Crystal growth by the travelling heater method using tapered crucibles and applied rotating magnetic field

Crystal growth experiments were carried out by the Travelling Heater Method using tapered growth ampoules with and without the application of a rotating magnetic field. The objective was to enhance its commercial potential by reducing the size of required seed crystals and increasing the growth rate. To this end, a number of GaSb crystals were grown using either 25 mm diameter straight, or 10 mm to 25 mm tapered growth ampoules. Growth rates of 2 mm/day and 5 mm/day were employed. The effect of rotating magnetic fields of several strengths and frequencies was examined. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of heat transfer for the modified markov method

In this work, we present a theoretical analysis of the behaviour of the temperature profile for the Modified Markov Method. We have assumed different materials and different thicknesses for the insulating element. Comparison of preliminary results with experimental evidences, demonstrate us the reliability of our model. Discussion among our calculated results are provided and future outlooks for the experimental work are fixed. (© 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.     

A method to model the transient performance of high frequency vibration in crystal growth

A general method to quantitatively study the transient fluctuation related to the high frequency vibration in crystal growth is developed. An example during the Czochralski crystal growth has been simulated, and the simulation coincides with the experimental date. On basis of the model, the influence of vibration frequency on the transient fluctuations for temperature and velocity are elucidated. Then an available approach to control the vibration effect is proposed.     

The role of inner and internal radiation on the melt growth of sapphire crystal

In this paper, for an inductively heated Czochralski furnace used to grow sapphire single crystal, influence of the inner (wall‐to‐wall) and crystal internal (bulk) radiation on the characteristics of the growth process such as temperature and flow fields, structure of heat transfer and crystal‐melt interface has been studied numerically using the 2D quasi‐steady state finite element method. The obtained results of global analysis demonstrate a strong dependence of thermal field, heat transport structure and crystal‐melt interface on both types of radiative heat transfer within the growth furnace.     

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)     

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)     

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)     

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.     

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.     

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 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)     

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.     

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)     

Optimization of thermal conditions during crystal growth in a multi‐zone resistance furnace

The thermal condition is one the of most important control parameters for crystal growth. In this paper we present an effective numerical method in detail for optimizing thermal conditions in multi‐zone crystal growth facilities, especially for crystal growth by the float zone (FZ) technique. A furnace function Ω is introduced to integrate the character of a growth furnace into a linear equation system. The desired power distribution can be therefore approached by solving the linear equation system iteratively. An expert systemlike algorithm has been developed in order to obtain a more suitable solution for practical applications. This method was used to investigate thermal parameters for experiments of SiGe/GeSi single crystal growth by the FZ technique. It is an individual program which can be combined with any commercial finite element/finite volume (FE/FV) program such as FIDAP^TM.     

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)     

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 study of radial temperature distribution in the AlN sublimation growth system

A large radial temperature gradient in the AlN sublimation growth system would lead to non‐uniform growth rate along the radial direction and introduce thermal stress in the as grown crystal. In this paper, we have numerically studied the radial thermal uniformity in the crucible of a AlN sublimation growth system. The temperature difference on the source top surface is insignificant while the radial temperature gradient on the lid surface is too large to be neglected. The simulation results showed that the crucible material with a large thermal conductivity is beneficial to obtain a uniform temperature distribution on the lid surface. Moreover, it was found that the temperature gradient on the lid surface decreases with increased lid thickness and decreased top window size.     

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)     

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.