(291d) Analysis of the Fluid Flow of Czochralski Crystal Growth Process Using Lagrangian Coherent Structures

The Czochralski (CZ) crystal growth process is the most important method for manufacturing crystals [1]. In the CZ process a crystal is pulled out vertically from a heated pool of melt contained in a cylindrical crucible. This process has the ability of meeting the stringent requirements for purity and crystallographic perfection [2]. The properties of the crystallized solid depend on the physical processes occurring during its formation [3]. The melt flow in the crucible during the crystal growth influences the quality of the grown crystal, impinges on microdefects and impacts striations appearing in it [2,4].

In CZ process there are several mechanisms of fluid motion of the melt in crucible. Gradients of density give rise to natural convection, while the rotation of the crystal during withdrawal causes fluid to spin out centrifugally. An additional mechanism of forced convection is possible rotation of the crucible, as well as the moving boundary of melt-gas interface. Flows driven by surface tension gradients may also exist [3]. The velocity and temperature field throughout the Czochralski melt are subject to significant and comparable amounts of both natural and forced convection [5]. Along this line, some previous studies reported flow patterns and time-dependent variations of axisymmetric melt flow condition in the crucible [6].

The computation of Lagrangian coherent structures (LCS) has become an important method for examining time-varying fluid transport field [7], (see Fig.1 which shows LCS based on the calculation of Finite Time Lyapunov Exponent (FTLE) field of the fluid particles for a time-varying periodic flow). Therefore, LCS can be utilized to better understand transport mechanisms within the CZ melt flow. In this work, the geometry of the CZ melt flow idealized and the melt-crystal and melt-ambient interfaces are considered to be with regular boundaries (see [6]), so that the melt domain is of rectangular shape with time-varying thermal and velocity boundary conditions. The finite element method is employed to simulate 2D melt flow dynamics [8,9], then the data are used to compute the LCS, which represent dynamically driven barriers to transport processes. These structures provide a novel insight into the mixing and transport properties of the system. In particular, it will be demonstrated that for a well-defined LCS, the flow flux over an LCS is shown to be negligible [10], which behaves as a transport barrier within the fluid flow field. The insight provided by the presence of LCS reveals the flow patterns which impact the purity of the grown crystal in the CZ crystal growth process.

References:

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[4] W. Miller, U. Rehse, K. Böttcher, Melt convection in a Czochralski crucible, Cryst. Res. Technol, 34, 1999.

[5] J. R. Carruthers, K. Nassau, Nonmixing cells due to crucible rotation during Czochralski crystal growth, Journal of Applied Physics, 39, 1968.

[6] P. A. Sackinger, R. A. Brown, A finite element method for analysis of fluid flow, heat transfer and free interfaces in Czochralski crystal growth, International Journal for Numerical Methods in Fluids, 9, 1989.

[7] S. C. Shadden, M. Astorino, J. F. Gerbeau, Computational analysis of an aortic valve jet with Lagrangian coherent structures, CHAOS, 20, 2010.

[8] J. J. Derby, R. A. Brown, Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth-I: Simulation, Journal of Crystal Growth, 74, 1986.

[9] J. J. Derby, R. A. Brown, Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth-II: Processing strategies, Journal of Crystal Growth, 75, 1986.

[10] S. C. Shadden, F. Lekien, J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212, 2005.