Floating Zone Silicon Crystal Growth Research Articles

3D numerical study of the asymmetric phenomenon in 200 mm floating zone silicon crystal growth

Abstract In this paper, we propose a three-dimensional model for a 200 mm floating zone silicon crystal growth process to investigate the fluid flow and solid–liquid interface. To study the effect of high-frequency (HF) electromagnetic (EM) heating on the melt flow and interface shape, HF-EM and heat transfer calculations were conducted in three dimensions. Through comparison of EM and Marangoni forces, EM force was found to have a larger effect than Marangoni force on the free surface flow. By considering 3D Marangoni and EM forces at the free surface, a more accurate melt flow distribution has been obtained. Moreover, the results showed that local growth rate became more inhomogeneous when the rotation speed of the crystal was increased. However, a more homogeneous three-phase line could be obtained with a high rotational crystal speed.

3D modeling of growth ridge and edge facet formation in [formula omitted] floating zone silicon crystal growth process

A 3D quasi-stationary model for crystal ridge formation in FZ crystal growth systems for silicon is presented. Heat transfer equations for the melt and crystal are solved, and an anisotropic crystal growth model together with a free surface shape solver is used to model the facet growth and ridge formation. The simulation results for 4″ and 5″ crystals are presented and compared to experimental ridge shape data.

Three-dimensional modelling of thermal stress in floating zone silicon crystal growth

During the growth of large diameter silicon single crystals with the industrial floating zone method, undesirable level of thermal stress in the crystal is easily reached due to the inhomogeneous expansion as the crystal cools down. Shapes of the phase boundaries, temperature field and elastic material properties determine the thermal stress distribution in the solid mono crystalline silicon during cylindrical growth. Excessive stress can lead to fracture, generation of dislocations and altered distribution of intrinsic point defects. Although appearance of ridges on the crystal surface is the decisive factor of a dislocation-free growth, the influence of these ridges on the stress field is not completely clear. Here we present the results of thermal stress analysis for 4” and 5” diameter crystals using a quasi-stationary three dimensional mathematical model including the material anisotropy and the presence of experimentally observed ridges which cannot be addressed with axis-symmetric models. The ridge has a local but relatively strong influence on thermal stress therefore its relation to the origin of fracture is hypothesized. In addition, thermal stresses at the crystal rim are found to increase for a particular position of the crystal radiation reflector.

3D numerical simulation of free surface shape during the crystal growth of floating zone (FZ) silicon

In FZ growth processes, the stability of the free surface is important in the production of single crystal silicon with high quality. To investigate the shape of the free surface in the FZ silicon crystal growth, a 3D numerical model that included gas and liquid phases was developed. In this present study, 3D Young-Laplacian equations have been solved using the Volume of Fluid (VOF) Model. Using this new model, we predicted the 3D shape of the free surface in FZ silicon crystal growth. The effect of magnetic pressure on shape of free surface has been considered. In particular, the free surface of the eccentric growth model, which could not be previously solved using the 2D Young-Laplacian equations, was solved using the VOF model. The calculation results are validated by the experimental results.

Application of enthalpy model for floating zone silicon crystal growth

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 4in 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 polycrystalline rod show the development of melt-filled grooves at the open melting front surface. The distance between the grooves is shown to grow with the increase of the skin-layer depth in the solid material.

3D modeling of doping from the atmosphere in floating zone silicon crystal growth

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.

Mathematical modelling of the feed rod shape in floating zone silicon crystal growth

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 of the asymmetric feed rod shape. The feed rod rotation is shown to have a smoothing effect on the shape of the open melting front.

Hydrodynamical aspects of the floating zone silicon crystal growth process

3D numerical modeling of dopant transport in the melt is carried out for the 100mm 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 experimental results. Differences between axis-symmetric and non-symmetric models are analyzed.

The Effect of Oxygen Partial Pressure on Marangoni-Flow-Induced Dopant Striations in Floating-Zone Silicon Crystals

A silicon crystal was grown after the floating-zone (FZ) method under oxygen partial pressures (Po2 in) of 2.0×10-8 MPa and 5.8×10-6 MPa, which were measured at the entrance of a furnace. It was found that dopant striations showed a single-peak spectrum at 0.19 Hz in this crystal, and a crystal grown at Po2 in of 2.0×10-8 MPa showed multiple frequencies. This result is explained by a decrease in the Marangoni number with increasing oxygen partial pressure. It has thus been concluded that Marangoni flow in FZ silicon crystal growth can be controlled by introducing oxygen gas, independently of the temperature field control in the vicinity of the crystal/melt interface.

Modelling of phase boundaries for large industrial FZ silicon crystal growth with the needle-eye technique

In order to facilitate the numerical calculations of the phase boundaries in large industrial floating zone silicon crystal growth with the needle-eye technique, the chain of improved mathematical models is developed. The phase boundaries are solved in a partly transient way and the modelling improvements cover the open melting front, the inner triple point and the free melt surface. The view factors model is applied for the radiative heat transfer. The electromagnetic field is calculated with account of a multiple-slit inductor.

Three-dimensional simulation of Marangoni flow and interfaces in floating-zone silicon crystal growth

Three-dimensional (3D) simulation is conducted for the floating-zone (FZ) growth of silicon in an ellipsoid mirror furnace, for the first time, simultaneously considering the time-dependent Marangoni flow, heat transfer, and moving interfaces. The numerical method is based on an efficient multigrid finite-volume method with front tracking. The growth of an 8 mm diameter silicon crystal under microgravity is considered. A half-zone configuration is also used for benchmarking, and the calculated bifurcation points and flow structures are in good agreement with previous results. However, quasi-periodic modes and angular waves are observed at higher Marangoni number. For full zone calculations, including the feeding and growth interfaces, the bifurcation behavior is similar, but the primary bifurcation is found to be subcritical and the 3D one-fold flow mode is dominant. Significant growth rate fluctuations and back melting are found for a typical growth condition as well. The major fluctuation frequency ranges from 0.1 to 0.3 Hz, and the severe back melting may be related to the angular waves.

Effect of axisymmetric magnetic fields on heat flow and interfaces in floating-zone silicon crystal growth

A robust finite-volume/Newton method is used to simulate the heat transfer, fluid flow, and interface shapes of floating-zone silicon growth under an axisymmetric magnetic field generated by coils around the growth axis. The global method allows all the field variables, both the heat flow and magnetic fields, and the interface variables, both the interfaces and the free surface, to be solved simultaneously in an efficient manner. Calculated results show that the suppression of convection by the applied magnetic fields is effective for both natural convection and forced convection due to rotation, but less effective for thermocapillary convection. As a result, the melt in the core region is quiescent and the effect of rotation also becomes trivial, while in the periphery the flow is still vigorous. The core size and flow patterns depend on the applied magnetic strength and distribution. The calculated core size and dopant segregation for a growth in a mono-ellipsoid furnace are in good agreement with measured ones. The effects of rotation and finite electrical conductivity on current distribution are also illustrated.