquid. The bifurcation points and the corresponding eigenfunctions are predicted precisely by solving appropriate extended systems of equations. For very small Prandtl numbers, i.e. Pr < 0.06, the instability is of hydrodynamical origin that breaks the azimuthal symmetry of the basic state. The critical Reynolds number, for unit aspect ratio and insulated free surface, tends to be constant, Re_c → 1784, as Pr → 0, the most dangerous mode being m = 2. While for Pr 0.1, the instability takes the form of a pair of hydrothermal waves traveling azimuthally. The most dangerous mode is m = 3 for 0.1 Pr 0.8 and m = 2 for Pr 0.9. Dependence of the critical Reynolds number on other parameters is also presented. Our results confirm in large part the recent linear-theory results of Wanschura et al. [7] and provide a more complete stability diagram for the finite half-zone with a nondeformable free surface.     

Melt stirring effect of a weak magnetic field on crystal growth

Melt stirring effect of a weak magnetic field for the natural convection of liquid metal in an electrically adiabatic cubic enclosure heated from one vertical wall and cooled from an opposing wall was studied by a fully transient three-dimensional numerical analyses and the reasoning for melt stirring effect was clarified from the numerical results. Similar techniques were applied for the melt convection in a cylindrical Czochralski crystal growing crucible with an application of a vertical magnetic field. In a static crucible, central fluid column rotated in a magnetic field and in a rotating crucible, central fluid column did not rotate in a magnetic field. These peculiar characteristics could have been explained due to the Lorentz force.     

The effects of argon gas flow rate and furnace pressure on oxygen concentration in Czochralski silicon single crystals grown in a transverse magnetic field

The effects of the argon gas flow rate and furnace pressure on the oxygen concentration in a transverse magnetic field applied Czochralski (TMCZ) silicon single crystals were examined through experimental crystal growth. A gas controller which had been proposed by Zulehner was used for this series of experiments. In the TMCZ gas-controlled crystals, a decrease in the oxygen concentration with a decrease in furnace pressure was found. A clear relationship between the oxygen concentration and the argon gas flow rate was not obtained due to the limited experimental conditions. The relationships between the oxygen concentration and the furnace pressure and the argon gas flow rate previously observed for Czochralski (CZ) crystals by a similar gas controller were confirmed by the present gas controller. The oxygen concentration changes in the TMCZ and the CZ crystals were analyzed in terms of the calculated flow velocity of the argon gas between the gas controller and the silicon melt surface. In contrast with the CZ gas-controlled crystals, the oxygen concentration was decreased with an increase in the flow velocity of argon gas in the TMCZ gas-controlled crystals. The surface temperature model and the melt flow pattern model which had been proposed in the previous report are discussed again in light of the present experimental results.     

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)     

Silicon transport under rotating and combined magnetic fields in liquid phase diffusion growth of SiGe

The effect of applied rotating and combined (rotating and static) magnetic fields on silicon transport during the liquid phase diffusion growth of SiGe was experimentally studied. 72‐hour growth periods produced some single crystal sections. Single and polycrystalline sections of the processed samples were examined for silicon composition. Results show that the application of a rotating magnetic field enhances silicon transport in the melt. It also has a slight positive effect on flattening the initial growth interface. For comparison, growth experiments were also conducted under combined (rotating and static) magnetic fields. The processed samples revealed that the addition of static field altered the thermal characteristics of the system significantly and led to a complete melt back of the germanium seed. Silicon transport in the melt was also enhanced under combined fields compared with experiments with no magnetic field. (© 2010 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.     

Influence of magnetic field on the nucleation rate control of mono paracetamol

Crystallization of paracetamol was carried out under magnetic field in pure aqueous solution at ambient condition for the first time. Solutions at different supersaturation levels σ = 0.28–1.6 were exposed to various magnetic flux densities in the range 160–510 Gauss and the grown crystals were compared with that grown in the absence of magnetic field. Both in the absence and presence of magnetic field, solution yields only stable mono paracetamol and does not favors the metastable ortho polymorph at all supersaturation levels. The presence of magnetic field reduces the rate of crystal nucleations as well as modifies the habit of the nucleated mono paracetamol. Crystals obtained in the presence of magnetic field were of good quality and highly transparent as compared to the crystals grown in the absence of magnetic field. The induction period of the nucleation in solution decreases significantly with increase in magnetic flux density at all supersaturation ranges. The internal structure and thermal stability of the grown paracetamol crystal was confirmed by PXRD and DSC analysis.     

Influence of reduced working frequencies on the floating‐zone growth of silicon single crystals

Lowering the working frequency in the inductively heated floating zone growth of Si Single crystals will reduce the risk of arcing at the induction coil. This is of particular interest in the growth of large diameter crystals. In the current paper we present results from growth experiments at lower frequencies, 2 MHz and 1.7 MHz. It is found that the growth of dislocation‐free crystals is possible at these frequencies and cause distinct changes in the interface deflection and radial resistivity profiles. Results from numerical simulation of the melt flow at different frequencies are presented. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of magnetic field effect on pressure in cylindrical and hemispherical silicon CZ crystal growth

The effect of axial magnetic field of different intensities on pressure in silicon Czochralski crystal growth is investigated in cylindrical and hemispherical geometries with rotating crystal and crucible and thermocapillary convection. As one important thermodynamic variable, the pressure is found to be more sensitive than temperature to magnetic field with strong dependence upon the vorticity field. The pressure at the triple point is proposed as a convenient parameter to control the homogeneity of the grown crystal. With a gradual increase of the magnetic field intensity the convection effect can be reduced without thermal fluctuations in the silicon melt. An evaluation of the magnetic interaction parameter critical value corresponding to flow, pressure and temperature homogenization leads to the important result that a relatively low axial magnetic field is required for the spherical system comparatively to the cylindrical one.     

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.     

Instability of the melt flow in VGF growth with a traveling magnetic field

The linear instability of a thermally stratified melt flow in the VGF configuration driven by a traveling magnetic field (TMF) is considered numerically and experimentally. The dependency of the instability threshold on the governing parameters is found for several cuts through the parameter space covering a wide range of possible applications. In a first approximation the linear instability occurs when the dimensionless TMF forcing parameter reaches the magnitude of the Grashof number. This is particularly true in a medium-sized crucible where the first instability is axisymmetric and sub-critical. As the Grashof number increases the flow develops self-similar boundary layers and the instability becomes three-dimensional. The instability originates in the bottom boundary layer where the convection tends to suppress the imposed temperature gradient in the central part of the melt zone. It is shown that the TMF may serve as a tool to control the phase interface shape without causing flow instationarity when the crucible diameter exceeds a certain value. This value is estimated to be around 6 cm for GaAs. The flow stays stable if the TMF is used for a reversal of the meridional flow with the aim to remove a possible dopant concentration peak on the axis.     

Application of rotating magnetic fields in Czochralski crystal growth

The physical principles of electromagnetic stirring with a rotating magnetic field are explained and a mathematical model to calculate the electromagnetic volume force, the fluid flow and the transport of heat and solutes is outlined. For the electromagnetic volume force and for the order of magnitude of the flow velocities approximative analytical expressions are given. A high flexibility in configuring the volume force in order to achieve a desired flow distribution is obtained by multi-frequency stirring that is by superposition of two or several magnetic fields with different frequencies and/or sense of rotation. Results of experimental investigation and mathematical simulation of multi-frequency stirring are given. Numerical simulation of the fluid flow, the temperature and the oxygen distribution in a Czochralski process crucible was performed including the effect of single mode and multi-frequency stirring. The results indicate that electromagnetic stirring should offer large potentials for the optimization of the flow configuration in a Czochralski process crucible. Finally examples from literature of practical application of rotating magnetic fields in crystal growth are presented.