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Numerical simulation and optimization of SiC single crystal growth by sublimation from the gas phase

Collaborator: J. Geiser, O. Klein, P. Philip (until August 2004), J. Sprekels

Cooperation with: Ch. Meyer, F. Tröltzsch (Technische Universität Berlin), K. Böttcher, D. Schulz, D. Siche (Institut für Kristallzüchtung (IKZ), Berlin), P. Philip (since September 2004 Institute for Mathematics and its Applications (IMA), Minneapolis, USA)

Supported by: DFG Research Center MATHEON, project C9

Description:

Owing to numerous technical applications in electronic and optoelectronic devices, the industrial demand for high quality silicon carbide (SiC) bulk single crystals remains large. The most successful and most widely used growth technique of recent years is the sublimation growth of SiC bulk single crystals via physical vapor transport (PVT), also known as the modified Lely method.

During PVT, a graphite crucible (see Figure 1) is placed in a low-pressure inert gas atmosphere consisting of argon. The crucible is then intensely heated, e.g., by induction heating, to temperatures up to 3000 K. Inside the crucible, polycrystalline SiC source powder sublimates, and the gaseous species diffuse through the cavity to the SiC seed. As the single-crystalline seed is kept at a temperature below that of the SiC source, the species crystallize on the seed, which thereby grows into the reactor.

The physical and mathematical modeling of the growth process leads to a highly nonlinear system of coupled partial differential equations. In addition to the kinetics of a rare gas mixture at high temperatures, one has to consider heat transport by conduction and radiation, reactive matter transport through porous and granular media, different kinds of chemical reactions and phase transitions, and the electromagnetic fields and heat sources produced by the induction heater. The main control parameters with respect to an optimization of the crystal growth process are the design of the growth apparatus, the position of the induction coil, the heating power, and the inert gas pressure.

Within the covered research period, the simulation software WIAS-HiTNIHS has been made more flexible, such that the program can now deal with more general geometries. During runtime, the program can read descriptions of geometries from ASCII files, in which the geometry is described by a sequence of points followed by a list of polygons. Moreover, some preparations have been made to be able to use the output of a program with a GUI allowing the input of geometry description and the creation of meshes which is currently under development in the Research Group ``Numerical Mathematics and Scientific Computing''.

To handle general geometries, the treatment of radiation has been generalized, such that also radiation regions with a boundary consisting of several connected components can be considered, i.e. such that one can also deal with radiation regions surrounding some opaque objects. For two points (r, z) and (s, y) on the (r, z) boundary of the radiation region one has to determine the visibility intervals, i.e. the intervals for $\tau$ such that the points (r,$\tau$, z) and (s, 0, y) (expressed in cylindrical coordinates) are mutually visible, since the ray connecting these points is not blocked by any part of the crucible. Following [1], this is done by considering the circular projection of these rays. In Figure 2, we present the circular projection of rays connected to the boundaries of the visibility intervals.

Fig. 2: Circular projections of rays corresponding to the angles being either the begin (blue lines) or the end (red lines) of visibility intervals for a radiation region surrounding four opaque objects

The handling of material parameters has been made more flexible, and WIAS-HiTNIHS now reads the material parameters during runtime from a material data file, see [2]. Moreover, the programming of a software with a GUI to create and edit such material data files has been started.

We have also taken into account the direction dependence of the thermal conductivity of the insulation of the crucible. Considering the coordinates (r, z), a heat flux $\vec{{q}}$ of the form $\vec{{q}}$ = - $\kappa$(T)$\left(\vphantom{ \begin{array}{c c} \alpha_r & 0 \\ 0 & \alpha_z \end{array} }\right.$$\begin{array}{c c} \alpha_r & 0 \\ 0 & \alpha_z \end{array}$$\left.\vphantom{ \begin{array}{c c} \alpha_r & 0 \\ 0 & \alpha_z \end{array} }\right)$grad T is used, where T is the temperature, $\kappa$ is a temperature-dependent thermal conductivity, and $\alpha_{r}^{}$, $\alpha_{z}^{}$ are given positive constants. A FVM discretization for this term has been derived and implemented in WIAS-HiTNIHS. Some results can be found in Figure 3.

Some mathematical optimization problems connected to the SiC growth process are considered in [3], [4]. There, semilinear elliptic equations with nonlocal interface conditions are treated, modeling the diffuse-gray conductive-radiative heat transfer within the growth apparatus. Based on a minimum principle for the semilinear equation, as well as L^$\infty$ estimates for the weak solution, the existence of an optimal solution and necessary optimality conditions have been established. The theoretical results are illustrated by results of numerical computations.

Fig. 3: Computation for isotropic and anisotropic insulations. In the left figure, we have an isotropic insulation with $\alpha_{r}^{}$ = $\alpha_{z}^{}$ = 1, in the middle figure, we have z-anisotropy with $\alpha_{z}^{}$ = 1000 for all insulation parts, and in the right figure for the upper and lower parts of the insulation, we have r anisotropy with $\alpha_{r}^{}$ = 1000, and for the remaining parts of the insulation, we have z anisotropy with $\alpha_{z}^{}$ = 1000.

References:

1. F. DUPRET, P. NICODEME, Y. RYCKMANS, P. WOUTERS, M.J. CROCHET, Global modelling of heat transfer in crystal growth furnaces,
   Int. J. Heat Mass Transfer, 33(9) (1990), pp.  1849-1871.

2. J. GEISER, O. KLEIN, P. PHILIP, WIAS-HiTNIHS: A software tool for simulation in sublimation growth of SiC single crystals: Applications and methods,
   to appear in the Proceedings of the International Congress of Nanotechnology and Nano World Expo 2004.

3. C. MEYER, P. PHILIP, F. TRÖLTZSCH, Optimal control of a semilinear PDE with nonlocal radiation interface conditions,
   WIAS Preprint no. 976, 2004 , submitted.

4. C. MEYER, P. PHILIP, Optimizing the temperature profile during sublimation growth of SiC single crystals: Control of heating power, frequency, and coil position, WIAS Preprint no. 895, 2003 , Crystal Growth & Design, 5 (2005), pp. 1145-1156.

5. M. PONS, M. ANIKIN, K. CHOUROU, J.M. DEDULLE, R. MADAR, E. BLANQUET, A. PISCH, C. BERNARD, P. GROSSE, C. FAURE, G. BASSET, Y. GRANGE, State of the art in the modelling of SiC sublimation growth,
   Materials Science and Engineering B, 61-62 (1999), pp. 18-28.

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Hysteresis operators

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Phase diagrams including surface tension and

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