ISSN:
1573-7691
Keywords:
Czochralski crystal growth
;
finite element method
;
thermal-capillary model
;
moving-boundary problem
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
Notes:
Abstract Numerical methods are presented for solution of the complex moving-boundary problem described by a thermal-capillary model for Czochralski crystal growth, which accounts for conduction through melt, crystal, and crucible and radiation between diffuse-gray body surfaces. Transients are included that are caused by energy transport, by changes in the shapes of the melt-crystal, melt-ambient phase boundaries and the moving crystal, and by the batchwise decrease of the melt volume in the crucible. Finite-element discretizations are used to approximate the moving boundaries and the energy equation in each phase. A two-level, implicit integration algorithm is presented for transient calculations. The temperature fields and moving boundaries are advanced in time by a trapezoid rule approximation with modified Newton's iterations to solve algebraic systems for effective ambient temperatures computed with diffuse-gray radiation. The implicit coupling between radiative exchange, interface shapes, and the temperature field is necessary for preserving the second-order accuracy of the integration method and is achieved by successive iterations between the radiation calculation and solution of the thermal capillary model. Analysis of a quasi-steady-state model (QSSM) demonstrates the inherent stability of the CZ process. Including either diffuse-gray radiation among crystal, melt, and crucible or a simple controller for maintaining constant radius can lead to oscillations in the crystal radius. The effects of these oscillations on batchwise crystal growth are addressed.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01061294
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