Czochralski Crystal Growth Research Articles

Model predictive temperature tracking in crystal growth processes

The temperature gradients and distribution evolution within the crystal domain in Czochralski crystal growth process have important role in produced crystal's quality. Precise and tight regulation of temperature distribution and gradients is the most promising approach to ensure the crystal quality. In this work, the coupled crystal pulling dynamics and heat transfer models in Czochralski crystal growth with a time-varying boundary is considered. The moving boundary finite element method is used to obtain the optimal reference temperature trajectory associated with the reference crystal shape taking into account constraints on input and the temperature gradients. The obtained reference trajectory is used to implement a model predictive control strategy to track the reference temperature despite uncertainties in crystal domain geometry evolution and disturbances. Finally, the proposed method is implemented on a high fidelity finite element process model with non-planar interface and results are presented to validate the success of the proposed methodology.

Nonlinear model-based control of the Czochralski process IV: Feedforward control and its interpretation from the crystal grower׳s view

The paper proposes a model-based feedforward control for the Czochralski crystal growth process. It completes the system for nonlinear model-based control presented by the authors in previous publications. From a system theoretical point of view such a feedforward control requires a dynamic lumped parameter model of the hydrodynamical–geometrical as well as of the thermal subsystem of the process. For the heater feedforward control one lacks the latter one with the necessary accuracy. A static approach circumventing this problem is presented here, based on a mathematical description regarding the most important qualitative dependencies between relative changes of manipulated and controlled variables during growth. In combination with feedback control, high precision tracking of the most important system variables, crystal diameter and crystal growth rate, is ensured. Additionally, a comprehensive discussion of the dependencies of the heater feedforward control on technological parameters, growth regimes and crystal shapes is given.

Computational Study of Convective and Radiative Heat Transport in Czochralski Growth of Oxides

For a Czochralski crystal growth of sapphire, influence of radiative heat transfer and gas convection on the temperature field of the system and the crystal-melt interface have been studied numerically using the 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. Comparison between the results presented here and those of previous works demonstrate the importance of heat radiation, gas flow and realistic heat transfer boundary conditions on the thermal field of oxide Czochralski growth systems.

Numerical Modeling of Induction Heating in Crystal Pulling Method Change in the Coil Radius

An induction coil with two different radiuses for oxide Czochralski crystal growth systems is considered and corresponding results of electromagnetic field and volumetric heat generation have been computed using a finite element method (FlexPDE package). The calculation results show the importance of the coil radius on the heat generation distribution in a CZ growth system.

Temperature distribution reconstruction in Czochralski crystal growth process

A mechanical geometric crystal growth model is developed to describe the crystal length and radius evolution. The crystal radius regulation is achieved by feedback linearization and accounts for parametric uncertainty in the crystal growth rate. The associated parabolic partial differential equation (PDE) model of heat conduction is considered over the time‐varying crystal domain and coupled with crystal growth dynamics. An appropriately defined infinite‐dimensional representation of the thermal evolution is derived considering slow time‐varying process effects. The computational framework of the Galerkin's method is used for parabolic PDE order reduction and observer synthesis for temperature distribution reconstruction over the entire crystal domain. It is shown that the proposed observer can be utilized to reconstruct temperature distribution from boundary temperature measurements. The developed observer is implemented on the finite‐element model of the process and demonstrates that despite parametric and geometric uncertainties present in the model, the temperature distribution is reconstructed with the high accuracy. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2839–2852, 2014

Anisotropic study of thermal stresses induced by diameter fluctuation during Czochralski silicon single crystal growth

Instabilities in the Czochralski crystal growth process result in deviation from the desired crystal diameter i.e. uneven crystal surface. The excessive thermoelastic stress induced by diameter perturbation is studied by mean of numerical simulations. A set of 3D simulation has been performed for an axisymmetric crystal. The crystal anisotropy is taken into account. The influence of crystal diameter fluctuation on stress field inside the crystal has been studied. The resolved shear stresses have been calculated for 12 slip systems for three crystal orientations. Accumulated excess stress from its critical value is calculated. Simulation results suggest that crystal surface undulation affects both thermal field and stress distribution inside the crystal. The crystal region with high risk of dislocation generation is discussed for three crystal orientation. The stress level in [111] crystal orientation is found to be lower than other orientations.

Numerical simulation of unsteady flows in Czochralski crystal growth by lattice Boltzmann methods

This study presents model extensions for a lattice Boltzmann (LB) approach to thermal axisymmetric flow including swirl or rotation. An incompressible axisymmetric lattice Boltzmann D2Q9 model was applied to solve the axial and radial velocities through inserting source terms into the two-dimensional lattice Boltzmann equation. The equations governing azimuthal (or swirling) velocity and the temperature were also solved by the LBM. It is found that this scheme is much more stable and consistent compared to previous hybrid schemes. It provides a significant advantage in simulation of melt flows with high Reynolds number and high Grashof number. The present scheme was validated by comparing the LB results with benchmark solutions for melt flow in Czochralski crystal growth. Unsteady flows with high Grashof numbers were studied in detail. The critical Grashof number for the onset of the oscillation is found to be about 2.5×106. The oscillation amplitude ψmax is proportional to (Gr-Grc)0.5 for 2.5×106<Gr<6×106. The frequencies and flow patterns of the unsteady flows are also analyzed. The distributions of the mean quantities of the temperature and rms of temperature at Grashof number as high as 6×107 is found to be similar to those obtained by 3D simulations.

Ultrasonic flow measurements in a model of a Czochralski puller

An experimental study of the buoyancy-induced flow in a model of a Czochralski crystal growth system was conducted. Ultrasonic velocimetry was used to measure fluid velocities. To approach the thermal boundary conditions in an industrial growth facility, a double-walled glass crucible, which is flown through by a heating fluid, was chosen to hold the fluid. Similarity of the heat transfer conditions was achieved by selecting a liquid metal as the fluid under investigation, which was the ternary alloy GaInSn having a Prandtl number of 0.021. Because of the double-walled crucible, measurements through the container wall are difficult if ever possible. Since the availability of relatively short ultrasonic transducers it is practicable to have the sensor immersed into the fluid. Measurements of the radial velocity component shortly below the melt surface across the entire diameter of the crucible at various azimuthal angles reveal the complex flow structure of natural convection in a Czochralski crucible. As it is not to be expected to grow high quality mono-crystalline crystals from such a non-axisymmetric flow, rotating magnetic fields (RMF) are often proposed to render the flow more axisymmetric. This paper also addresses the question what happens to the buoyancy-driven flow when such an RMF is applied.

Solidification Modeling: Evolution, Benchmarks, Trends in Handling Turbulence, and Future Directions

A systematic evolution of the solidification modeling is presented in this article. An approach starting from the basic governing equations to the intricate modeling of the alloy solidification using different approaches has been reviewed. Important advantages and issues related to different formulations and the use of fixed/moving grids for the modeling of solidification have been discussed. This article outlines the important solidification modeling approaches used in the literature. The mathematical description of the most frequently employed methods for modeling of solidification has been presented providing adequate references for other solidification models. This article highlights an important subdomain of solidification modeling, namely, the modeling of solidification processes having significant turbulence (such as welding, casting, and Czochralski crystal growth). A review of the use of different turbulence models along with the state-of-the-art techniques in these areas is presented. The paper also describes the important benchmarking studies (both experimental and numerical modeling results) used for the validation of solidification of both pure metals and alloys. Finally, the physical and numerical complexities associated with the solidification modeling phenomena along with the important challenges and future directions are presented.

The phenomenon of “cold plume” instability in Czochralski hydrodynamic model: Physical and numerical simulation

The oscillatory modes of thermal gravitational-capillary convection were simulated experimentally and numerically in an application to hydrodynamic model of Czochralski crystal growth. The experiments and calculations were carried out with using an ethanol as modelling fluid. The convection was dependent on the various temperature differences between the lateral crucible wall and the disk modelling the crystal: ΔT=1–15K. Numerical simulation was based on complete Navier–Stokes–Boussinesq equations for axisymmetrical and three-dimensional cases. The features of transition from a stationary flow and heat transfer to its unstable modes accompanying with an occurrence and development of temperature pulsations were investigated. The clear graphic example of “cold plume” lifetime (its formation, development and pull-off under a crystal surface) has been shown. The spectral powers of temperature pulsations for different ΔT have been analysed. The unstable hydrodynamic modes were presented in a comparison with corresponding experimental data and its agreement has been established.

Experimental investigation on the electromagnetically controlled buoyancy-induced flow in a model of a Czochralski puller

A great deal of work with respect to both physical and numerical modelling of Czochralski crystal growth has been conducted in the generic Rayleigh-Benard system. In order to come closer to the conditions in a real Czochralski puller, specific effects such as the influence of a rounded crucible bottom, deviations of the thermal boundary conditions from the generic case, crucible and/or crystal rotation, and the influence of magnetic fields are often studied separately. Within this paper we present a model experiment focusing on investigations of the impact of magnetic fields on the flow in a Czochralski puller. To achieve similar thermal boundary conditions as in an industrial growth facility, a double-walled rounded bottom glass crucible was chosen to hold the fluid. Similarity of the heat transfer conditions was guaranteed by selecting the ternary alloy GaInSn as the model fluid. Measurements of the fluid flow have been conducted by means of the ultrasound Doppler velocimetry. The results reveal the complex flow structure of natural convection in a Czochralski crucible. Because the growth of high quality mono-crystalline crystals is impeded by such a non-axisymmetric flow, rotating magnetic fields (RMF) are often proposed to render the flow more axisymmetric. To study the effect of an RMF on the natural convection in a Czochralski system the experimental apparatus was mounted inside the home-made MULTIpurpose MAGnetic field facility (MULTIMAG). In the present contribution the three-dimensional convective patterns as well as the resulting temperature fluctuations will be discussed both for the pure buoyant case and for the application of an RMF.

Review of some Aspects of Single Crystal Growth Using Czochralski Crystal Growth Technique

Abstract Single crystal that semiconductor industry thrive on, are grown using Czochralski (CZ) crystal growth technique. Over the years many researchers have investigated the problems related to CZ crystal growth system and number of studies related to different aspects of CZ crystal growth is available in open literature. Some of the important aspects related to Czochralski (CZ) crystal growth system used for growing single crystal are discussed. Origin of melt convection due to buoyant forces, forced convection due to crystal and crucible rotation and Marangoni convection due to thermo-capillary forces, and their effect on heat and species transport phenomena in the melt pool has been reviewed. Flow in present day large size crystal growth using CZ crystal growth system becomes oscillatory. Oscillations in melt can be damped using externally applied magnetic field. Effect of axial, transverse and inhomogeneous (CUSP) magnetic field on different melt convection has been a subject of interest for various research efforts. Oxygen release from silica crucible during growth of silicon crystal and mechanism for its incorporation into the growing crystal has also been reviewed. Conditions leading to growth of low oxygen concentration silicon crystal are elaborated.

Evolution of interface configuration in sapphire single crystal growth via Czochralski method

The evolution of interface configuration during sapphire crystal growth by Czochralski method was studied on the basis of chemical bonding conditions at crystal/melt interface. On the basis of chemical bonding theory of single crystal growth, when sapphire single crystal was grown along the c axis direction in Czochralski system, crystal/metal interface is composed of six crystallographically equivalent {110} interfaces thermodynamically. With the rotation of sapphire single crystal in the growth process, the interface configuration developed from hexagon to circle due to the appearance of {100} surfaces and {h10} microfacets at the intersection of two adjacent interfaces. The (h10) microfacet is made of (100) and (110) planes. Lower chemical bonding densities at the {100} surfaces and {h10} microfacets relative to that at {110} surfaces thermodynamically drive the formation of round interface configuration in sapphire Czochralski crystal growth along the c axis direction.

Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process

This paper considers the optimal control problem for a class of convection-diffusion-reaction systems modelled by partial differential equations (PDEs) defined on time-varying spatial domains. The class of PDEs is characterised by the presence of a time-dependent convective-transport term which is associated with the time evolution of the spatial domain boundary. The functional analytic description of the PDE yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite-dimensional function space. The properties of the time-varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis. An industrial application of the crystal temperature regulation problem for the Czochralski crystal growth process is considered and numerical simulation results are provided.

A Review of the Automation of the Czochralski Crystal Growth Process

On the occasion of the centennial of the invention of the Czochralski crystal growth process by the Polish scientist Jan Czochralski, a review of selected strategies for the automatic control of this process is given. This review provides a sketch of the fundamental challenges of controlling the Czochralski process and the basic concepts of feedback control. Both early and modern approaches to the control of the Czochralski process are described. The discussion focuses on questions related to feed-forward control, feedback control, and state estimation. The presented methods rely on simple mathematical process models in contrast to the nite element model-based approaches typically used in crystal growth process design and analysis. Such mathematical models motivate both the structure and parameters of the chosen controller. A comprehensive list of references to background literature on this topic completes this survey.

Order‐reduction of parabolic PDEs with time‐varying domain using empirical eigenfunctions

A novel methodology for the order‐reduction of parabolic partial differential equation (PDE) systems with time‐varying domain is explored. In this method, a mapping functional is obtained, which relates the time‐evolution of the solution of a parabolic PDE with time‐varying domain to a fixed reference domain, while preserving space invariant properties of the initial solution ensemble. Subsequently, the Karhunen–Loève decomposition is applied to the solution ensemble on fixed spatial domain resulting in a set of optimal eigenfunctions. Further, the low dimensional set of empirical eigenfunctions is mapped on the original time‐varying domain by an appropriate mapping, resulting in the basis for the construction of the reduced‐order model of the parabolic PDE system with time‐varying domain. This methodology is used in three representative cases, one‐ and two‐dimensional (1‐D and 2‐D) models of nonlinear reaction‐diffusion systems with analytically defined domain evolutions, and the 2‐D model of the Czochralski crystal growth process with nontrivial geometry. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4142–4150, 2013

Model experiments for the Czochralski crystal growth technique

A lot of the physical and the numerical modeling of Czochralski crystal growth is done on the generic Rayleigh-Benard system. To better approximate the conditions in a Czochralski puller, the influences of a rounded crucible bottom, deviations of the thermal boundary conditions from the generic case, crucible and/or crystal rotation, and the influence of magnetic fields are often studied separately. The present contribution reviews some of these topics while concentrating on studies of the flow and related temperature fluctuations in systems where a rotating magnetic field (RMF) was applied. The three-dimensional convective patterns and the resulting temperature fluctuations will be discussed both for the mere buoyant case and for the application of an RMF. It is shown that a system between a Rayleigh-Benard and a more realistic configuration, which is still cylindrical but whose surface is partially covered by a crystal model, behaves much the same as a Rayleigh-Benard system. An RMF can be used to damp the temperature fluctuations. Secondly, a more Czochralski-like system is examined. It turns out that the RMF does not provide the desired damping of the temperature fluctutions in the parameter range considered.

Global simulation of the Czochralski silicon crystal growth in ANSYS FLUENT

Silicon crystals for high efficiency solar cells are produced mainly by the Czochralski (CZ) crystal growth method. Computer simulations of the CZ process established themselves as a basic tool for optimization of the growth process which allows to reduce production costs keeping high quality of the crystalline material. The author shows the application of the general Computational Fluid Dynamics (CFD) code ANSYS FLUENT to solution of the static two-dimensional (2D) axisymmetric global model of the small industrial furnace for growing of silicon crystals with a diameter of 100mm. The presented numerical model is self-sufficient and incorporates the most important physical phenomena of the CZ growth process including latent heat generation during crystallization, crystal–melt interface deflection, turbulent heat and mass transport, oxygen transport, etc. The demonstrated approach allows to find the heater power for the specified pulling rate of the crystal but the obtained power values are smaller than those found in the literature for the studied furnace. However, the described approach is successfully verified with the respect to the heater power by its application for the numerical simulations of the real CZ pullers by “Bosch Solar Energy AG”.

Numerical investigation of magnetic field effect on pressure in cylindrical and hemispherical silicon CZ crystal growth

AbstractThe 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.

Sensitivity analyses of furnace material properties in the Czochralski crystal growth method for silicon

Reliability of the numerical simulation results strongly depends on the input data, in particular on the thermo-physical properties of the furnace components, e.g. graphite thermal conductivity and steel surface emissivity. Uncertainties are always involved in the measurement of these parameters. A set of global 2D simulations has been carried out in order to investigate the impact of each property on the growth conditions in Czochralski silicon growth furnaces. The results indicate that steel emissivity and felt conductivity have significant impact on the calculated thermal field and energy consumption.