Bulk crystal growth has major high-tech applications among which three particularly important uses should be highlighted: (i) Integrated Circuit (IC) components such as DRAMs, transistors, etc., which all over fill our everyday life, are manufactured on mono-crystalline silicon or other semi-conductors; (ii) most solar cells are made on mono- or multi-crystalline silicon wafers; (iii) mono-crystalline sapphire is the key substrate of blue LEDs as devoted to completely replace traditional lighting in the near future.  In general single- and multi-crystals represent strategic products to drive the progress of new and clean technologies for the forthcoming years.

Most of industrial bulk crystals are grown from the melt except in those peculiar cases where this technique is not applicable.  In particular the Czochralski (Cz) process is principally used to grow mono-crystalline silicon boules for IC and solar applications.  However the production of large diameter high quality wafers generally requires making use of rigid magnetic fields of various shapes (transverse or configured) to stabilize the melt flow.  An alternative method to grow silicon single crystals is the Floating Zone (FZ) process, which is more expensive but whose use is needed to produce oxygen-free wafers for power devices.  The Cz process may also be applied to grow sapphire single crystals in competition with alternative techniques such as the Heat Exchanger Method (HEM) or the Kyropoulos (Ky) technique.  Finally the Directional Solidification System (DSS) is a rapidly developing process to produce silicon multi-crystals for solar applications.  Nowadays, at a lower cost, the efficiency of DSS-produced cells almost reaches that of Cz-produced cells, especially when the mono-cast technique is used.

All the above examples may be used to illustrate the privileged interest of numerical simulation to optimize crystal growth and other forming processes.  Using a trial and error approach the software user’s objective is to perform so-called numerical experiments in order to investigate the effect of any change of the geometry (and especially the hot zone design), the selected material properties and the processing conditions (heater power, crystal growth and rotation rates, etc.) on the resulting crystal quality and process yield.  To achieve this goal two requirements have to be fulfilled by the simulation tool: (i) the use of global and, if possible, time-dependent simulations is a need to predict the detail of the crystal and melt thermal evolution, the solidification front shape and the melt velocity field at any stage of the growth process; (ii) the software must have the ability to accurately predict crystal quality and any quantity measuring the process yield as resulting from these global calculations. 

FEMAGSoft S.A. is a spin-off company of the Université catholique de Louvain (Belgium) whose activity is to develop crystal growth simulation software for industrial use.  From its creation in 2003 most of the company research activity has borne on the development of physical models and numerical algorithms to simulate bulk crystal growth from the melt in collaboration with the university [1-5].  Accordingly, on the one hand, various programs have been developed to model the global heat transfer in the overall furnace, the melt and ambient gas flows and the solidification process in order to accurately predict the thermal, mechanical and geometrical evolution of the melt and crystal for Cz, FZ, DSS... processes.  On the other hand, additional programs have been developed to predict the resulting crystal quality as measured by point- and micro-defect densities in the crystal (for mono-crystalline silicon or germanium), by thermal stresses and dislocation density (e.g. for mono-crystalline sapphire), and by the concentration of any species characterizing crystal stoichiometry or incorporated into the crystal during the growth process (dopants, impurities, oxygen in silicon growth, etc.).  Improving the software prediction accuracy together with reducing the simulation time has constantly governed FEMAGSoft’s development strategy.

In general, numerical methods, programming techniques and computer hardware have experienced an accelerating progression rate during the last decades and this has driven a very steep increase of quality of the numerical tools used by the FEMAG family of products.  In particular the development of a very efficient linear solver (in terms of computation cost and memory) together with the use of up-to-date numerical methods allow the FEMAG programs to provide well-resolved solutions to the user in a time not exceeding 1-2 days.  However getting accurate results remains a complex and hard task in view of the high difficulty to model the involved physical phenomena.  This holds true for both global heat transfer and crystal quality modeling.  Correlating the resulting crystal quality and process yield to the furnace and process design nonetheless requires to have an accurate physical model at one’s disposal.  Therefore the current objective of FEMAGSoft’s R&D team is to improve the predictive capabilities of the FEMAG models wherever this is possible.  This approach is illustrated by several examples in the following paragraphs.

A first application is provided by the growth of large diameter mono-crystalline silicon ingots under the effect of a strong transverse magnetic field [6].  Using this technique is mandatory in view of the very high crystal quality requirements imposed by IC technology.  The numerical problem is solved by means of a combined FEM - spectral method with accurate resolution of the very thin Hartmann boundary layers.  A key problem is to well-model the flow turbulence, which is rapidly weakened and becomes 2D without being completely cancelled out when the magnetic field is sufficiently strong.  To this end a non-isotropic mixing length tensor is introduced in order to correctly model the momentum transfer resulting from the 2D turbulence. 

A second application is provided by the growth of silicon multi-crystals by DSS process for the production of solar cells.  Here again a combined FEM - spectral technique is applied to efficiently solve the 3D problem.  Main issues are to model the growth transitions between columnar and equi-axial grain structures, and also to accurately predict the distribution of species (especially carbon and oxygen) in the crystal, taking into account the effects of the layered flow structure, of Marangoni convection, of the gas flow and of segregation at the solid-liquid interface. 

A last application is provided by the numerical modeling of FZ single crystal growth.  Several difficult problems are posed.  First, the high-frequency magnetic field has to be accurately calculated including its thermal and mechanical effects which are modeled as a heat flux, magnetic pressure and shear stress acting along the melt surface.  This approach is favored to the equivalent but more expensive modeling of Joule heating and Lorentz force by use of refined boundary layer meshes in the melt [7, 8].  Secondly the open melting front is governed by complex physical effects since molten silicon flows along the feed-rod in the form of small droplets or as a non-uniform thin film whose material properties must be correctly averaged.  Finally a main issue in FZ growth results from the absence of turbulent mixing.  As an undesired technical consequence of this effect the distribution of dopants and of the associated resistivity in the crystal may be strongly segregated and various techniques are used to counteract this problem.  These techniques generally result in a loss of the system axisymmetry or quasi-steadiness and hence require appropriate modeling. 

References

[1] F. Dupret, P. Nicodème, Y. Ryckmans, P. Wouters, M.J. Crochet, International Journal of Heat and Mass Transfer 33 (1990), pp. 1849-1871.

[2] F. Dupret, N. Van den Bogaert, “Modelling Bridgman and Czochralski growth”, Handbook of Crystal Growth, Chapter 15, Vol. 2, D.T.J. Hurle editor, Elsevier, 1994, pp. 875-1010.

[3, 4] N. Van den Bogaert, F. Dupret, Journal of Crystal Growth, 171 (1997), pp. 65-76, 77-93.

[5] T. Sinno, E. Dornberger, R.A. Brown, W. von Ammon, F. Dupret, “Defect engineering of Czochralski single-crystal silicon”, Materials Science and Engineering: R Reports, 28 (2000), pp. 149-198.

[6] Y. Collet, O. Magotte, N. Van den Bogaert, R. Rolinsky, F. Loix, M. Jacot, V. Regnier, J.-M. Marchal, F. Dupret, to be published in Journal of Crystal Growth (2012).

[7, 8] F. Bioul, F. Dupret, IEEE Transactions on Magnetics, 41 N° 9 (2005), pp. 2496-2505, 2506-2514.

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