0000000000008789
AUTHOR
A. Muiznieks
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 …
Mathematical modelling of the industrial growth of large silicon crystals by CZ and FZ process
The present paper gives an overview of the complex mathematical modelling of industrial Czochralski (CZ) and floating‐zone (FZ) processes for the growth of large silicon single crystals from melt. Extensive numerical investigations of turbulent Si‐melt flows in large diameter CZ crucibles, global thermal calculations in growth facilities and analysis of the influence of various electromagnetic fields on CZ process are presented. For FZ process, a complex system of coupled 2D and 3D mathematical models is presented to show the possibilities of modelling from the calculation of the molten zone shape till the resistivity distribution in the grown crystal. A special developed program code is pr…
Numerical modelling of the industrial silicon single crystal growth processes
Silicon wafers produced from the silicon single crystals are the basic material for the manufacturing of various kinds of electronic devices determining the everyday life of the modern society. Silicon single crystals industrially are mainly grown by two methods - by the Czochralski and by the floating zone technique. Both of them involve various physical processes with complex interactions which makes the experimental optimization of the growth techniques a rather hard and expensive task. Therefore, mathematical modelling supported by the rapid increase of the computer power has become an effective means in the development of the industrial crystal growth. (© 2007 WILEY-VCH Verlag GmbH & C…
Crystal shape 2D modeling for transient CZ silicon crystal growth
Abstract A non-stationary axisymmetric model of Czochralski silicon single crystal growth is presented. The model describes transient behavior of crystal–melt, melt–gas and crystal–gas interfaces in connection with PID-based control of crystal diameter by changing crystal pulling velocity and heater power. To calculate significant crystal shape changes, unstructured finite element mesh is used in crystal and melt together with automatic element size control. Heater temperature changes are modeled with a simplified integral model. A numerical simulation example of start cone growth is given.
Warming of water in a glass
The article focuses on the process of water warming from 0 °C in a glass. An experiment is performed that analyses the temperature in the top and bottom layers of water during warming. The experimental equipment is very simple and can be easily set up using devices available in schools. The temperature curves obtained from the experiment help us to understand the process of convection in the glass and to determine the temperature at which the density of water is maximum. In addition, computational fluid dynamics—CFD modeling is carried out to facilitate better comprehension of the phenomenon observed in the experiment.
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…
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…
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…
Study of silicon crystal surface formation based on molecular dynamics simulation results
Abstract The equilibrium shape of 〈 110 〉 -oriented single crystal silicon nanowire, 8 nm in cross-section, was found from molecular dynamics simulations using LAMMPS molecular dynamics package. The calculated shape agrees well to the shape predicted from experimental observations of nanocavities in silicon crystals. By parametrization of the shape and scaling to a known value of { 111 } surface energy, Wulff form for solid-vapor interface was obtained. The Wulff form for solid–liquid interface was constructed using the same model of the shape as for the solid–vapor interface. The parameters describing solid–liquid interface shape were found using values of surface energies in low-index dir…
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 study of silicon crystal ridge growth
Abstract The size of the ridge-like protrusions appearing on the external surface of dislocation-free 〈 100 〉 silicon crystals grown from a melt was studied theoretically. According to existing models the growth of the ridges is caused by the presence of { 111 } crystal planes at the crystal–melt interface. They affect the height of triple phase line, free surface orientation and the crystal growth angle. A numerical 2-dimensional model was proposed for the calculation of the size of the crystal ridges. The model included the effect of the undercooling of the crystal–melt interface on the crystal growth angle. The numerical model estimated the effect of the ridge size on the free surface at…
Numerical study and comparisons with experimental data for transient behaviour of phase boundaries during industrial FZ process for silicon crystal growth
Abstract In our numerical transient model developed previously for the industrial FZ crystal growth process with the needle-eye technique, the meshing algorithms are essentially improved and a significant amount of numerical studies are carried out for model verification. Transient modelling for the experimental growth process with step-like time dependences of inductor current and feed rod velocity has shown that time dependencies of the crystal radius and zone height calculated numerically agree with the data from praxis. The fully transient simulation for growth process of crystal starting cone has shown that the model is capable of performing the simulation even if the crystal diameter …
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 …
Simplified Monte Carlo simulations of point defects during industrial silicon crystal growth
Abstract The paper proposes Monte-Carlo method-based 2D and 3D models of vacancies and interstitials in a cubic crystal. The model exploits the concept of lattice gas with covalent bounds between neighbour nodes. Two lattices shifted by half-period serve as nodes for atoms of the main crystal and interstitials. Distribution of particles between both lattices characterizes the entropy of the crystal. Successfully chosen interaction energies between main and sub-lattices allows the authors to detect a phase transition solid–liquid as well as to study the production of crystal defects/their agglomeration as a function of cooling/heating rate. Although the introduced 3D modification of the mode…
Influence of the three dimensionality of the HF electromagnetic field on resistivity variations in Si single crystals during FZ growth
Abstract Three-dimensional numerical modelling is carried out to analyse the floating zone crystal growth with the needle-eye technique used for the production of high-quality silicon single crystals with large diameters ( ⩾100 mm ). Since the pancake inductor has only one turn, the EM field and the distribution of heat sources and EM forces are only roughly axisymmetric. The non-symmetry together with crystal rotation reflects itself on the hydrodynamic, thermal and dopant concentration fields in the molten zone and causes variations of resistivity in the grown single crystal, which are known as the so-called rotational striations. The non-symmetric high-frequency electromagnetic field of …
Convective phenomena in large melts including magnetic fields
The set of characteristic parameters which describe modern large industrial CZ silicon single crystal growth systems is introduced. The main melt flow driving mechanisms are considered, and the characteristic density values of various in the melt acting forces are estimated. The analysis is illustrated with examples of numerical simulation and comparisons with experiments.