3D modeling of doping from the atmosphere in floating zone silicon crystal growth
Abstract Three-dimensional numerical simulations of the inert gas flow, melt flow and dopant transport in both phases are carried out for silicon single crystal growth using the floating zone method. The mathematical model allows to predict the cooling heat flux density at silicon surfaces and realistically describes the dopant transport in case of doping from the atmosphere. A very good agreement with experiment is obtained for the radial resistivity variation profiles by taking into account the temperature dependence of chemical reaction processes at the free surface.
Simulation of the influence of gas flow on melt convection and phase boundaries in FZ silicon single crystal growth
Abstract Axisymmetric calculations of inert gas flow in a floating zone puller are carried out using an open source software package OpenFOAM. Transient axisymmetric melt flow in liquid silicon and quasi-stationary shape of silicon phase boundaries are calculated using a specialized program FZone. Additional heat losses at silicon surfaces caused by the gas flow are taken into account for argon and helium, while maintaining the height of molten zone by adjusting inductor current. Cooling causes an increase of electromagnetic force, heat sources and more intense melt flow, while crystallization interface deflection decreases. The shear stress of gas flow is found to be an order of magnitude …
Analysis of the dopant segregation effects at the floating zone growth of large silicon crystals
Abstract A computer simulation is carried out to study the dopant concentration fields in the molten zone and in the growing crystal for the floating zone (FZ) growth of large (> 100mm) Si crystals with the needle-eye technique and with feed/crystal rotation. The mathematical model developed in the previous work is used to calculate the shape of the molten zone and the velocity field in the melt. The influence of melt convection on the dopant concentration field is considered. The significance of the rotation scheme of the feed rod and crystal on the dopant distribution is investigated. The calculated dopant concentration directly at the growth interface is used to determine the normalized …
Application of enthalpy model for floating zone silicon crystal growth
Abstract A 2D simplified crystal growth model based on the enthalpy method and coupled with a low-frequency harmonic electromagnetic model is developed to simulate the silicon crystal growth near the external triple point (ETP) and crystal melting on the open melting front of a polycrystalline feed rod in FZ crystal growth systems. Simulations of the crystal growth near the ETP show significant influence of the inhomogeneities of the EM power distribution on the crystal growth rate for a 4 in floating zone (FZ) system. The generated growth rate fluctuations are shown to be larger in the system with higher crystal pull rate. Simulations of crystal melting on the open melting front of the pol…
Numerical 3D modelling of turbulent melt flow in a large CZ system with horizontal DC magnetic field. II. Comparison with measurements
This paper presents a comparison between numerically calculated and measured temperature distributions in turbulent flow in a laboratory model for a CZ large silicon single crystal industrial growth system with a horizontal DC magnetic field. The laboratory model consists of an electrically heated 20” crucible with low-temperature InGaSn melt, a water-cooled metallic crystal model, and a magnet system creating a horizontal magnetic field in the range 0–. Distributions of time-averaged temperature values in various cross sections in the melt are obtained from measurements by a multichannel thermocouple system. A 3D numerical model for the scalar potential induced in the melt by the velocity …
Physical modelling of the melt flow during large-diameter silicon single crystal growth
Abstract The reported investigations concern physical modelling of Czochralski growth of silicon large-diameter single crystals. InGaSn eutectic was used as a modelling liquid, employing actual criteria of the real process (Prandtl, Reynolds, Grashof numbers, etc.) and geometric similarity. A multi-channel measuring system was used to collect and process the temperature and flow velocity data. The investigations were focused on the study of heat transfer, in particular, the instability of the “cold zone” of the melt at the crystallization front.
Modelling of thermal field and point defect dynamics during silicon single crystal growth using CZ technique
Abstract Silicon single crystal growth by the Czochralski (CZ) technique is studied numerically using non-stationary mathematical models which allow to predict the evolution of the CZ system in time, including Dash neck, cone and cylindrical growth stages. The focus is on the point defect dynamics, also considering the effect of the thermal stresses. During the cylindrical stage, the crystal pull rate is temporarily reduced as in experiments by Abe et al. The crystal radius and heater power change is explicitly considered in the calculations for crystal diameters of 50, 100 and 200 mm and the agreement with experiments is discussed.
Numerical model of turbulent CZ melt flow in the presence of AC and CUSP magnetic fields and its verification in a laboratory facility
The paper describes a numerical simulation tool for heat and mass transfer processes in large diameter CZ crucibles under the influence of several non-rotating AC and CUSP magnetic fields. Such fields are expected to provide an additional means to influence the melt behaviour, particularly in the industrial growth of large diameter silicon crystals. The simulation tool is based on axisymmetric 2D models for the AC and CUSP magnetic fields in the whole CZ facility and turbulent hydrodynamics, temperature and mass transport in the melt under the influence of the electromagnetic fields. The simulation tool is verified by comparisons to experimental results from a laboratory CZ setup with eutec…
Application of dynamic and combined magnetic fields in the 300mm silicon single-crystal growth
Abstract The increase of diameter in the silicon single crystal growth from 200 to 300 mm for industrial application, and to 400 or 450 mm for research, respectively, has triggered off the development of numerous new technologies like crystal-growth-supporting systems, low-power hot zones, high strength of static magnetic fields and new quartzglas qualities. At Wacker Siltronic, new kinds of magnetic fields have been developed for 300 mm CZ growth. In this paper, the results of dynamic and combined (static and alternating) magnetic fields are discussed. Instead of buoyancy-driven convection, a magnetic-field-controlled melt flow has been obtained in large melt volumes. The crucible wall tem…
Experimental and Numerical Analysis of Air Flow, Heat Transfer and Thermal Comfort in Buildings with Different Heating Systems
Abstract Monitoring of temperature, humidity and air flow velocity is performed in 5 experimental buildings with the inner size of 3×3×3 m3 located in Riga, Latvia. The buildings are equipped with different heating systems, such as an air-air heat pump, air-water heat pump, capillary heating mat on the ceiling and electric heater. Numerical simulation of air flow and heat transfer by convection, conduction and radiation is carried out using OpenFOAM software and compared with experimental data. Results are analysed regarding the temperature and air flow distribution as well as thermal comfort.
Floating Zone Growth of Silicon
Abstract The floating zone (FZ) technique changed from a crucible-free purification method into a growth technique mainly for high purity silicon crystals. The melt zone is inductively heated by the high frequency magnetic field of a sophisticated one-turn induction coil being the heart of the FZ growth. The needle-eye technique allows for crystals with large diameters beyond the capillary limitations of a cylindrical zone, but both electric breakthrough at the coil slit and bursting of the crystal by thermomechanical stress presently limit the diameter to 200 mm. A novel gFZ concept is depicted that works with granular silicon feedstock instead of expensive feed rods. The automation of the…
Mathematical modelling of the feed rod shape in floating zone silicon crystal growth
Abstract A three-dimensional (3D) transient multi-physical model of the feed rod melting in the floating zone (FZ) silicon single-crystal growth process is presented. Coupled temperature, electromagnetic (EM), and melt film simulations are performed for a 4 inch FZ system, and the time evolution of the open melting front is studied. The 3D model uses phase boundaries and parameters from a converged solution of a quasi-stationary axisymmetric (2D) model of the FZ system as initial conditions for the time dependent simulations. A parameter study with different feed rod rotation, crystal pull rates and widths of the inductor main slit is carried out to analyse their influence on the evolution …
Prediction of the growth interface shape in industrial 300mm CZ Si crystal growth
Abstract A model approach for a modification of the effective heat conductivity in the turbulent melt flow simulation for 28″ Si CZ crucibles is presented, which helped to overcome deficiencies in the growth interface shape prediction for industrial 300 mm Si CZ growth. The model has been incorporated into a CZ simulation tool based on the simulation software codes FEMAG for the global heat transfer and CFD-ACE for the turbulent melt flow simulation. The model predictions are compared to results from 300 mm Si CZ growth experiments with 200 kg charge weight in 28″ crucibles in a growth parameter range covered by standard industrial processes. The model is an engineering approach. Neverthele…
Towards a System to Monitor the Virus’s Aerosol-Type Spreading
Recent scientific studies indicate that attention should be paid to the indoor spread of the Covid-19 virus. It is recommended to reduce the number of visitors to the premises and to provide frequent ventilation of the premises. The problem is that it is not known what the risk of infection is in a particular room at a specific time, when and what actions should be taken to reduce the risk. We offer a system that helps monitor the conditions in the premises with the help of sensors, calculate the risk of infection and provide information to reduce the infection risk. We give an insight into the created prototype with data collection from public spaces and data visualization according to use…
Validation of a 3D mathematical model for feed rod melting during floating zone Si crystal growth
Abstract A mathematical model of global 3D heat transfer in floating zone silicon single crystal growth process is used to predict the shape of the open melting front of the feed rod. The model is validated using measurement data from research-scale growth experiments. Shape profiles of the open melting front are obtained from the feed rod leftover using a movable dial gauge. Azimuthal asymmetry of the rim of the open melting front is revealed in both simulations and measurements, quantitatively indicating the influence of the main slit of the inductor.
Numerical analysis of the influence of ultrasonic vibration on crystallization processes
The challenge in the future fabrication of semiconductor bulk crystals is the improvement of the crystal quality with a simultaneous increase of the yield. For that, a proper control of mass transfer within the fluid phase is required. Besides the damping of violent convective fluctuations, the thickness of the diffusion boundary layer, causing morphological instability, has to be decreased. The influence of ultrasound in molten Germanium was analyzed by numerical simulations. The simulations were provided by applying commercial software packages ANSYS ® and FLUENT ® . ANSYS ® was used to model the ultrasonic wave propagation in the whole growth system consisting of melt and crystal, crucib…
Hydrodynamical aspects of the floating zone silicon crystal growth process
Abstract 3D numerical modeling of dopant transport in the melt is carried out for the 100 mm floating zone silicon single crystal growth process. The axis-symmetric shape of the molten zone is calculated with the program FZone considering the coil and the high frequency (HF) electromagnetic (EM) field in 3D. Time dependent melt flow, temperature and dopant concentration fields are modeled using a specialized solver based on the open source code library OpenFOAM ® . The influence of the Marangoni coefficient in the boundary conditions on the melt velocity field is analyzed. The obtained shapes of the crystallization interface and resistivity profiles in the grown crystal are compared with ex…
Numerical modelling for the diameter increase of silicon crystals grown with the pedestal method
Abstract The pedestal method is one of crucible-free crystal growth methods, that has been less researched than the well-known floating zone (FZ) method. However, the pedestal method may be a cost-effective alternative to FZ, if large diameter feed rods are available. The investigated system contains two electromagnetic inductors: high-frequency inductor for pedestal top surface melting and middle-frequency inductor for pedestal side heating. The present work describes recent advances in numerical modelling of heat transfer and phase boundaries in axially symmetrical approximation, neglecting the melt flow. The shape of high-frequency inductor was optimized with the algorithm of gradient de…
Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal
Prediction and adjustment of point defect (vacancies and self-interstitials) distribution in silicon crystals is of utmost importance for microelectronic applications. The simulation of growth processes is widely applied for process development and quite a few different sets of point defect parameters have been proposed. In this paper the transient temperature, thermal stress and point defect distributions are simulated for 300 mm Czochralski growth of the whole crystal including cone and cylindrical growth phases. Simulations with 12 different published point defect parameter sets are compared to the experimentally measured interstitial–vacancy boundary. The results are evaluated for stand…
Float zone single crystals for testing rods, pulled under electron beam heating
Effect of process parameters and crystal orientation on 3D anisotropic stress during CZ and FZ growth of silicon
Abstract Simulations of 3D anisotropic stress are carried out in and oriented Si crystals grown by FZ and CZ processes for different diameters, growth rates and process stages. Temperature dependent elastic constants and thermal expansion coefficients are used in the FE simulations. The von Mises stress at the triple point line is ~5–11% higher in crystals compared to crystals. The process parameters have a larger effect on the von Mises stress than the crystal orientation. Generally, the crystal has a higher azimuthal variation of stress along the triple point line (~8%) than the crystal (~2%). The presence of a crystal ridge increases the stress beside the ridge and decreases it on the ri…
Experimental and numerical investigation of laboratory crystal growth furnace for the development of model-based control of CZ process
Abstract The presented study is focused on laboratory Czochralski crystal growth experiments and their mathematical modelling. The developed small-scale CZ crystal growth furnace is described as well as the involved automation systems: crystal radius detection by image recognition, temperature sensors, adjustable heater power and crystal pull rate. The CZ-Trans program is used to model the experimental results – transient, 2D axisymmetric simulation software primarily used for modelling of the industrial-scale silicon crystal growth process. Poor agreement with the experimental results is reached; however, the proven ability to perform affordable, small-scale experiments and successfully mo…
Validation of mathematical model for CZ process using small-scale laboratory crystal growth furnace
The present material is focused on the modelling of small-scale laboratory NaCl-RbCl crystal growth furnace. First steps towards fully transient simulations are taken in the form of stationary simulations that deal with the optimization of material properties to match the model to experimental conditions. For this purpose, simulation software primarily used for the modelling of industrial-scale silicon crystal growth process was successfully applied. Finally, transient simulations of the crystal growth are presented, giving a sufficient agreement to experimental results.
Numerical 2D modelling of turbulent melt flow in CZ system with dynamic magnetic fields
Abstract The paper presents results of 2D axisymmetric mathematical modelling of laboratory CZ model facility that corresponds well to a large industrial silicon CZ growth system. The purpose of the investigation is to examine turbulent melt flow features that develop in the model crucible, when various dynamic magnetic fields (travelling, alternating) are applied, and to test the applicability of the modified low Re k–e turbulence model for the calculation of flows in these cases by extensive comparisons between calculated and measured data. The electromagnetic field is calculated with a self-developed program, and the calculations of the melt motion are carried out with the user modified …
Stress-induced dislocation generation in large FZ- and CZ-silicon single crystals—numerical model and qualitative considerations
When growing silicon crystals with higher diameter (presently up to 300 mm) the thermal stresses and possible dislocation generation in single crystals become a serious problem for both FZ- and CZ-methods. A two-dimensional problem oriented code for the FEM-package ANSYS has been developed to calculate the temperature field in the growing crystal considering radiation exchange with reflectors and environment and thermal stresses. Comparing calculated stresses with critical stresses, the dislocated zone is determined. A qualitative concept for the occurrence of dislocations using the metastable state is developed. In a parametric study for different thermal boundary conditions and crystal ge…