Numerical modelling of heat transfer in GaSb Czochralski crystal growth (CZ and LEC)

By the hand of a system-oriented steady-state model of the GaSb crystal growth (CZ and LEC) the crystal-melt interface, the heat losses due to cooling, and the required crucible temperatures are calculated as a response to the boundary conditions for temperature and heat flows, and to the chosen geometry. Of special interest is a phantom of an after-heater influencing the cooling of the crystal.     

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.     

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.     

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.     

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.     

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)     

Thermal convection and related instabilities in models of crystal growth from the melt on earth and in microgravity: Past history and current status

The paper presents a comparative study of a number of theoretical/experimental/numerical results concerning the dynamics of natural (gravitational), Marangoni and related mixed convection in various geometrical models of widely‐used technologies for the production of single‐crystalline materials (Horizontal and vertical Bridgman growth, Czochralski method, Floating Zone Technique). Emphasis is given to fundamental knowledge provided over the years by landmark analyses as well as to very recent contributions. Such a knowledge is of paramount importance since it is validating new, more complex models, accelerating the current trend towards predictable and reproducible phenomena and finally providing an adequate scientific foundation to industrial processes which are still conducted on a largely empirical basis. A deductive approach is followed with fluid‐dynamic systems of growing complexity being treated as the discussion progresses. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Transient and quasi‐stationary simulation of heat and mass transfer in Czochralski silicon crystal growth

The formation of grown‐in defects in silicon crystals is controlled by the concentration of intrinsic point defects. Under steady state conditions the type of the prevailing point defect species is linked to the ratio of pull rate and temperature gradient in the crystal at the solidification front. It has been shown that this ratio as well as computed point defect distributions are in good agreement with experimental data. In this paper we compare a coupled transient heat transfer and transient point defect transport model with quasi steady state simulations at various time steps. Both simulations show the same qualitative results, quantitative differences in temperature are less than 1 %. But already for constant pull rates the defect distributions show qualitative differences between transient and quasi steady state simulations. Therefore, for a detailed understanding how defects are related to growth conditions, the thermal history should not be neglected.     

Numerical investigation of heat transport and fluid flow during the seeding process of oxide Czochralski crystal growth Part 2: rotating seed

In this paper, the role of seed rotation on the characteristics of the two‐dimensional temperature and flow field in the oxide Czochralski crystal growth system has been studied numerically for the seeding process. Based on the finite element method, a set of two‐dimensional quasi‐steady state numerical simulations were carried out to analyze the seed‐melt interface shape and heat transfer mechanism in a Czochralski furnace with different seed rotation rates: ω_seed = 5‐30 rpm. The results presented here demonstrate the important role played by the seed rotation for influencing the shape of the seed‐melt interface during the seeding process. The seed‐melt interface shape is quite sensitive to the convective heat transfer in the melt and gaseous domain. When the local flow close to the seed‐melt interface is formed mainly due to the natural convection and the Marangoni effect, the interface becomes convex towards the melt. When the local flow under the seed‐melt interface is of forced convection flow type (seed rotation), the interface becomes more concave towards the melt as the seed rotation rate (ω_seed) is increased. A linear variation of the interface deflection with respect to the seed rotation rate has been found, too. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of heat transport and fluid flow during the seeding process of oxide Czochralski crystal growth Part 1: non‐rotating seed

For the seeding process of oxide Czochralski crystal growth, the flow and temperature field of the system as well as the seed‐melt interface shape have been studied numerically using the finite element method. The configuration usually used initially in a real Czochralski crystal growth process consists of a crucible, active afterheater, induction coil with two parts, insulation, melt, gas and non‐rotating 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 source the fluid flow and temperature field were determined in the whole system. We have considered two cases with respect to the seed position: (1) before and (2) after seed touch with the melt. It was observed that in the case of no seed rotation (ω_seed = 0), the flow pattern in the bulk melt consists of a single circulation of a slow moving fluid. In the gas domain, there are different types of flow motion related to different positions of the seed crystal. In the case of touched seed, the seed‐melt interface has a deep conic shape towards the melt. It was shown that an active afterheater and its location with respect to the crucible, influences markedly the temperature and flow field of the gas phase in the system and partly in the melt. (© 2007 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.     

Change of symmetry and rotation of thermal field as a new method of control of heat and mass transfer in crystal growth (by example of β-BaB<Subscript>2</Subscript>O<Subscript>4</Subscript>)

It is suggested to change the symmetry and rotation of thermal field as a method of contact-free control of the heat and mass transfer in crystal growth. By the example of growth of the low-temperature barium borate (β-BaB₂O₄) phase, a technically important crystal with nonlinear-optical properties, it is shown that the use of the suggested method allows one to grow larger crystals of a higher quality.