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In the literature, numerical computations of the Czochralski process for crystal growth are conducted using a quasi-steady state assumption for the crystallization neglecting time-dependent effects. In the present work, an algorithm is developed, which allows to calculate the transient behavior of the crystallization interface including the movement of the three-phase boundary and the free surface of the melt. Thus, in conjunction with the computation of the turbulent melt flow and heat transfer, more realistic predictions of the crystal growth can be achieved. Test cases show that with the algorithm, realistic phenomena like crystal diameter increase and decrease during the growth process could be reproduced.
In the literature, numerical computations of the Czochralski process for crystal growth are conducted using a quasi-steady state assumption for the crystallization neglecting time-dependent effects. In the present work, an algorithm is developed, which allows to calculate the transient behavior of the crystallization interface including the movement of the three-phase boundary and the free surface of the melt. Thus, in conjunction with the computation of the turbulent melt flow and heat transfer, more realistic predictions of the crystal growth can be achieved. Test cases show that with the algorithm, realistic phenomena like crystal diameter increase and decrease during the growth process could be reproduced.
A DNS of the turbulent flow and heat transfer in an idealized Czochralski configuration was conducted using a very fine grid. The results show the irregular forming of buoyant thermal plumes as well as the occurrence of a large, stable vortical structure in the bulk of the melt. In the averaged flow field a B´enard-cell-like pattern can be recognized. The DNS data served also as a reference for the validation of the LES method. The LES computations were conducted using different grid sizes, SGS models, and discretization methods. For relatively fine grids and central differences, the results agree very well with the DNS. Using an upwind discretization introduces numerical errors. In combination with coarser grids, this leads to large deviations and even qualitative differences. However, overall the computational effort could be reduced significantly by LES. Thus a compromise between accuracy achieved and effort required has to be found.
A numerical algorithm is presented which allows three-dimensional time-dependent simulations of Czochralski and other crystal growth processes. Besides the computation of turbulent flow and heat transfer in the melt, it includes the movement of the crystallization interface between melt and crystal, the dynamics of the melt free surface with the meniscus and the variation of the three-phase boundary. Thus, there is the potential that crystal diameter variations, which occur during growth of the start and the end cone can be predicted. This is illustrated by results of sample computations.
For numerical simulations of the Czochralski crystal growth process, in the literature the quasi-steady state approach is used, i.e. the diameter of the crystal is kept constant. However, this leads to errors in the computation of the phase interface. In the present work, for the first time in 3D, the three-phase junction between melt, crystal and atmo-sphere is allowed to move in horizontal direction. Thus, diameter changes during the crystal growth process could be simulated accordingly. In the future, the vertical movement will be allowed as well, considering the free surface stresses. This will enable realistic predictions of the transient behavior of the crystallization front.
3D computation of oxygen transport in Czochralski crystal growth of silicon considering evaporation
(2007)
For Czochralski (Cz) growth of silicon single crystals, 3D unsteady simulations of the oxygen transport in the turbulent melt flow were conducted using a new formulation for the evaporation at the free surface found by Sakai et al. [Electrochem. Solid-State Lett. 5 (8) (2002) G72]. Compared with 2D results from literature, the computed oxygen concentration now fits the experimental data better, showing lower values in the critical region beneath the crystal than in previous simulations, which mostly arises from the more accurate prediction of the flow field. Further computations of Cz configurations show how the oxygen distribution can be influenced by rotation of the crucible and crystal.