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.     

Numerical simulation of thermal and mass transport during Czochralski crystal growth of sapphire

The thermal and flow transport in an inductively heated Czochralski crystal growth furnace during a crystal growth process is investigated numerically. The temperature and flow fields inside the furnace, coupled with the heat generation in the iridium crucible induced by the electromagnetic field generated by the RF coil, are computed. The results indicate that for an RF coil fixed in position during the growth process, although the maximum value of the magnetic, temperature and velocity fields decrease, the convexity of the crystal‐melt interface increases for longer crystal growth lengths. The convexity of the crystal‐melt interface and the power consumption can be reduced by adjusting the relative position between the crucible and the induction coil during growth. (© 2010 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.     

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)     

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.     

Connection of the Thermocapillary Flow Characteristics and the Impurity Distribution Pattern in Floating Zone Molten Molybdenum Single Crystals

The results of the impurity distribution in W-doped Molybdenum single crystals grown by electron beam floating zone melting are related to the characteristics of computed steady thermocapillary flow and impurity distribution within the molten zone. Particularly the influence of the number and the strength of eddies in the molten zone on the impurity distribution pattern in the grown crystal for different aspect ratios is considered.     

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 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)     

System of Mathematical Models for the Analysis of Industrial FZ-Si-Crystal Growth Processes

A system of coupled mathematical models and the corresponding program package is developed to study the interface shape, heat transfer, thermal stresses, fluid flow as well as the transient dopant segregation in the floating zone (FZ) growth of large silicon crystals (diameter more than 100mm) grown by the needle-eye technique. The floating zone method with needle-eye technique is used to produce high-purity silicon single crystals for semiconductor devices to overcome the problems resulting from the use of crucibles. The high frequency electric current induced by the pancake induction coil, the temperature gradients and the feed/crystal rotation determine the free surface shape of the molten zone and cause the fluid motion. The quality of the growing crystal depends on the shape of the growth interface, the temperature gradients and corresponding thermal stresses in the single crystal, the fluid flow, and especially on the dopant segregation near the growth interface. From the calculated transient dopant concentration fields in the molten zone the macroscopic and microscopic resistivity distribution in the single crystal is derived. The numerical results of the resistivity distributions are compared with the resistivity distributions measured in the grown crystal.     

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。     

Influence of the crucible bottom shape on the heat transport and fluid flow during the seeding process of oxide Czochralski crystal growth

For the seeding process of oxide Czochralski crystal growth, influence of the crucible bottom shape on the heat generation, temperature and flow field of the system and the seed‐melt interface shape have been studied numerically using the finite element method. The configuration usually used in a real Czochralski crystal growth process consists of a crucible, active afterheater, induction coil with two parts, insulation, melt, gas and 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 generation in the crucible wall as a source the fluid flow and temperature field of the entire system as well as the seed‐melt interface shape were determined. We have considered two cases, flat and rounded crucible bottom shape. It was observed that using a crucible with a rounded bottom has several advantages such as: (i) The position of the heat generation maximum at the crucible side wall moves upwards, compared to the flat bottom shape. (ii) The location of the temperature maximum at the crucible side wall rises and as a result the temperature gradient along the melt surface increases. (iii) The streamlines of the melt flow are parallel to the crucible bottom and have a curved shape which is similar to the rounded bottom shape. These important features lead to increasing thermal convection in the system and influence the velocity field in the melt and gas domain which help preventing some serious growth problems such as spiral growth. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of induction heating and heat transfer in a SiC growth system

A global simulation model is applied for a silicon carbide growth system heated by induction coils. A finite‐volume method (FVM) and a global model are applied to solve the equations for electromagnetic field, conductive and radiative heat transfer. The growth rate is predicted by Hertz‐Knudsen equation and one‐dimensional mass transfer equation. Further, simulations for five different coil positions are carried out to investigate the effects of coil position on temperature distribution in the furnace. The numerical results reveal that the variation of temperature in the radial direction along the substrate surface and the temperature difference between the powder and substrate are greatly affected by the coil position. The predicted growth rate along the substrate surface for five coil positions is also studied. Finally, a reasonable range of coil positions maintaining a balance between large‐diameter crystal, high growth rate, temperature limitation of material and lower electrical power consumption is obtained. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of the dopant segregation effects at the floating zone growth of large silicon crystals

A computer simulation is carried out to study the dopant concentration fields in the molten zone and in the growing crystal for the floating zone (FZ) growth of large (> 100 mm) Si crystals with the needle-eye technique and with feed/crystal rotation. The mathematical model developed in the previous work is used to calculate the shape of the molten zone and the velocity field in the melt. The influence of melt convection on the dopant concentration field is considered. The significance of the rotation scheme of the feed rod and crystal on the dopant distribution is investigated. The calculated dopant concentration directly at the growth interface is used to determine the normalized lateral resistivity distribution in the single crystal. The calculated resistivity distributions are compared with lateral spreading resistivity measurements in the single crystal.     

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.     

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.     

Direct observation and numerical simulation of molten silicon flow during crystal growth under magnetic fields by x-ray radiography and large-scale computation

Control of melt flow during Czochralski (CZ) crystal growth by application of magnetic fields is an important technique for large-diameter (>300 mm) silicon single crystals. Melt convection under magnetic fields is an interesting problem for electromagnetic-hydrodynamics. This paper reviews the effects of a vertical magnetic field and a cusp-shaped magnetic field on melt flow during CZ crystal growth. Melt flow in vertical magnetic fields or cusp-shaped magnetic fields was investigated by the direct observation method based on X-ray radiography and by numerical simulation. The first part of this review shows the result of direct observation of molten silicon flow under magnetic fields. It also compares the results of experimental and numerical simulation. The second part shows the details of the numerical simulation of the behavior of molten silicon in magnetic fields.     

Singular ring‐shaped distribution of Nd in NdxY1‐xVO4 crystals grown by floating zone method

A singular ring‐shaped distribution of high Nd concentration was observed in Nd‐doped YVO₄ single crystals grown by the floating zone (FZ) method. The ring‐shaped distribution appeared 500‐1000 μm inside from the rim of the crystals. Results of growth experiments by the anisotropic heating floating zone (AHFZ) method showed that the Nd concentration was high at the high‐temperature side of the grown crystals. We found a small concave projection at a part of the convex solid‐liquid interface by quenching the molten zone during growth. The cause of the singular ring‐shaped distribution of the Nd‐rich area was discussed in relation with the concave projection at the interface and the convection in the molten zone. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Three-dimensional thermocapillary and buoyancy convections and interface shape in horizontal Bridgman crystal growth

Computer simulation is conducted to study three-dimensional (3D) thermocapillary and buoyancy convections and their effects on the growth interface for horizontal Bridgman crystal growth. The free-boundary model is based on a finite volume approximation of continuity, momentum, and energy equations on a collocated grid. Crystal growth of GaAs is used as an example. From calculated results, it is observed that the effect of buoyancy convection on the growth interface is significant. With the thermocapillary effect, the 3D flow structures are not changed much, but its effect on the growth interface is not trivial. Due to the convections, the growth interface is always concave, and its deflection is affected significantly by the growth rate and thermal environment. A simple strategy of interface control is illustrated. Furthermore, slight crucible tilting can also affect the 3D flows leading to an asymmetric growth interface.     

Dislocations and dislocation reduction in space grown GaSb

The behaviour of dislocations in GaSb crystals grown in space both from a stoichiometric melt (floating zone method, FZ) and a Bi solution (floating solution zone, FSZ) respectively, is studied. Predominantly straight 60° dislocations with Burgers vectors of the type b = a/2 <110> in (111) glide planes are identified. In the 20 mm long FZ single crystal the linear growing out of the dislocations is observed which reduces the dislocation density in the centre of the crystal to values below 300 cm^–2. The Bi incorporation in the FSZ crystal results in a misfit between seed and grown crystal and in a network of misfit dislocations at the interface. Thermocapillary convection during growth as well as the surface tension may be the reasons for the presence of curved dislocations and the higher dislocation density within a 1 – 2 mm border region at the edges of both of the crystals. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)