Heat transfer and convection in Czochralski growth of large BGO Crystals

In the present work, numerical modeling has been performed to analyze heat transfer and melt convection during bismuth germanate Bi₄Ge₃O₁₂ (BGO) crystal growth by the Czochralski growth method. In addition to global heat-transfer modeling, the suggested model accounts for the radiative heat exchange in the crystal and melt convection together with the crystallization front formation. The model helped to analyze the modification of the growth setup made by including additional heater. The numerical predictions obtained with CGSim software agree well with available experimental data.     

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.     

Physical and physicochemical processes accompanying powder synthesis,growth of PbMoO<Subscript>4</Subscript> crystals,and their annealing in various media: III. Methods of controlling the crystallization front shape

Different methods of controlling the crystallization front shape are considered: variation in a crucible position in a heater, rotation of a crystal, introduction of impurities absorbing melt radiation, and formation of forced convection under dominant free convection in the melt.     

Study of melt dynamics in crystal growth of Pb1−xSnxTe by the inverted Bridgman method

Experiments with crystal growth of Pb_1−xSn_xTe by the inverted Bridgman method confirm the flow transition in the melt through succeeding stages of unsteady (turbulent) convection, periodically unsteady convection, one-vortex steady motion, two-vortex motion and in the end, lack of any detectable convection — in the range of Rayleigh numbers and ratios of melt height to ampoule diameter given by Müller et al. [J. Crystal Growth 70 (1984) 78]. Additionally, quasi-steady axially asymmetric flow between the first two stages has been observed. Its direction is opposite to the direction of the vortex formed in the subsequent stage of crystal growth. The two-vortex flow, beginning when the melt height equals the ampoule radius, is unsteady and periodically changes the liquid-solid interface shape.     

Mathematical modeling and experimental investigation of the effect of temperature gradients on crystallization processes under terrestrial and space conditions

Mathematical modeling of the processes of heat and mass transfer during directed crystallization under terrestrial and space conditions is performed on the basis of experimental data on the temperature distribution (boundary conditions). Convective processes are described by the system of Oberbeck-Boussinesq equations together with the heat-conduction equation (the Stefan problem). A dependence of the intensity of thermal gravitational convection on the radial and axial temperature gradients is established. It is shown that one of the necessary conditions for the growth of homogeneous semiconductor crystals under both terrestrial and zero-gravity (on board spacecraft) conditions is the absence of the free surface of a melt (the Marangoni convection) and optimization of the temperature gradients (first of all, the radial gradient).     

Modeling analysis of liquid encapsulated Czochralski growth of GaAs and InP crystals

The results of three‐dimensional unsteady modeling of melt turbulent convection with prediction of the crystallization front geometry in liquid encapsulated Czochralski growth of InP bulk crystals and vapor pressure controlled Czochralski growth of GaAs bulk crystals are presented. The three‐dimensional model is combined with axisymmetric calculations of heat and mass transfer in the entire furnace. A comprehensive numerical analysis using various two‐dimensional steady and three‐dimensional unsteady models is also performed to explore their possibilities in predicting the melt/crystal interface geometry. The results obtained with different numerical approaches are analyzed and compared with available experimental data. It has been found that three‐dimensional unsteady consideration of heat and mass transfer in the crystallization zone provides a good reproduction of the solidification front geometry for both GaAs and InP crystal growth.     

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)     

Modulating dopant segregation in floating-zone silicon growth in magnetic fields using rotation

The feasibility of modulating dopant segregation using rotation for floating-zone silicon growth in axisymmetric magnetic fields is investigated through computer simulation. In the model, heat and mass transfer, fluid flow, magnetic fields, melt/solid interfaces, and the free surface are solved globally by a robust finte-volume/Newton's method. Different rotation modes, single- and counter-rotations, are applied to the growth under both axial and cusp magnetic fields. Under the magnetic fields, it is observed that dopant mixing is poor in the quiescent core region of the molten zone, and the weak convection there is responsible for the segregation. Under an axial magnetic field, moderate counter-rotation or crystal rotation improves dopant uniformity. However, excess counter-rotation or feed rotation alone results in more complicated flow structures, and thus induces larger radial segregation. For the cusp fields, rotation can enhance more easily the dopant mixing in the core melt and thus improve dopant uniformity.     

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.     

热屏优化对大直径单晶硅生长影响的数值模拟

通过对28英寸热场生长300 mm硅单晶过程中结晶速率、固液界面形状、晶体中热应力及晶体中氧含量的数值计算提出了在该热场条件下热屏的优化方案。数值计算结果表明:对热屏底端与晶体表面和熔体自由液面的距离以及热屏材料(优化前热屏使用单一石墨材料,优化后采用辐射率较高的内壁材料结合反射率较高的外壁材料组成复合式热屏)的优化可以减少主加热器对晶体的热辐射使得固液界面更加平坦,藉此增加结晶速率,减小晶体内热应力和熔体中氧含量。     

Capillary shaping of crystals pulled downward from a melt by the crucibleless AHP method

A model is proposed for capillary shaping of a crystal in the crucibleless variant of the AHP (axial heat flux close to the phase interface) method when the melt in the form of a film flowing over the AHP heater is fed to a meniscus. The meniscus and the film of the melt are described by the same equation with a discontinuous right-hand side. The dependences of the crystal radius and the thickness of the melt film on the parameters of the process are numerically investigated, and the capillary stability of the pulling process is analyzed. It is demonstrated that, in this method, the thickness of the melt layer between the crystal and the heater can be considerably larger than the capillary constant.     

Effect of the growth rate on thermal conditions during the growth of single crystals with melt feeding

An attempt is made to establish a correlation between the radial and axial growth rates and the change in the conditions of heat transfer from a growing crystal to the atmosphere of the water-cooled vacuum furnace for the growth of large alkali halide single crystals. It is found experimentally that an increase in the growth rate leads to an increase in the automatic compensation of the melt temperature by the main heater. In this case, the thickness of the layer of melt condensate on the end face and the lateral surface of the crystal decreases. It is revealed that the possibility of growing infinitely long ingots in the presence of intense melt evaporation is restricted by the possibilities of the heat transport through the boundary between the furnace atmosphere and the cooled furnace walls, onto which melt condensate deposits.     

The effect of mass transfer on the temperature gradient of a crystal growing from melt

The change in the temperature gradient on the crystal side while the rate of crystal growth from melt is varied has long been debated. Abe and Takahashi have recently reported an unambiguous experimental demonstration that the temperature gradient is a decreasing function of the growth rate, which is different from previous theories, experimental results, and widely held notion of other researchers. The present paper provides a theoretical basis for this seemingly peculiar effect of the growth rate on the temperature gradient. The essential matter is the effect of mass transfer, the role of which had been commonly disregarded in old studies. Although the rate of mass transfer is not large compared to that of heat conduction, it is proven that the temperature gradient is subjected to the mass transfer in a definite manner. Our analysis shows that the effect becomes significant when the crystal diameter is large, which is consistent with the experimental observation. Another effect of the mass transfer is the change in the shape of melt/crystal interface. In old studies, the temperature gradient was determined by Stefan's equation; however, this treatment confuses the cause and effect. The temperature gradient should be determined by the fundamental equation of heat conduction. When the gradient is determined in this way, the shape of the melt/crystal interface spontaneously adjusts to satisfy Stefan's equation.     

Nd:YVO4 crystal growth by Czochralski technique with a submerged plate

This paper is to investigate the growth of Nd:YVO₄ (yttrium vanadate) crystal by the modified Czochralski technique with a submerged plate. Numerical studies are performed to examine melt convection and heat transfer during Nd:YVO₄ growth. The attention is paid to study the effects of initial elevation of the submerged plate, crystal diameter, and melt level on melt inclusions. It is found that the increase in crystal rotation rate and crystal diameter, and the decrease in melt level will increase the axial temperature gradient at the edge and in the center of the crystal, and change the interface shape from convex to flat. The experiments are also carried out to confirm the feasibility of the proposed new technique for controlling melt inclusions in Nd:YVO₄ crystal growth.     

半导体单晶生长过程中的位错研究

阐述了现有的半导体单晶位错模型,即临界切应力模型和粘塑性模型的基本理论及应用状况.分析了熔体法单晶生长过程中影响位错产生、增殖的各种因素,以及抑制位错增殖的措施.与熔体不润湿、与晶体热膨胀系数相近的坩埚材料,低位错密度的籽晶可有效地抑制生长晶体的位错密度;固液界面的形状及晶体内的温度梯度是降低位错密度的关键控制因素,而两因素又受到炉膛温度梯度、长晶速率、气体和熔体对流等晶体生长工艺参数的影响.最后,对熔体单晶生长过程的位错研究进行了展望.     

Global simulation of a Czochralski furnace for silicon crystal growth against the assumed thermophysical properties

In order to understand the effects of the thermophysical properties of the melt on the transport phenomena in the Czochralski (Cz) furnace for the single crystal growth of silicon, a set of global analyses of momentum, heat and mass transfer in small Cz furnace (crucible diameter: 7.2 cm, crystal diameter: 3.5 cm, operated in a 10 Torr argon flow environment) was carried out using the finite‐element method. The global analysis assumed a pseudosteady axisymmetric state with laminar flow. The results show that different thermophysical properties will bring different variations of the heater power, the deflection of the melt/crystal interface, the axial temperature gradient in the crystal on the center of the melt/crystal interface and the average oxygen concentration along the melt/crystal interface. The application of the axial magnetic field is insensitive to this effect. This analysis reveals the importance of the determination of the thermophysical property in numerical simulation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ACRT forced convection and its effects on solute segregation and heat and mass transfer during single crystal growth

A numerical simulation study was carried out for CdZnTe vertical Bridgman method crystal growth with the accelerated crucible rotation technique (ACRT). The convection, heat and mass transfer in front of the solid‐liquid interface, and their effects on the solute segregation of the grown crystal can be characterized with the following. ACRT brings about a periodic forced convection in the melt, of which the intensity and the incidence are far above the ones of the natural convection without ACRT. This forced convection is of multiformity due to the changes of the ACRT parameters. It can result in the increases of both the solid‐liquid interface concavity and the temperature gradient of the melt in front of the solid‐liquid interface, of which magnitudes vary from a little to many times as the ACRT wave parameters change. It also enhances the mass transfer in the melt in a great deal, almost results in the complete uniformity of the solute distribution in the melt. With suitable wave parameters, ACRT forced convection decreases the radial solute segregation of the crystal in a great deal, even makes it disappear completely. However, it increases both the axial solute segregation and the radial one notably with bad wave parameters. An excellent single crystal could be gotten, of which the most part is with no segregation, by adjusting both the ACRT wave parameters and the crystal growth control parameters, e.g. the initial temperature of the melt, the temperature gradient, and the crucible withdrawal rate. (© 2008 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.     

Physical and physicochemical processes accompanying powder synthesis,growth of PbMoO<Subscript>4</Subscript> crystals,and their annealing in various media: II. Shape of the crystal-melt interface caused by melt evaporation and supercooling

The temperature distribution in a melt of PbMoO₄ is investigated using a modified thermocouple and a scanning device. It is shown that the PbMoO₄ melt whose thermal field is significantly inhomogeneous at different depths demonstrates a strong (below the melting temperature) supercooling at the surface due to the evaporation. The position of the isotherm corresponding to the melting temperature of PbMoO₄ in the melt is determined by the degree of the melt supercooling at different growth parameters. The results obtained show that the conditions of mass and heat exchange in the melt are mainly determined by convection. The free convection remains dominant even at intense rotation of a crystal and the most thorough thermal isolation of the crystallization unit.     

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.