Modelling of the isothermal melt flow due to rotating magnetic fields in crystal growth

The use of a rotating magnetic field promises the feature of a contactless flow control in crystal growth especially for configurations where an increase of the material transport in a definite way is desired. This paper gives the comparison of numerically calculated and experimentally obtained results on the flow due to a rotating magnetic field as well as numerical results on the influence of the field parameters (frequency, amplitude) on the fluid flow in the melt.     

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.     

Study of the hydrodynamic instabilities in a differentially heated horizontal circular cylinder corresponding to a Bridgman growth configuration

We are interested in determining the origin of the instabilities occurring in a metallic liquid (Prandtl number Pr=0.026) contained in horizontal circular cylinders heated from the end-walls. Our approach by direct numerical simulation (DNS) allows the determination of the transition thresholds for different aspect ratios varying from 1.5 to 10 as well as a precise characterization of the nature and structure of the new flow regimes close to the thresholds. In order to understand the mechanisms of flow transition, fluctuating energy analyses close to the threshold have been performed. The main contributions have been determined and localized in the cavity: shear has been found as the main instability factor but the way it acts is different according to the aspect ratio.     

Numerical simulation of MHD rotator action on hydrodynamics and heat transfer in single crystal growth processes

The article presents the results of the mathematical and physical simulations of the influence of a rotating magnetic field (RMF) on the hydrodynamics and heat transfer in processes of large semiconductor single crystal growth in ampoules. Different versions of the RMF are considered, in particular, for symmetric and asymmetric positions of a RMF inductor with regard to the melt in the ampoule, for two counter-rotating magnetic fields, for different geometrical ratios in the “RMF inductor - liquid melt” system, and for different electrical conductivities of the hard walls at their contact with the melt. The interconnection between the distribution of the electromagnetic forces in the liquid volume and the formed velocity patterns, temperature distribution and shape of the solidification front is studied. An original method for the definition of the electromagnetic forces, which considers finite dimensions of the RMF inductor and melt, was used to calculate real conditions of the RMF influence on growth processes. The numerical results obtained are compared to the data of model experiments. Their satisfactory agreement permits us to propose this calculation method for the definition of the optimal parameters of a growth process under specific conditions and to select the most rational type of RMF influence.     

Computational analysis of three-dimensional flow and mass transfer in a non-standard configuration for growth of a KDP crystal

Computational analysis of three-dimensional flow and mass transfer in a non-standard configuration for growth of a KDP crystal was conducted. The results show that the surface shear stress is mainly affected by the inlet velocity, and the distribution of the surface supersaturation is determined by the bulk supersaturation and the inlet velocity. By adjusting the inlet velocity, the homogeneity of surface supersaturation can be improved, which is helpful for reducing the occurrence of inclusions and enhancing the crystal quality. The thickness of solute boundary layer is closely related to the flow intensity, but it is almost free from the impact of the bulk supersaturation.     

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.     

Applicability of LES turbulence modeling for CZ silicon crystal growth systems with traveling magnetic field

To examine the applicability of LES turbulence modeling for CZ silicon crystal growth systems with traveling magnetic fields, LES calculations with Smagorinsky–Lilly turbulence model and van Driest damping at the solid walls are carried out. The program package for the calculations was developed on the basis of the open-source code library OpenFOAM^®${\mathrm{OpenFOAM}}^{\circledR }$. A previously published laboratory model with low temperature melt InGaSn, a 20” crucible, and process parameters corresponding to industrial Czochralski silicon systems is considered. Flow regimes with two crystal and crucible rotation rates and with different strengths of the traveling magnetic field “down” are analyzed. The calculated distributions of averaged temperature and standard temperature deviations are compared with measured ones in the laboratory system, and a relatively good agreement between them is shown. The influence of chosen time steps and grid sizes is analyzed by comparing Fourier spectra of temperature time-autocorrelation functions and temperature spatial distributions, and it is shown that the used moderate meshes of few hundred thousand cells can be applied for practical calculations.     

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.     

Effect of a vertical magnetic field on the dendrite morphology during Bridgman crystal growth of Al–4.5 wt% Cu

The effect of a vertical high magnetic field (up to 10 T) on the dendrite morphology has been investigated during Bridgman growth of Al–4.5 wt%Cu alloys experimentally. It is found that the field causes disorder in dendrites and their tilt in orientation. Along with the increase of the magnetic field and decrease of the growth velocity, the dendrites became broken and orientated in 1 1 1 along the direction of solidification instead of 1 0 0. The field also enlarged the primary dendrite spacing and promoted the branching of the dendrites to form high-order arms. Above phenomena are attributed to the thermoelectromagnetic convection effect and orientation caused by the high magnetic field.     

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.     

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.     

Levitation of metallic melt by using the simultaneous imposition of the alternating and the static magnetic fields

This study developed a new levitation method, which used the simultaneous imposition of static and alternating magnetic fields. Dynamic behavior was measured for pure Cu and pure Ni melts levitated by the proposed method. The oscillation due to surface tension and convection in levitated Cu melts were hardly observed at static magnetic fields exceeding 1 T. Only the rotation of this axis parallel to the static magnetic field was observed under high static magnetic fields. The proposed method demonstrated that metallic melt could be statically levitated like a solid sphere. It was also found that stable levitation of paramagnetic Ni melt was rather difficult at static magnetic fields exceeding 5 T, because of the magnetization force.     

Numerical simulation and validation of the Peltier pulse marking of solid/liquid interfaces

We describe a numerical approach of the solidification of binary alloys to study the motion of a crystal/melt interface submitted to current pulses involving a modification of the dopant concentration field. For the thermal aspect, the Thomson effect, the Peltier effect and Joule heating have been included in the heat flow. For the solutal segregation, our model is based on mass transports which occur in the liquid phase, namely diffusion and convection. Numerical computations are validated by comparison with experimental data and thus could find applications in the prediction of the effects of Peltier pulse marking in crystal growth.     

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.     

Influence of the flow velocity,supersaturation and temperature on the crystal growth from solutions

This paper is a review of our experimental research on the influence of the supersaturation, flow velocity and temperature on the linear crystallization rate of different faces of model crystal systems. The obtained experimental results are discussed in the light of the new theoretical treatments on crystal growth from low temperature solutions.     

Numerical and experimental studies on crack formation in LiNbO3 single crystal

A global analysis of heat transfer was carried out in an inductively heated Czochralski (CZ) furnace which was actually used to grow LiNbO₃ single crystals, and then the temperature profiles obtained were used to calculate the three-dimensional thermal stress field in the crystal. By comparing the numerical results with the experimental ones, it was found that controlling the thermal environment in the CZ furnace so that the thermal stresses at the crystal surface might not exceed a certain value is important to realize the cracking free growth operation. In this study, this was accomplished through some modifications in the furnace design such as insertion of an after-heater into the furnace. These findings were verified by additional numerical simulations and crystal growth experiments for some growth conditions.     

Growth of BiSbTe3 -Mixed Crystals using a Bridgman Configuration of the TITUS Furnace - Numerical Simulation of Convection and Segregation

A standard Bridgman configuration of the TITUS facility was used to grow BiSbTe ₃ -mixed crystals at normal and at reduced gravity. The growth experiments in space, including a dynamical registration of the temperature distribution of the furnace, were performed during the MIR'97 mission. The transient temperature profiles have been analysed to get thermal boundary conditions for numerical simulations of convection and segregation with the FEM package FIDAP. The calculations have been done within the frame of a 3D model close to the real growth conditions. The aim of the paper is to discuss the resulting Buoyancy driven flow configuration in the melt and its influence on the radial and axial segregation depending on the gravity level.     

Axial Solute Transport during Bridgman Growth of BiSbTe3‐Mixed Crystals ‐ a comparison of analytical and numerical results

2D/3D‐transient finite‐element computer simulations of heat and mass transport including convection have been performed for a Bridgman configuration close to real growth conditions. The results for the axial distribution of the excessive tellurium in BiSbTe₃ semiconductor crystals grown from the melt are compared with the predictions of analytical segregation models.It is shown that Favier's model can be successfully applied for quantitatively estimating model parameters of segregation. Finally, the transition from normal gravity to microgravity conditions is discussed.