Introduction

Although Floating Zone (FZ) process is widely known and has been developed for many years, it is necessary to improve numerical models of it. Computational resources allow considering many physical aspects of a system, for example, hydrodynamical (HD) processes in liquid silicon. In the present work HD approach is used for two purposes: obtaining more precise shape of phase boundaries and calculating radial resistivity profiles from dopant concentration field in melt. This allows to investigate influence of surface tension γ and its temperature coefficient M = ∂γ/∂T (i.e., Marangoni coefficient) on FZ process. It is essential for better understanding of modelling nuances because of Marangoni coefficient’s strong dependence on oxygen content in atmosphere.

Description of the modelling software

The most important programs, which were used in the present study, are:

• FZone. This is a program for 2D axisymmetric modelling of FZ crystal growth. It performs iteratively coupled calculations of temperature field (HD in melt can be included) and thermal radiation (considering all surfaces as optically gray). High frequency 3D electromagnetic field, that induces current in thin silicon skin layer, is azimuthally averaged and then taken into account as a heat source distribution [1].
• FZSiFOAM. This is a program for 3D HD calculations in melt, based on OpenFOAM code library. 3D finite volume mesh is created on the base of shape of phase boundaries, which were obtained via FZone. FZSiFOAM considers non-stationary, incompressible, laminar, buoyancy driven flow; with Marangoni and EM tangential surface stresses. Dopant concentration, melt temperature and pressure fields are considered as well [2].

Results of the study

As an example characteristic 4” FZ system from ICG, Berlin is used. Crystal pulling rate is 3.4 mm/min and zone height is 34.87 mm. The current frequency was reduced from 3 MHz to 2.2 and 1.6 MHz in order to compare calculation results with experimental results [3]. Realistic shape of phase boundaries can be obtained with Marangoni coefficient equal to –1.6·10^-4 N/(m·K) (see Fig. 1).

Then, it was investigated how does M influence radial resistivity distribution in single crystal. Wide range of M was examined – from 0.75 to 12·10^-4 N/(m·K), that correspond to strong variations of oxygen concentration [4]. At last, averaged resistivity profiles were obtained and compared with experimental data (few of them are shown in Fig. 2).

Figure 1. Shapes of phase boundaries obtained for 4” ICG system via FZone, using different values of Marangoni coefficient and inductor current frequency.

Figure 2.  Averaged in time radial resistivity profiles. They are obtained from dopant concentration distribution on crystallization interface.

Acknowledgements

The present work is carried out at the University of Latvia and has been supported by the European Regional Development Fund, project contract No. 2011/0245/ 2DP/2.1.1.1.0/10/APIA/VIAA/114 and 2011/0002/2DP/2.1.1.1.0/10/APIA/VIAA/085.

References

[1] G. Ratnieks, A. Muiznieks, A. Muhlbauer, Modelling of phase boundaries for large industrial FZ silicon crystal growth with the needle-eye technique, JCG, 255, 227-240 (2003)
[2] A. Rudevics, K. Lacis, A. Muiznieks, N. Jekabsons, B. Nacke, Using of open source code library OpenFOAM for 3D magnetohydrodynamic calculations in semiconductor crystal growth technologies, Modelling for Electromagnetic Processing (2008)
[3] H.-J. Rost, R.Menzel, A.Luedge, H.Riemann, Float-Zone silicon crystal growth at reduced RF frequencies, JCG, 360, 43–46 (2012)
[4] Z. F. Yuan, K. Mukai, W. L. Huang, Surface tension and its temperature coefficient of molten silicon at different oxygen potentials, Langmuir, 18(6):2054-2062 (2002)

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1. Modelling of 3D features of molten zone in FZ silicon crystal growth
2. Numerical study of silicon crystal ridge growth