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  • 1
    Electronic Resource
    Electronic Resource
    Finite-element methods for analysis of the dynamics and control of Czochralski crystal growth (1987)
    Springer
    Journal of scientific computing 2 (1987), S. 297-343 
    Details
    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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  • 2
    Electronic Resource
    Electronic Resource
    Microwave driven convection in a square cavity (1994)
    Hoboken, NJ : Wiley-Blackwell
    AIChE Journal 40 (1994), S. 1268-1272 
    Details
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/aic.690400718
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  • 3
    Electronic Resource
    Electronic Resource
    Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains (1994)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 18 (1994), S. 1185-1209 
    Details
    ISSN: 0271-2091
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Two approaches which employ the finite element method to solve for large-scale, coupled, incompressible flows through adjacent porous and open domains are developed and evaluated in a model for the spontaneous ignition of coal stockpiles. Both formulations employ the Navier-Stokes equations do describe flow in the open region; two different descriptions, Darcy's law and the Brinkman equation, are employed to model flows within the porous region. The formulation which uses Darcy's law employs the Beavers-Joseph slip condition and a novel implementation of the interfacial conditions. The other approach invokes the Brinkman equation: this considerably simplifies the implementation of matching conditions at the interface between the porous and open fluid domains, but also results in velocity boundary layers in the porous region adjacent to this interface which can be difficult to resolve numerically. A direct comparison of model results shows that the Darcy-slip formulation produces solutions which are more accurate and more economical to compute than those obtained using the Brinkman formulation.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650181205
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  • 4
    Electronic Resource
    Electronic Resource
    A finite element method for analysis of fluid flow, heat transfer and free interfaces in Czochralski crystal growth (1989)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 9 (1989), S. 453-492 
    Details
    ISSN: 0271-2091
    Keywords: Czochralski crystal growth ; Finite element method ; Free boundary problem ; Incompressible fluid flow ; Heat transfer ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: A finite element algorithm is presented for simultaneous calculation of the steady state, axisymmetric flows and the crystal, melt/crystal and melt/ambient interface shapes in the Czochralski technique for crystal growth from the melt. The analysis is based on mixed Lagrangian finite element approximations to the velocity, temperature and pressure fields and isoparametric approximations to the interface shape. Galerkin's method is used to reduce the problem to a non-linear algebraic set, which is solved by Newton's method. Sample solutions are reported for the thermophysical properties appropriate for silicon, a low-Prandtl-number semiconductor, and for GGG, a high-Prandtl-number oxide material. The algorithm is capable of computing solutions for both materials at realistic values of the Grashof number, and the calculations are convergent with mesh refinement. Flow transitions and interface shapes are calculated as a function of increasing flow intensity and compared for the two material systems. The flow pattern near the melt/gas/crystal tri-junction has the asymptotic form predicted by an inertialess analysis assuming the meniscus and solidification interfaces are fixed.
    Additional Material: 27 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650090406
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  • 5
    Electronic Resource
    Electronic Resource
    Parallel computation of radiation view factors between two arbitrarily oriented surfaces (1996)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 12 (1996), S. 43-50 
    Details
    ISSN: 1069-8299
    Keywords: view factor ; radiation ; parallel computation ; heat transfer ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Two implementations for parallel computation of radiation heat transfer view factors are formulated and tested for a model problem. Using a sufficiently large number of processors and a suitable communications paradigm, the solution time for the problem considered here scales linearly with the number of surface elements in the parallel implementation rather than the quadratic scaling obtained using a serial approach. For larger problems, significantly shorter solution times are obtained using the massively parallel Connection Machine 5 than those obtained on a single processor of the Cray C-90, a traditional vector supercomputer. Parallel performance was degraded for implementations using too few processors or data layouts leading to inefficient processor communication.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
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  • 6
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    Microwave driven convection in a square cavity (1994)
    Wiley
    Details
    Publication Date: 1994-07-01
    Print ISSN: 0001-1541
    Electronic ISSN: 1547-5905
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Published by Wiley on behalf of American Institute of Chemical Engineers.
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    PAPER CURRENT
  • 7
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    Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth (1986)
    In:  Other Sources
    Details
    Publication Date: 2011-08-19
    Description: Results are presented from finite element analysis of the Czochralski (CZ) and Liquid Encapsulated Czochralski (LEC) crystal growth processes based on a thermal-capillary model which governs the heat transfer in the system simultaneously with setting the shapes of the melt/solid interface, the melt and encapsulant menisci, and the radius of a steadily growing crystal. Calculations are performed for the small-scale growth of silicon (CZ) and gallium arsenide (LEC). The effects of melt volume and crucible position relative to the heater on the radius of the crystal and the shape of the melt/solid interface are predicted for the CZ system, and the importance of including an accurate representation of the melt meniscus for modeling the process is demonstrated. The additional effect of an encapsulant layer on heat transfer is treated for the LEC method for the cases of totally transparent and opaque encapsulant. The responses of these LEC prototype systems are examined for changes in pull rate and encapsulant volume.
    Keywords: SOLID-STATE PHYSICS
    Type: Journal of Crystal Growth (ISSN 0022-0248); 74; 605-624
    Format: text
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  • 8
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    Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth. II - Processing strategies (1986)
    In:  Other Sources
    Details
    Publication Date: 2011-08-19
    Description: The pseudosteady-state heat transfer model developed in a previous paper is augmented with constraints for constant crystal radius and melt/solid interface deflection. Combinations of growth rate, and crucible and bottom-heater temperatures are tested as processing parameters for satisfying the constrained thermal-capillary problem over a range of melt volumes corresponding to the sequence occuring during the batchwise Czochralski growth of a small-diameter silicon crystal. The applicability of each processing strategy is judged by the range of existence of the solution, in terms of melt volume and the values of the axial and radial temperature gradients in the crystal.
    Keywords: SOLID-STATE PHYSICS
    Type: Journal of Crystal Growth (ISSN 0022-0248); 75; 227-240
    Format: text
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  • 9
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    Thermal-capillary model for Czochralski growth of semiconductor materials (1985)
    In:  CASI
    Details
    Publication Date: 2016-06-07
    Description: The success of efficiently calculating the temperature field, crystal radius, melt mensicus, and melt/solid interface in the Czochralski crystal growth system by full finite-element solution of the government thermal-capillary model is demonstrated. The model predicts realistic response to changes in pull rate, melt volume, and the thermal field. The experimentally observed phenomena of interface flipping, bumping, and the difficulty maintaining steady-state growth as the melt depth decreases are explained by model results. These calculations will form the basis for the first quantitative picture of Cz crystal growth. The accurate depiction of the melt meniscus is important in calculating the crystal radius and solidification interface. The sensitivity of the results to the equilibrium growth angle place doubt on less sophisticated attempts to model the process without inclusion of the meniscus. Quantitative comparison with experiments should be possible once more representation of the radiation and view factors in the thermal system and the crucible are included. Extensions of the model in these directions are underway.
    Keywords: SOLID-STATE PHYSICS
    Type: JPL Proceedings of the Flat-Plate Solar Array Project Workshop; p 195-214
    Format: application/pdf
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  • 10
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    Segregation effects during growth of pseudo-binary system with large liquidus-solidus separation (1982)
    In:  Other Sources
    Details
    Publication Date: 2019-07-27
    Description: Results are reported for an experimental study of the seeded growth, of PbSnTe single crystals in a vertical Bridgman-Stockbarger (B-S) furnace. Profiles of axial and radial segregation of Sn are presented for crystal growth from melts with aspect ratios of 15 and 6.8; the profiles indicate intense convective mixing. A one-dimensional model of heat and mass transfer in an idealized B-S furnace with diffusion control is used to examine the importance of liquidus-solidus separation in determining growth-rate and composition transients in a pseudobinary system. It is shown that the coupling of heat and mass transfer in a pseudobinary system with a large liquidus-solidus separation causes the transient behavior in a nondilute system to differ markedly from such behavior in a dilute system and that PbSnTe crystals with a reasonable diameter can be grown under nearly convectionless conditions in a microgravity environment.
    Keywords: SOLID-STATE PHYSICS
    Type: IAF PAPER 82-130
    Format: text
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