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)     

Three‐dimensional numerical investigation of the effects of an open window on the convective heat transfer of an oxide Czochralski system

A three‐dimensional numerical analysis was carried out for a real Czochralski crystal growth furnace containing only gas and without any melt and crystal in order to investigate the effects of a small observation window on the temperature and flow field of the system. For this approach, the induction heating equations, the Navier‐Stokes equation with Boussinesq approximation, the continuity and energy equations have been solved in cylindrical coordinates using the finite element method. It has been found that the flow and thermal fields in the system are obviously three‐dimensional and non‐axisymmetric. The gas enters the system through the window is directed towards the opposite side wall where it is divided into two parts of vertical direction as well as expands in horizontal direction. Consequently, there is a spiral gas flow in the crucible and afterheater which rotates upwards in azimuthal direction along the walls. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 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)     

Numerical investigation of carbon and silicon carbide contamination during the melting process of the Czochralski silicon crystal growth

Carbon contamination in single crystalline silicon is detrimental to the minority carrier lifetime, one of the critical parameters for electronic wafers. In order to study the generation and accumulation of carbon contamination, transient global modeling of heat and mass transport was performed for the melting process of the Czochralski silicon crystal growth. Carbon contamination, caused by the presence of carbon monoxide in argon gas and silicon carbide in the silicon feedstock, was predicted by the fully coupled chemical model; the model included six reactions taking place in the chamber. A simplified model for silicon carbide generation by the reaction between carbon monoxide and solid silicon was proposed using the closest packing assumption for the blocky silicon feedstock. The accumulation of carbon in the melted silicon feedstock during the melting and stabilization stages was predicted. Owing to this initial carbon content in the melt, controlling carbon contamination before the growth stage becomes crucial for reducing the carbon incorporation in a growing crystal.     

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 modelling of instability and supercritical oscillatory states in a Czochralski model system of oxide melts

The motivation for this study is the need for accurate numerical models of melt flow instabilities during Czochralski growth of oxides. Such instabilities can lead to undesirable spiralling shapes of the bulk crystals produced by the growing process. The oxide melts are characterized by Prandtl numbers in the range 5<Pr <20, which makes the oxide melt flow qualitatively different from the intensively studied flows of semiconductors characterized by smaller Prandtl numbers Pr <0.1. At the same time, these flows can be modelled experimentally by many transparent test fluids (e.g. water, silicon oils, salt melts), which have similar Prandtl numbers, but allow one to avoid the extremely high melting‐point temperatures of the oxide materials. Most previous studies of melt instabilities for Prandtl numbers larger than unity suffer from a lack of accuracy that is caused by the use of coarse grids. Recent convergence studies made for a series of simplified problems and for a hydrodynamic model of Czochralski growth showed that for a second order finite volume method reliable stability results can be obtained on grids having at least 100 nodes in the shortest spatial direction. The obvious numerical difficulties call for an extensive benchmark exercise, which is proposed here on the basis of recently published experimental and numerical data, as well as some preliminary results of this study. The calculations presented are performed by two independent numerical approaches, which are based on second‐order finite volume and finite element discretizations. We start our comparison from the steady states, whose parametric dependencies sometimes exhibit turning points and multiplicity. We then compare the critical temperature differences corresponding to the onset of instability, and finally compare calculated supercritical oscillatory states and phase plots. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermocapillary‐buoyancy flow of silicon melt in a shallow annular pool

In order to understand the nature of surface spoke patterns on silicon melt in industrial Czochralski furnaces, a series of unsteady three‐dimensional numerical simulations were conducted for thermocapillary‐buoyancy flow of silicon melt in annular pool (inner radius r_i = 15 mm, outer radius r_o = 50 mm, depth d = 3 mm). The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces either are adiabatic or allow heat transfer in the vertical direction. Results show that a small temperature difference in the radial direction generates steady roll‐cell thermocapillary‐buoyancy flow. With large temperature difference, the simulation can predict three‐dimensional oscillatory flow, which is characterized by spoke patterns traveling in the azimuthal direction. The small vertical heat flux (3 W/cm²) does not have significant effects on the characteristics of this oscillatory flow. Details of the flow and temperature disturbances are discussed and the critical conditions for the onset of the oscillatory flow are determined. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical optimization of czochralski sapphire single crystal growth using orthogonal design method

Crystal quality during Czochralski (Cz) growth is influenced significantly by the convexity of solid/liquid (S/L) interface, which is related to operating conditions, such as Radio‐Frequency (RF) coil position, crystal rotation and crucible rotation. Generally, a flat interface shape is preferred for high‐quality crystal growth. It is difficult to achieve the optimized conditions even from numerical modeling due to the large computational load from examining all of the affecting factors. Orthogonal design/test method, fortunately, provides an efficient way to organize the study of multiple factors with the minimization of computational load. In the paper, this method is adopted to investigate the affecting factors of Cz‐sapphire single crystal growth based on the coupled calculation of thermal field and melt flows. The orthogonal analysis clearly reveals the optimized growth conditions to achieve a relative flat S/L interface under the current ranges of the parameters.     

Influence of internal radiation on the heat transfer during growth of YAG single crystals by the Czochralski method

Heat and mass transfer taking place during growth of Y₃Al₅O₁₂ (YAG) crystals by the Czochralski method, including inner radiation, is analyzed numerically using a Finite Element Method. For inner radiative heat transfer through the crystal the band approximation model and real transmission characteristics, measured from obtained crystals, are used. The results reveal significant differences in temperature and melt flow for YAG crystals doped with different dopands influencing the optical properties of the crystals. When radiative heat transport through the crystal is taken into account the melt‐crystal interface shape is different from that when the radiative transport is not included. Its deflection remains constant over a wide range of crystal rotation rates until it finally rapidly changes in a narrow range of rotation rates. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of crystal growth process of bulk Si and nitrides – a review

Heat and mass transfer during crystal growth of bulk Si and nitrides by using numerical analysis was studied. A three‐dimensional analysis was carried out to investigate temperature distribution and solid‐liquid interface shape of silicon for large‐scale integrated circuits and photovoltaic silicon. The analysis enables prediction of the solid‐liquid interface shape of silicon crystals. The result shows that the interface shape became bevel like structure in the case without crystal rotation. We also carried out analysis of nitrogen transfer in gallium melt during crystal growth of gallium nitride using liquid‐phase epitaxy. The result shows that the growth rate at the center was smaller than that at the center. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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)     

Numerical optimization of thermal field in CdTe crystal growth by travelling heater method

Cadmium telluride (CdTe) and his compounds play a leading role in X‐ray and γ‐ray detector technology. One of the most used methods for growing these crystals is the travelling heater method (THM). The ingots obtained by using this technique show excellent composition uniformity, but the structural quality is affected by the presence of large grains which appear because of large curvatures of the solid‐liquid interface during the solidification process. This numerical work investigate the thermal field and melt convection in CdTe processing by THM in order to elucidate the mechanism of growing these crystals. The influence of the furnace geometry on the interface shape and temperature gradient in liquid is analyzed for samples with small (1 cm) and large (5 cm) diameters. The computations include flow effects on thermal field in the melted zone. The thermal conditions are optimized for THM growth of CdTe crystals at high solidification temperatures. A new multi‐zone furnace configuration for growing crystals of large diameter and flattened interface is proposed in this work.     

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)     

Influence of active afterheater on the induction heating process in oxide Czochralski systems

Different shapes and orientations of an active afterheater for oxide Czochralski crystal growth systems are considered and corresponding results of electromagnetic field and volumetric heat generation have been computed using a finite element method (Flex‐PDE package). For the calculations, the eddy current in the induction coil (i.e. the self‐inductance effect) has been taken into account. The calculation results show the importance of an active afterheater, its shape as well as its geometry and position with respect to the crucible on the heat generation distribution in a CZ growth system. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Measurement of physical properties of DyScO3 melt

Rare earth scandate crystals ReScO₃ (Re=La, Ce, Pr, Nd, Sm, Eu, Gd, Tb and Dy) can be grown from the melt at temperatures of about 2100 °C. The needs of thermal insulation of the whole system are very high in order to reach the melting point and to control the thermal gradients which are required by the Czochralski (Cz) method. The consequence is that in‐vivo system observations are practically almost impossible or very hard to perform. Therefore numerical investigations using a mathematical model of the real system can be very helpful. However, numerical models need some physical properties of the considered real system (e.g. density, viscosity, thermal expansion coefficients, thermal conductivity). Most material properties of high melting oxides are not referenced in the literature or they are incomplete and inaccurate. Because the accuracy of qualitative and quantitative results of numerical simulations depend on the used physical properties, we performed corresponding measurements in an adapted Cz configuration at a temperature around the melting point of DyScO₃ (2060 °C). The results are presented and discussed in this work. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.