Global simulation of coupled carbon and oxygen transport in a Czochralski furnace for silicon crystal growth

For accurate prediction of carbon and oxygen impurities in a single crystal produced by the Czochralski method, global simulation of coupled oxygen and carbon transport in the whole furnace was implemented. Both gas-phase transportation and liquid-phase transportation of oxygen and carbon were considered. With five chemical reactions considered, SiO and CO concentrations in gas and C and O atom concentrations in silicon melt were solved simultaneously. The simulation results show good agreement with experimental data.     

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.     

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)     

Large eddy simulation of Marangoni convection in Czochralski crystal growth

Large eddy simulation model is used to simulate the fluid flow and heat transfer in an industrial Czochralski crystal growth system. The influence of Marangoni convection on the growth process is discussed. The simulation results agree well with experiment, which indicates that large eddy simulation is capable of capturing the temperature fluctuations in the melt. As the Marangoni number increases, the radial velocity along the free surface is strengthened, which makes the flow pattern shift from circumferential to spiral. At the same time, the surface tension reinforces the natural convection and forces the isotherms to curve downwards. It can also be seen from the simulation that a secondary vortex and the Ekman layer are generated. All these physical phenomena induced by Marangoni convection have great impacts on the shape of the growth interface and thus the quality of the crystal. (© 2011 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)     

Characterization of a Czochralski grown silicon crystal doped with 1020 cm‐3 germanium

A dislocation‐free silicon single crystal doped with 10²⁰ cm^‐3 germanium (Ge) has been grown using the Czochralski (CZ) growth technique. The Ge concentration in the seed‐end and tang‐end of the crystal was 8×10¹⁹cm^‐3and 1.6×10²⁰ cm^‐3, respectively. The effective segregation coefficient of Ge, the distribution of flow pattern defects (FPDs) and the wafer warpage have been characterized. Both the effective segregation coefficient and the equilibrium segregation coefficient of Ge in silicon were evaluated. Then, the density of FPDs was traced from seed‐end to tang‐end of the ingot, a suppression of FPDs by Ge doping was shown. That is probably because the Ge atoms consume free vacancies and thus a higher density of smaller voids is formed. Furthermore, the mechanical strength of wafers has also been characterized by batch warpage analysis. The warpage in the seed‐end was larger than that in the tang‐end of the ingot, showing that the mechanical strength of wafers is enhanced by Ge doping. Such improvement is interpreted by an enhanced dislocation pinning effect associated with the enhanced nucleation of grown‐in oxygen precipitates in the Ge‐doped silicon wafers. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于数值模拟的CZSi单晶收尾控制参数研究

采用理论分析和有限元数值模拟相结合的方法研究了晶体收尾过程中的温度分布与热量传递.解释了"倒草帽"形收尾(收尾前段晶体直径猛然收缩)的成因,指出了改进收尾形状的方法,研究结果对于合理设定直拉硅单晶收尾工艺参数具有一定的参考意义.通过对比试验,仿真计算结果与单晶炉实验测试结果相吻合.     

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)     

Three‐dimensional study of the pressure field and advantages of hemispherical crucible in silicon Czochralski crystal growth

The effects of several growth parameters in cylindrical and spherical Czochralski crystal process are studied numerically and particularly, we focus on the influence of the pressure field. We present a set of three‐dimensional computational simulations using the finite volume package Fluent in two different geometries, a new geometry as cylindro‐spherical and the traditional configuration as cylindro‐cylindrical. We found that the evolution of pressure which is has not been studied before; this important function is strongly related to the vorticity in the bulk flow, the free surface and the growth interface. It seems that the pressure is more sensitive to the breaking of symmetry than the other properties that characterize the crystal growth as temperature or velocity fields. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boron segregation control in silicon crystal ingots grown in Czochralski process

Boron-doped silicon single crystals of 207 mm diameter with various growing conditions are grown from a large amount of the melt in the cusp-magnetic Czochralski method, and the effects of growing parameters on dopant concentrations in the crystals are experimentally investigated. Equilibrium distribution coefficient of boron calculated by BPS model is 0.716. With the crystal rotation (ω) of 13 rpm and the crucible rotation of , the effective distribution coefficient (k_e) is 0.751 in zero magnetic strength and increases up to 0.78 in the magnetic strength of 640 G. For , there is no significant influence of ω on k_e. With , k_e is almost unity. The experimental results are compared with theory.     

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)     

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.     

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.     

Combined effects of crucible geometry and Marangoni convection on silicon Czochralski crystal growth

In order to understand the influence of crucible geometry combined with natural convection and Marangoni convection on melt flow pattern, temperature and pressure fields in silicon Czochralski crystal growth process, a set of numerical simulations was conducted. We carry out calculation enable us to determine temperature, pressure and velocity fields in function of Grashof and Marangoni numbers. The essential results show that the hemispherical geometry of crucible seems to be adapted for the growth of a good quality crystal and the pressure field is strongly affected by natural and Marangoni convection and it is more sensitive than temperature. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two and three‐dimensional numerical modeling of induction heating in oxide Czochralski systems

In this article we report on a set of two‐dimensional and three‐dimensional numerical calculations for three different oxide Czochralski configurations in order to compare the results of the electromagnetic fields and the heat generation distribution. Two configurations without and with a gap between the crucible and active afterheater have axisymmetric conditions while the configuration with an open observation window in the afterheater is characterized by a non‐axisymmetric geometry. It has been found that in the non‐axisymmetric configuration under the influence of the observation window is located at the crucible side wall after a short distance behind the window cut out. Besides this influence the volumetric distribution of heat generation in the system is about symmetric. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

直拉单晶硅生长时空洞演化的相场模拟

为研究大直径直拉硅生长时空洞的演化规律,建立了与有限元模型所模拟的晶体生长温度场相结合的空洞演化相场模型,并应用该模型模拟研究了空洞形貌及其分布状态的变化过程以及不同初始点缺陷浓度对空洞演化的影响规律.结果表明:直拉硅单晶生长过程中,空洞的演化经历了孕育-形核-长大-稳定四个阶段,其形貌和分布状态亦由孤立的球形向偏聚的串珠形转变;与较低的点缺陷浓度相比,初始点缺陷浓度较高时,空洞的数目、平均尺寸、面积分数普遍较大,孕育阶段缩短、形核和长大阶段延长;空洞的偏聚及合并、长大的现象显著;当温度低于980 K时,大直径的空洞数目不再增加.     

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.     

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)     

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)