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)     

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 flow in a thin annular layer of silicon melt with bidirectional temperature gradients

Bidirectional temperature gradients coexist virtually in surface tension driven flows. However, the simulations have been performed to the flow with only one temperature gradient. A series of 3 D numerical simulations are conducted to investigate the Marangoni‐thermocapillary flow of silicon melt in a thin annular layer with bidirectional temperature gradients. The temperature gradients are produced by the temperature difference ΔT between walls and the constant heat flux q on the bottom, respectively. When changing q, the melt presents different state evolutions at different ΔT. Furthermore, two critical q are found, one makes the minimum melt temperature higher than the crystallization temperature and the other makes the flow unsteady. Both of the critical heat fluxes decrease with increasing ΔT. q contributes more to the elevation of the melt temperature, while ΔT contributes more to the enhancement of the melt instability. In addition, the melt on the free surface flows mainly along the radial direction.     

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)     

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)     

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)     

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)     

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.     

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)     

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 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 study of influences of buoyancy and solutal Marangoni convection on flow structures in a germanium‐silicon floating zone

This paper presents a numerical study of Marangoni flows in a floating zone of germanium‐silicon crystals, which was performed by using a commercial finite element program FIDAD^TM. The numerical results point out that for fluids with a small Pr number the influence of buoyancy forces cannot be ignored in the numerical model. Furthermore, the competition between the thermocapillary (TC) and solutocapillary (SC) flows in the floating zones was qualitatively examined. If the TC flow is as strong as that in the Si‐rich floating zone, the SC flow may be restricted to the bottom area near the free surface. Otherwise, the SC flow may overcome the TC flow and induce a surface transfer of species. The numerical predictions agree well with the previous experiment results. (© 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)     

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)     

Numerical Simulation of Temperature and Flow Field in the Melt for the Vapour‐pressure‐ controlled Czochralski Growth of GaAs

The influence of the melt flow on the temperature field and interface during the vapour‐pressure‐controlled growth of GaAs was studied numerically with the commercial general‐purpose program FIDAP^TM. The thermal boundary conditions for the domain of seed, crystal, boron oxide and crucible were taken from a global calculation for an equipment used at the IKZ to grow 6″ crystals. Due to the large melt volume the buoyancy forces become rather strong and have to be counteracted by reasonable rotation rates. Preliminary results have been obtained for iso‐ and counter‐rotation showing that the flow field exhibits structures on small scales. High rotation rates are needed to counteract the buoyancy flow efficiently and to achieve a smooth flat interface. Even if the the flow structure is not resolved in detail, the interface shape can be deduced form the calculations.     

勾形磁场下提拉法生产单晶硅的数值模拟

本文给出了提拉单晶硅时,勾形磁场强度的计算公式,并对单晶硅在有无勾形磁场情况下熔体内流场和氧的浓度分布进行了数值模拟,计算出磁场作用下磁场强度和洛伦兹力及有无磁场时流函数、垂直截面处的速度场和氧的浓度分布.通过分析表明,勾形磁场能使流动更为平稳,能有效地降低熔体内及生长界面氧的浓度,并对产生这一现象的机理作了理论分析.     

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.     

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 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.