The current importance of rare-earth scandate crystals (ReScO₃; Re=Y, La, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm and Lu) is initiated by commercial interest. For this reason a deeper investigation of Czochralski crystal growth from the melt at the Institute for Crystal Growth in Berlin (IKZ) is promoted. Rare-earth scandate crystals show good chemical and physical properties to be used as substrates of ferroelectric materials (e.g. non-volatile FeRAMs). Unfortunately most scandate crystals tend to undesired spiral growth, i.e. symmetry breaking of an initially axisymmetric behaviour.
The melting point of scandate crystals is about 2000°C. Therefore internal measurements are hard to do and a numerical approach is mandatory. Nevertheless, we can monitor the temperature during the growth (Czochralski method) in a complicated setup at specific locations very close to the crucible. We use this data for comparison with numerical results.
Our objective is to figure out the reasons of spiral growth using a hydrodynamic approach. The hypothesis is that this undesired spiral instabilities are being initiated by heat and momentum disturbances in the melt [1]. The crystal grower is interested in getting stable parameters which do not lead to spiral instabilities.
To get such stable parameters the theoretical approach will lead to a full 3D model, while currently a 2D axisymmetric solution is superposed with a 3D disturbance. And this leads to a large scale eigenvalue problem which has to be solved efficiently. For this reason we are improving permanently our simulation tools (solver, eigensolver) in order to be efficient(time/cost).
Experiments have shown that some scandate crystals (e.g. GdScO₃) which are transparent with respect to thermal radiation do not tend to spiral growth. Therefore the numerical model must be extended to take into account the internal thermal radiation.
Using effective branch following techniques (bifurcation problem) it is possible to detect oscillatory behaviour, ambiguity of solutions and the general solution behaviour of our strongly non-linear system. This helps us to decide where an extended eigenvalue analysis should be performed in order to well characterize the solution type.
Further experiments have shown, that a change of crystal growth rotation direction implies a change of the direction of grown spiral. This behaviour is confirmed by numerical results also.
We are collaborating with our partners in Israel [2] in order to compare numerical results.
Our future work will focus on doing a full 3D approach using a more accurate model. This will require more computational performance and therefore we have to work out further improvements in our software.

References:
[1] Numerical study of hydrodynamic instabilities during growth of dielectric crystals from the melt, H. Wilke, N. Crnogorac, K. A. Cliffe, Journal of Crystal Growth (2006), In press
[2] A. Yu Gelfgat, School of Mechanical Engineering, Tel-Aviv University, Israel

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1. Numerical study of crucible bottom shape on the heat transport and fluid flow during the seeding process of oxide Czochralski crystal